{"text":"\\section{Introduction}\n\\label{sec:Intro}\n\n\n    With the increased use of electric vehicles (EVs) over the past decade, a large network of EV charging stations has been installed on the electric grid.\n    Data collected from city-wide deployment of EV charging stations can be used for both academic and industrial purposes, e.g.,  through statistical analysis \\cite{nasrinEV2016} and modeling. Studies require reliable sessions data for understanding behaviors and exploring flexibility.  The scarcity of reliable data has been discussed previously \\cite{EVDSreview}, and its necessity has been pointed out for further research purposes. Even when data exists, it may be protected under confidentiality for private collectors, and not freely available for academic or public use. Lack of wide-spread availability and accessibility of realistic EV charging sessions data pose a significant hurdle, impeding further research in the field.\n    \n    We propose a Synthetic Data Generator (SDG), that can be used to generate a sample of EV charging sessions. This implies temporal statistical modeling of arrivals and modeling of departures and the associated electrical load (i.e.,  required energy). We define a parametric model that can be trained from a real-world dataset. This trained model can be subsequently used to generate realistic data samples of EV sessions. \n    In this paper, we contribute with:\n        \\begin{itemize}[topsep=0pt]\n            \\setlength\\itemsep{0em}\n            \\item An approach to model sessions data for EVs over a group of charging stations defined as the SDG (\\secref{sec:SDG});\n            \\item An outline of the model parameters, and discussion over benefits and drawbacks of different models;\n            \\item Answers to our main research questions, being:\n           \n        \n        \t\\begin{question}\\label{q:SDGdefination}~Which parametric models can be used to describe sessions of EVs. What are the input parameters and latent variables for these models? \\end{question}\n        \n        \t\\begin{question}\\label{q:generate}~How can we generate synthetic samples of EV sessions data from these parametric models? \\end{question}\n    \t   \n        \\end{itemize}\n\n\\section{Synthetic Data Generator}\n\\label{sec:SDG}\n\nEach EV session can be described using three parameters:\n\\begin{enumerate*}[(i)]\n        \\item \\textit{Arrival time},\n        \\item \\textit{Departure time}, and \n        \\item \\textit{Energy charged} (in kWh).\n\\end{enumerate*}\nWe can define three models for each of these as:\n\\begin{itemize}[topsep=0pt]\n    \\item \\textit{Arrivals} = $AM$(Horizon)\n    \\item \\textit{Connected times} = $MM_{c}$(Arrivals) \n    \\item \\textit{Energy} = $MM_{e}$(Arrivals)\n\\end{itemize}\nWhere $AM$ means arrival model, $MM_{c}$ means mixture model for connection times and $MM_{e}$ means mixture model for the energy charged. `Horizon' is the time horizon for which the data needs to be generated and is an input parameter. Departure time in an EV charging session can be calculated as the sum of its arrival time and connected time.\n\n\\subsection{Arrival Models}\n\\label{sec:arrivalmodel}\n    \n    Arrivals of EVs in a group of charging stations can be considered as events on a continuous timescale. Supported by our large-scale dataset, we assume that the inter-arrival time of EVs\n    follows an exponential distribution,\n    characterized by a parameter $\\lambda$ representing the arrival rate (EVs per time unit).\n    This means we can either model the times in between successive events, or rather the number of events in a given time interval:\n    \\begin{itemize}[topsep=0pt]\n        \\item \\textit{Exponential IAT distribution:} we generate the arrival of the next EV using $t_{i}  = t_{i-1}  + \\Delta t $. Where $t_{i} $ is the time of $i^{th}$ arrival. $\\Delta t $ is the time difference between the $i^\\textrm{th}$ and ${i-1}^\\textrm{th}$ arrivals. The probability density of $\\Delta t $ (inter-arrival time, IAT) is exponentially distributed, characterized by $\\lambda = f(\\textrm{month}, \\textrm{daytype}, \\textrm{time-of-day})$, where `daytype' is either weekday or weekend.\n        \\item \\textit{Poisson process:} we generate the number of arrivals in a given time slot (e.g.,  slots of 60 minutes). The number of arrivals $N_\\textit{arr}$ in a given time slot can be generated using sampling from a Poisson distribution\n        with mean $\\lambda \\cdot T$ (with $T$ the duration of a timeslot).\n    \\end{itemize}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n        \n        \n   \n        \n   \n    \n    \n    \\noindent In \\figref{fig:expdensity} we plot density of inter-arrival times where the original data refers to real-world data collected by ELaadNL. A fitted exponential distribution validates this assumption that inter-arrival times are exponentially distributed: Kolmogorov\u2013Smirnov test results for all combinations of month, daytype and time-of-day slots have $p$-value $< 0.05$.\n    \n    \\begin{figure}[!t]\n        \\centering\n        \\includegraphics[width=\\columnwidth,height=6cm]{Resources\/exponential-dist-excel.png}\n       \n        \\caption{Inter-arrival time probability density for real world data collected by ELaadNL.\n        \\label{fig:expdensity}\n        \\setlength{\\belowcaptionskip}{-00pt}\n    \\end{figure} \n    \n\n    We can model $\\lambda$ (in function of month, daytype, time-of-day) using averaging (discontinuous) over each timeslot or rather fitting a continuous function of time (continuous). While fitting continuous curves, capturing the peaks during the day becomes very important. Also, fitted curves could result in negative values of $\\lambda$, thus we need to impose lower and upper bounds on it.\n    \n    \n    Using the IAT approach, the time of next EV arrival is determined relatively to the time of the previous EV. In case $\\lambda$ at a given time-of-day is very low, this implies the next EV arrival may be be generated very late (with a large $\\Delta t $), thus skipping several consecutive timeslots. This is problematic if those next timeslots exhibit a much higher $\\lambda$, and thus have a high probability of EV arrivals --- which would not be generated because of the very large  $\\Delta t $.\n   \n   \n    \n   \n    The second modeling approach, as a Poisson process, circumvents the above problem, since when certain timeslots have low $\\lambda$, we will likely generate 0 arrivals, but still proceed to the immediately next timeslot (with possibly high $\\lambda$ to generate possible arrivals there). A potential drawback still is that the variance and mean of the Poisson distribution are equal. In case this assumption does not hold, we adopt a negative binomial distribution instead.\n \n\n\n\n\n\n\n\n\n\n\n\\subsection{Mixture Models for Connected Times and Energy} \n\\label{sec:departuremodel}\n\nAside from arrival events, also EV departure events, as well as the EV charging load (or total energy charged) need to be modeled.\nArrivals are conditional on previous arrivals, obviously, and we model the probability distribution for connected times ($t_\\textit{depart} - t_\\textit{arr}$) as Gaussian Mixture Models (GMMs). Also for EV charging energy we use GMMs.\nAs in the arrival generation models, we define and fit different models for each month and time-of-day.\n\n\n\n\n\n\nIn summary, answering \\qref{q:SDGdefination} (Which models can be used in SDG?), we propose\n\\begin{enumerate*}[(i)]\n    \\item exponential IAT distribution or Poisson distribution for arrivals per timeslot ($\\textit{AM}$),\n    \\item GMMs for both the duration of EV sessions ($\\textit{MM}_c$) and \n    \\item their associated energy charged ($\\textit{MM}_e$).\n\\end{enumerate*}\n\n\n\n\\section{Generating Samples}\n\\label{sec:training}\n\nAfter fitting the SDG on a real-world data, we have $AM$, $MM_{c}$ and $MM_{e}$ as parametric models for arrival times, connection times and energy required. For synthetic generation of arrival times, e.g.,  $\\lambda$ values can be used to generate $\\Delta t $ (and hence a series of arrivals). For connection times and energy required, a sample from the PDFs can be used as a synthetically generated connection time and required energy. This answers \\qref{q:generate}, on how we can generate samples.\n\nWith our SDG (AM, $MM_{c}$ and $MM_{e}$) can be saved as a separate file. These models will be supplied with the code (that we plan to make publicly available) to generate new synthetic data samples. These models do not include the actual real-world EV session data that the SDG was trained on. Hence, these models can be shared without violating confidentiality constraints. Generated samples from these models can be subsequently used for flexibility analysis, load balancing and other research purposes.  \n\n\n\n\\section{Conclusion and Future Work}\n\\label{sec:future work}\n    Our paper summarized the modeling approach of EV charging sessions. \n    We adopted these models for training a synthetic data generator (SDG) with real-world data, that can thus generate synthetic samples of EV sessions data.\n   \n    We plan to release the source code including the \\emph{training} scripts, but also \\emph{generation} code (including the settings thereof reflecting the trained model characteristics based on a large-scale real-world dataset) to produce synthetic EV session data reflecting real-world behavior. We believe this fits a strong need of researchers in both academic and industrial settings.\n   \n    \n    As future work, we aim to propose modeling methods for the time-varying arrival distribution model parameters.\n    Further, we will tackle the following challenges in depth:\n    \\begin{enumerate*}[label={(\\arabic*)}]\n    \\item Studying the properties of the real-world data with the goal to define evaluations metrics for comparing real-world data with generated samples.\n    \\item Correctly modeling peaks of arrivals during the day; study effective methods that avoid negative values in continuous $\\lambda$ curves. \n\\end{enumerate*}\n    \n    \n\\begin{acks}\nThis research received funding from the Flemish Government under the \"Onderzoeksprogramma Artifici\u00eble Intelligentie (AI) Vlaanderen\" programme.\n\\end{acks}\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for all short- and\nfull-length articles, and optional for two-page abstracts.\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{The 1907 Franklin Model D roadster.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader. Figure captions go below the figure. Your figures should\n{\\bfseries also} include a description suitable for screen readers, to\nassist the visually-challenged to better understand your work.\n\nFigure captions are placed {\\itshape below} the figure.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  \n\\section{Introduction}\nACM's consolidated article template, introduced in 2017, provides a\nconsistent \\LaTeX\\ style for use across ACM publications, and\nincorporates accessibility and metadata-extraction functionality\nnecessary for future Digital Library endeavors. Numerous ACM and\nSIG-specific \\LaTeX\\ templates have been examined, and their unique\nfeatures incorporated into this single new template.\n\nIf you are new to publishing with ACM, this document is a valuable\nguide to the process of preparing your work for publication. If you\nhave published with ACM before, this document provides insight and\ninstruction into more recent changes to the article template.\n\nThe ``\\verb|acmart|'' document class can be used to prepare articles\nfor any ACM publication --- conference or journal, and for any stage\nof publication, from review to final ``camera-ready'' copy, to the\nauthor's own version, with {\\itshape very} few changes to the source.\n\n\\section{Template Overview}\nAs noted in the introduction, the ``\\verb|acmart|'' document class can\nbe used to prepare many different kinds of documentation --- a\ndouble-blind initial submission of a full-length technical paper, a\ntwo-page SIGGRAPH Emerging Technologies abstract, a ``camera-ready''\njournal article, a SIGCHI Extended Abstract, and more --- all by\nselecting the appropriate {\\itshape template style} and {\\itshape\n  template parameters}.\n\nThis document will explain the major features of the document\nclass. For further information, the {\\itshape \\LaTeX\\ User's Guide} is\navailable from\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}.\n\n\\subsection{Template Styles}\n\nThe primary parameter given to the ``\\verb|acmart|'' document class is\nthe {\\itshape template style} which corresponds to the kind of publication\nor SIG publishing the work. This parameter is enclosed in square\nbrackets and is a part of the {\\verb|documentclass|} command:\n\\begin{verbatim}\n  \\documentclass[STYLE]{acmart}\n\\end{verbatim}\n\nJournals use one of three template styles. All but three ACM journals\nuse the {\\verb|acmsmall|} template style:\n\\begin{itemize}\n\\item {\\verb|acmsmall|}: The default journal template style.\n\\item {\\verb|acmlarge|}: Used by JOCCH and TAP.\n\\item {\\verb|acmtog|}: Used by TOG.\n\\end{itemize}\n\nThe majority of conference proceedings documentation will use the {\\verb|acmconf|} template style.\n\\begin{itemize}\n\\item {\\verb|acmconf|}: The default proceedings template style.\n\\item{\\verb|sigchi|}: Used for SIGCHI conference articles.\n\\item{\\verb|sigchi-a|}: Used for SIGCHI ``Extended Abstract'' articles.\n\\item{\\verb|sigplan|}: Used for SIGPLAN conference articles.\n\\end{itemize}\n\n\\subsection{Template Parameters}\n\nIn addition to specifying the {\\itshape template style} to be used in\nformatting your work, there are a number of {\\itshape template parameters}\nwhich modify some part of the applied template style. A complete list\nof these parameters can be found in the {\\itshape \\LaTeX\\ User's Guide.}\n\nFrequently-used parameters, or combinations of parameters, include:\n\\begin{itemize}\n\\item {\\verb|anonymous,review|}: Suitable for a ``double-blind''\n  conference submission. Anonymizes the work and includes line\n  numbers. Use with the \\verb|\\acmSubmissionID| command to print the\n  submission's unique ID on each page of the work.\n\\item{\\verb|authorversion|}: Produces a version of the work suitable\n  for posting by the author.\n\\item{\\verb|screen|}: Produces colored hyperlinks.\n\\end{itemize}\n\nThis document uses the following string as the first command in the\nsource file:\n\\begin{verbatim}\n\\documentclass[sigconf]{acmart}\n\\end{verbatim}\n\n\\section{Modifications}\n\nModifying the template --- including but not limited to: adjusting\nmargins, typeface sizes, line spacing, paragraph and list definitions,\nand the use of the \\verb|\\vspace| command to manually adjust the\nvertical spacing between elements of your work --- is not allowed.\n\n{\\bfseries Your document will be returned to you for revision if\n  modifications are discovered.}\n\n\\section{Typefaces}\n\nThe ``\\verb|acmart|'' document class requires the use of the\n``Libertine'' typeface family. Your \\TeX\\ installation should include\nthis set of packages. Please do not substitute other typefaces. The\n``\\verb|lmodern|'' and ``\\verb|ltimes|'' packages should not be used,\nas they will override the built-in typeface families.\n\n\\section{Title Information}\n\nThe title of your work should use capital letters appropriately -\n\\url{https:\/\/capitalizemytitle.com\/} has useful rules for\ncapitalization. Use the {\\verb|title|} command to define the title of\nyour work. If your work has a subtitle, define it with the\n{\\verb|subtitle|} command.  Do not insert line breaks in your title.\n\nIf your title is lengthy, you must define a short version to be used\nin the page headers, to prevent overlapping text. The \\verb|title|\ncommand has a ``short title'' parameter:\n\\begin{verbatim}\n  \\title[short title]{full title}\n\\end{verbatim}\n\n\\section{Authors and Affiliations}\n\nEach author must be defined separately for accurate metadata\nidentification. Multiple authors may share one affiliation. Authors'\nnames should not be abbreviated; use full first names wherever\npossible. Include authors' e-mail addresses whenever possible.\n\nGrouping authors' names or e-mail addresses, or providing an ``e-mail\nalias,'' as shown below, is not acceptable:\n\\begin{verbatim}\n  \\author{Brooke Aster, David Mehldau}\n  \\email{dave,judy,steve@university.edu}\n  \\email{firstname.lastname@phillips.org}\n\\end{verbatim}\n\nThe \\verb|authornote| and \\verb|authornotemark| commands allow a note\nto apply to multiple authors --- for example, if the first two authors\nof an article contributed equally to the work.\n\nIf your author list is lengthy, you must define a shortened version of\nthe list of authors to be used in the page headers, to prevent\noverlapping text. The following command should be placed just after\nthe last \\verb|\\author{}| definition:\n\\begin{verbatim}\n  \\renewcommand{\\shortauthors}{McCartney, et al.}\n\\end{verbatim}\nOmitting this command will force the use of a concatenated list of all\nof the authors' names, which may result in overlapping text in the\npage headers.\n\nThe article template's documentation, available at\n\\url{https:\/\/www.acm.org\/publications\/proceedings-template}, has a\ncomplete explanation of these commands and tips for their effective\nuse.\n\n\\section{Rights Information}\n\nAuthors of any work published by ACM will need to complete a rights\nform. Depending on the kind of work, and the rights management choice\nmade by the author, this may be copyright transfer, permission,\nlicense, or an OA (open access) agreement.\n\nRegardless of the rights management choice, the author will receive a\ncopy of the completed rights form once it has been submitted. This\nform contains \\LaTeX\\ commands that must be copied into the source\ndocument. When the document source is compiled, these commands and\ntheir parameters add formatted text to several areas of the final\ndocument:\n\\begin{itemize}\n\\item the ``ACM Reference Format'' text on the first page.\n\\item the ``rights management'' text on the first page.\n\\item the conference information in the page header(s).\n\\end{itemize}\n\nRights information is unique to the work; if you are preparing several\nworks for an event, make sure to use the correct set of commands with\neach of the works.\n\n\\section{CCS Concepts and User-Defined Keywords}\n\nTwo elements of the ``acmart'' document class provide powerful\ntaxonomic tools for you to help readers find your work in an online\nsearch.\n\nThe ACM Computing Classification System ---\n\\url{https:\/\/www.acm.org\/publications\/class-2012} --- is a set of\nclassifiers and concepts that describe the computing\ndiscipline. Authors can select entries from this classification\nsystem, via \\url{https:\/\/dl.acm.org\/ccs\/ccs.cfm}, and generate the\ncommands to be included in the \\LaTeX\\ source.\n\nUser-defined keywords are a comma-separated list of words and phrases\nof the authors' choosing, providing a more flexible way of describing\nthe research being presented.\n\nCCS concepts and user-defined keywords are required for all short- and\nfull-length articles, and optional for two-page abstracts.\n\n\\section{Sectioning Commands}\n\nYour work should use standard \\LaTeX\\ sectioning commands:\n\\verb|section|, \\verb|subsection|, \\verb|subsubsection|, and\n\\verb|paragraph|. They should be numbered; do not remove the numbering\nfrom the commands.\n\nSimulating a sectioning command by setting the first word or words of\na paragraph in boldface or italicized text is {\\bfseries not allowed.}\n\n\\section{Tables}\n\nThe ``\\verb|acmart|'' document class includes the ``\\verb|booktabs|''\npackage --- \\url{https:\/\/ctan.org\/pkg\/booktabs} --- for preparing\nhigh-quality tables.\n\nTable captions are placed {\\itshape above} the table.\n\nBecause tables cannot be split across pages, the best placement for\nthem is typically the top of the page nearest their initial cite.  To\nensure this proper ``floating'' placement of tables, use the\nenvironment \\textbf{table} to enclose the table's contents and the\ntable caption.  The contents of the table itself must go in the\n\\textbf{tabular} environment, to be aligned properly in rows and\ncolumns, with the desired horizontal and vertical rules.  Again,\ndetailed instructions on \\textbf{tabular} material are found in the\n\\textit{\\LaTeX\\ User's Guide}.\n\nImmediately following this sentence is the point at which\nTable~\\ref{tab:freq} is included in the input file; compare the\nplacement of the table here with the table in the printed output of\nthis document.\n\n\\begin{table}\n  \\caption{Frequency of Special Characters}\n  \\label{tab:freq}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Non-English or Math&Frequency&Comments\\\\\n    \\midrule\n    \\O & 1 in 1,000& For Swedish names\\\\\n    $\\pi$ & 1 in 5& Common in math\\\\\n    \\$ & 4 in 5 & Used in business\\\\\n    $\\Psi^2_1$ & 1 in 40,000& Unexplained usage\\\\\n  \\bottomrule\n\\end{tabular}\n\\end{table}\n\nTo set a wider table, which takes up the whole width of the page's\nlive area, use the environment \\textbf{table*} to enclose the table's\ncontents and the table caption.  As with a single-column table, this\nwide table will ``float'' to a location deemed more\ndesirable. Immediately following this sentence is the point at which\nTable~\\ref{tab:commands} is included in the input file; again, it is\ninstructive to compare the placement of the table here with the table\nin the printed output of this document.\n\n\\begin{table*}\n  \\caption{Some Typical Commands}\n  \\label{tab:commands}\n  \\begin{tabular}{ccl}\n    \\toprule\n    Command &A Number & Comments\\\\\n    \\midrule\n    \\texttt{{\\char'134}author} & 100& Author \\\\\n    \\texttt{{\\char'134}table}& 300 & For tables\\\\\n    \\texttt{{\\char'134}table*}& 400& For wider tables\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\section{Math Equations}\nYou may want to display math equations in three distinct styles:\ninline, numbered or non-numbered display.  Each of the three are\ndiscussed in the next sections.\n\n\\subsection{Inline (In-text) Equations}\nA formula that appears in the running text is called an inline or\nin-text formula.  It is produced by the \\textbf{math} environment,\nwhich can be invoked with the usual\n\\texttt{{\\char'134}begin\\,\\ldots{\\char'134}end} construction or with\nthe short form \\texttt{\\$\\,\\ldots\\$}. You can use any of the symbols\nand structures, from $\\alpha$ to $\\omega$, available in\n\\LaTeX~\\cite{Lamport:LaTeX}; this section will simply show a few\nexamples of in-text equations in context. Notice how this equation:\n\\begin{math}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{math},\nset here in in-line math style, looks slightly different when\nset in display style.  (See next section).\n\n\\subsection{Display Equations}\nA numbered display equation---one set off by vertical space from the\ntext and centered horizontally---is produced by the \\textbf{equation}\nenvironment. An unnumbered display equation is produced by the\n\\textbf{displaymath} environment.\n\nAgain, in either environment, you can use any of the symbols and\nstructures available in \\LaTeX\\@; this section will just give a couple\nof examples of display equations in context.  First, consider the\nequation, shown as an inline equation above:\n\\begin{equation}\n  \\lim_{n\\rightarrow \\infty}x=0\n\\end{equation}\nNotice how it is formatted somewhat differently in\nthe \\textbf{displaymath}\nenvironment.  Now, we'll enter an unnumbered equation:\n\\begin{displaymath}\n  \\sum_{i=0}^{\\infty} x + 1\n\\end{displaymath}\nand follow it with another numbered equation:\n\\begin{equation}\n  \\sum_{i=0}^{\\infty}x_i=\\int_{0}^{\\pi+2} f\n\\end{equation}\njust to demonstrate \\LaTeX's able handling of numbering.\n\n\\section{Figures}\n\nThe ``\\verb|figure|'' environment should be used for figures. One or\nmore images can be placed within a figure. If your figure contains\nthird-party material, you must clearly identify it as such, as shown\nin the example below.\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\linewidth]{sample-franklin}\n  \\caption{1907 Franklin Model D roadster. Photograph by Harris \\&\n    Ewing, Inc. [Public domain], via Wikimedia\n    Commons. (\\url{https:\/\/goo.gl\/VLCRBB}).}\n  \\Description{The 1907 Franklin Model D roadster.}\n\\end{figure}\n\nYour figures should contain a caption which describes the figure to\nthe reader. Figure captions go below the figure. Your figures should\n{\\bfseries also} include a description suitable for screen readers, to\nassist the visually-challenged to better understand your work.\n\nFigure captions are placed {\\itshape below} the figure.\n\n\\subsection{The ``Teaser Figure''}\n\nA ``teaser figure'' is an image, or set of images in one figure, that\nare placed after all author and affiliation information, and before\nthe body of the article, spanning the page. If you wish to have such a\nfigure in your article, place the command immediately before the\n\\verb|\\maketitle| command:\n\\begin{verbatim}\n  \\begin{teaserfigure}\n    \\includegraphics[width=\\textwidth]{sampleteaser}\n    \\caption{figure caption}\n    \\Description{figure description}\n  \\end{teaserfigure}\n\\end{verbatim}\n\n\\section{Citations and Bibliographies}\n\nThe use of \\BibTeX\\ for the preparation and formatting of one's\nreferences is strongly recommended. Authors' names should be complete\n--- use full first names (``Donald E. Knuth'') not initials\n(``D. E. Knuth'') --- and the salient identifying features of a\nreference should be included: title, year, volume, number, pages,\narticle DOI, etc.\n\nThe bibliography is included in your source document with these two\ncommands, placed just before the \\verb|\\end{document}| command:\n\\begin{verbatim}\n  \\bibliographystyle{ACM-Reference-Format}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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}%\n}\n\n\\newcommand{w}{w}\n\\newcommand{\\mathfrak{p}}{\\mathfrak{p}}\n\\newcommand{$g$-equivalent}{$g$-equivalent}\n\\newcommand{$g$-equivalence}{$g$-equivalence}\n\\newcommand{G^g}{G^g}\n\\newcommand{\\Psi}{\\Psi}\n\\newcommand{\\Upsilon}{\\Upsilon}\n\\newcommand{(\\sieve,\\siftable)}{(\\Psi,\\Upsilon)}\n\n\\newenvironment{epigraph}\n{\\hfill\\begin{minipage}{0.6\\linewidth}\\raggedleft\\footnotesize}{\\end{minipage}\\bigskip\\bigskip}\n\n\n\\newcommand{\\mathcal{M}}{\\mathcal{M}}\n\\newcommand{\\mathcal{L}}{\\mathcal{L}}\n\\newcommand{\\mathcal{S}}{\\mathcal{S}}\n\\newcommand{\\mathcal{C}}{\\mathcal{C}}\n\\newcommand{\\mathcal{P}}{\\mathcal{P}}\n\\newcommand{\\mathrm{P}}{\\mathrm{P}}\n\\newcommand{\\mathrm{L}}{\\mathrm{L}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\\newcommand{\\mathcal{Q}}{\\mathcal{Q}}\n\\newcommand{\\mathcal{K}}{\\mathcal{K}}\n\\newcommand{\\mathcal{R}}{\\mathcal{R}}\n\\newcommand{\\mathcal{J}}{\\mathcal{J}}\n\\newcommand{\\mathcal{I}}{\\mathcal{I}}\n\\newcommand{\\mathcal{G}}{\\mathcal{G}}\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{E}}{\\mathcal{E}}\n\n\\newcommand{\\mathfrak{a}}{\\mathfrak{a}}\n\\newcommand{\\mathfrak{p}}{\\mathfrak{p}}\n\n\\newcommand{\\lambda_f}{\\lambda_f}\n\\newcommand{\\rho_f}{\\rho_f}\n\\newcommand{\\lambda_g}{\\lambda_g}\n\\newcommand{\\rho_g}{\\rho_g}\n\n\\newcommand{\\varphi}{\\varphi}\n\n\\renewcommand{\\geq}{\\geqslant}\n\\renewcommand{\\leq}{\\leqslant}\n\\renewcommand{\\Re}{\\mathfrak{Re}\\,}\n\\renewcommand{\\Im}{\\mathfrak{Im}\\,}\n\n\\newcommand{\\eqref}{\\eqref}\n\\newcommand{\\backslash}{\\backslash}\n\n\\newcommand{\\ov}[1]{\\overline{#1}}\n\n\\newcommand{\\norm}[1]{\\|{#1}\\|}\n\n\\newcommand{\\peter}[1]{\\langle{#1}\\rangle}\n\\newcommand\\sumsum{\\mathop{\\sum\\sum}\\limits}\n\\newcommand\\sumsumsum{\\mathop{\\sum\\sum\\sum}\\limits}\n\\newcommand\\sumsumnd{\\mathop{{\\sum\\sum}^{nd}}\\limits}\n\\newcommand\\delval{1\/8}\n\\newcommand\\delvaln{1\/16}\n\\newcommand\\finalexponent{1\/24}\n\\newcommand\\rpfree{1\/144}\n\n\\begin{document}\n\n\\title{Periodic twists of $\\GL_3$-automorphic forms}\n\n\\author{Emmanuel  Kowalski}\n\\address{ETHZ, Switzerland }\n\\email{kowalski@math.ethz.ch}\n\n\\author{Yongxiao Lin}\n\\address{EPFL\/MATH\/TAN, Station 8, CH-1015 Lausanne, Switzerland }\n\\email{yongxiao.lin@epfl.ch}\n\n\\author{Philippe Michel}\n\\address{EPFL\/MATH\/TAN, Station 8, CH-1015 Lausanne, Switzerland }\n\\email{philippe.michel@epfl.ch}\n\n\\author{Will Sawin}\n\\address{Columbia University, USA }\n\\email{sawin@math.columbia.edu}\n\n\\date{\\today,\\ \\thistime} \n\n\\subjclass[2010]{11F55,11M41,11L07, 11T23, 32N10}\n\n\\keywords{Automorphic forms on $\\GL_3$, Fourier coefficients, Hecke\n  eigenvalues, discrete Fourier transform, trace functions,\n  subconvexity}\n\n\\begin{abstract}\n  We prove that sums of length about $q^{3\/2}$ of Hecke eigenvalues of\n  automorphic forms on~$\\SL_3(\\mathbf{Z})$ do not correlate with $q$-periodic\n  functions with bounded Fourier transform. This generalizes the earlier\n  results of Munshi and Holowinsky--Nelson, corresponding to\n  multiplicative Dirichlet characters, and applies in particular to\n  trace functions of small conductor modulo primes.\n\\end{abstract}\n\n\\thanks{Y. L., Ph.\\ M.\\ and E.\\ K.\\ were partially supported by a\n  DFG-SNF lead agency program grant (grant number\n  200020L\\_175755). W. S. was partially supported by the Clay Mathematics\n  Institute. \\today\\ \\currenttime}\n\n\\maketitle \n\n\n\\section{Introduction}\n\nLet $\\varphi$ be a cusp form for $\\SL_3(\\mathbf{Z})$ which is an eigenfunction\nof all Hecke operators.\nFor any prime number~$q$ and any primitive Dirichlet character~$\\chi$\nmodulo~$q$, we can then define the twisted $L$-function\n$L(\\varphi\\otimes\\chi,s)$, which is an entire function satisfying a\nfunctional equation relating $s$ to $1-s$.\nIn a recent breakthrough, Munshi~\\cite{Munshi,Munshi1} solved the\nsubconvexity problem for these twisted $L$-functions\n$L(\\varphi\\otimes \\chi,s)$ in the conductor aspect:\n\n\\begin{theorem}[Munshi]\\label{th-munshi}\n  Let~$s$ be a complex number such that $\\Re s=1\/2$. For any\n  prime~$q$, any primitive Dirichlet character~$\\chi$ modulo~$q$, and\n  for any~$\\varepsilon>0$, we have\n\\begin{equation}\\label{eq:subconvex}\n  L(\\varphi\\otimes \\chi,s)\\ll q^{3\/4-1\/308+\\varepsilon},\n\\end{equation}\nwhere the implied constant depends on $\\varphi$, $s$ and~$\\varepsilon$.\n\\end{theorem}\n\nThis result was recently analyzed in depth by Holowinsky and\nNelson~\\cite{HN}, who discovered a remarkable simplification (and\nstrenghtening) of Munshi's ideas. They proved:\n\n\\begin{theorem}[Holowinsky--Nelson]\\label{th-hn}\n  With notation and assumptions as in Theorem~\\emph{\\ref{th-munshi}},\n  we have\n  \\begin{equation}\\label{eq:hn}\n    L(\\varphi\\otimes \\chi,s)\\ll q^{3\/4-1\/36+\\varepsilon}\n  \\end{equation}\n  where the implied constant depends on $\\varphi$, $s$ and~$\\varepsilon$.\n\\end{theorem}\n\n\\begin{remark}\n  We mention further variants, simplifications and improvements, by\n  Aggarwal, Holowinsky, Lin and Sun~\\cite{AHLS}, Holowinsky, Munshi\n  and Qi~\\cite{HMQ}, Lin~\\cite{Lin}, Sun and Zhao ~\\cite{SZ}.\n\\end{remark}\n\nLet $(\\lambda(m,n))$ denote the Hecke-eigenvalues of~$\\varphi$.  By\nthe approximate functional equation for the twisted $L$-functions, the\nbound \\eqref{eq:hn} is essentially equivalent to the bound\n\\begin{equation}\\label{eq:sumbound}\n  \\sum_{n\\geq 1}\\lambda(1,n)\\chi(n)\n  V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr)\\ll q^{3\/2-\\delta},\n\\end{equation}\nfor~$\\delta<1\/36$, where $V$ is any smooth compactly supported\nfunction and the implied constant depends on~$\\varphi$, $\\delta$\nand~$V$.\n\nFrom the perspective of such sums, motivated by the previous work of\nFouvry, Kowalski and Michel~\\cite{FKM1}, which relates to automorphic\nforms on~$\\GL_2$, it is natural to ask whether this\nbound~\\eqref{eq:sumbound} holds when $\\chi$ is replaced by a more\ngeneral trace function $K:{\\mathbf{F}_q}\\to \\mathbf{C}$. Our main result shows that this\nis the case, and in fact extends the result to a much wider range of\n$q$-periodic functions by obtaining estimates only in terms of the\nsize of the discrete Fourier transform modulo~$q$.\n\nPrecisely, for any function~$V$ with compact support on~$\\mathbf{R}$, we set\n\\begin{equation}\\label{defSKX}\n  S_{V}(K,X):=\\sum_{n\\geq 1}\\lambda(1,n)K(n)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{equation}\n\nWe will assume that $V:\\mathbf{R}\\to \\mathbf{C}$ satisfies the following conditions\nfor some parameter~$Z\\geq 1$:\n\\begin{equation}\\label{eq:Vprop}\n  \\mathrm{supp}(V)\\subset ]1,2[,\\text{ and }V^{(i)}(x)\\ll Z^i\\text{\n    for all $i\\geq 0$},\n\\end{equation}\nwhere the implied constant depends only on~$i$.\n\nFor any integer~$q\\geq 1$ and any $q$-periodic function\n$K\\colon \\mathbf{Z}\\to \\mathbf{C}$, we denote by\n\\begin{equation}\\label{eq:fourierK}\n  \\widehat K(n)=\\frac{1}{q^{1\/2}}\\sum_{x\\in{\\mathbf{F}_q}}K(x)e\\Bigl(\\frac{nx}{q}\\Bigr),\n\\end{equation}\nfor~$n\\in\\mathbf{Z}$, its (unitarily normalized) discrete Fourier transform\nmodulo~$q$.  We write~$\\norm{\\widehat{K}}_{\\infty}$ for the maximum of\n$|\\widehat{K}(n)|$ for~$n\\in\\mathbf{Z}$. We then have the discrete Fourier\ninversion formula\n$$\nK(x)=\\frac{1}{q^{1\/2}}\\sum_{n\\in{\\mathbf{F}_q}} \\widehat{K}(n)\ne\\Bigl(-\\frac{nx}{q}\\Bigr)\n$$\nfor~$x\\in\\mathbf{Z}$.\n\nOur main result is a general bound for \\eqref{defSKX} which matches\nprecisely the bound of Holowinsky--Nelson \\cite{HN} in the case of a\nmultiplicative character:\n\n\\begin{theorem}\\label{thm:main}\n  Let $\\varphi$ be an $\\SL_3(\\mathbf{Z})$-invariant cuspidal Hecke-eigenform\n  with Hecke eigenvalues $(\\lambda(m,n))$.  Let~$q$ be a prime number,\n  and $K\\colon \\mathbf{Z}\\to\\mathbf{C}$ be a $q$-periodic function.\n \n \n  Let $V$ be a smooth, compactly supported function satisfying\n  \\eqref{eq:Vprop} for some $Z\\geq 1$.  Assume that\n$$\n    Z^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}.\n$$\t\n For any~$\\varepsilon>0$, we have\n \\begin{equation}\\label{eq:sumboundK}\n   S_V(K,X) \\ll\n   \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6},\n \\end{equation}\n where the implied constant depends only on~$\\varepsilon$, on~$\\varphi$, and on\n the implicit constants in~\\emph{(\\ref{eq:Vprop})}.\n\\end{theorem}\n\n\\begin{remark}\n  (1)  Suppose that we vary~$q$ and apply this bound with functions~$K$\n  modulo~$q$ that have absolutely bounded Fourier transforms.  Take\n  $X=q^{3\/2}$. We then obtain the bound\n  $$\n  S_V(K,q^{3\/2}) \\ll Z^{10\/9}q^{3\/2-1\/36+\\varepsilon}\n  $$\n  for any~$\\varepsilon>0$.\n  \\par\n  (2) For the bound \\eqref{eq:sumboundK} to be non-trivial (i.e.,\n  assuming~$K$ to be absolutely bounded, better than $X$), it is\n  enough that\n  $$\n  X\\geq Z^{20\/3}q^{4\/3+\\delta}\n  $$\n  for some~$\\delta>0$.\n  \\par\n  (3) As in the paper~\\cite{short-sums} of Fouvry, Kowalski, Michel,\n  Raju, Rivat and Soundararajan, where the main estimate is also\n  phrased in Fourier-theoretic terms only,\\footnote{Although the size\n    of~$K$ enters in~\\cite{short-sums} as well as that of its Fourier\n    transform.} the motivating examples of functions~$K$ satisfying\n  uniform bounds on their Fourier transforms are the trace functions\n  of suitable $\\ell$-adic sheaves modulo~$q$. The simplest example\n  is~$K(n)=\\chi(n)$, which recovers the bound of Munshi (up to the\n  value of the exponent) and Holowinsky--Nelson, since the values of\n  the Fourier transform are normalized Gauss sums of modulo~$\\leq\n  1$. We recall some other examples below in\n  Section~\\ref{sec-examples}.\n\\end{remark}\n\nWe can  deduce from Theorem~\\ref{thm:main} a weak but non-trivial\nbound for the first moment of the twisted central $L$-values, with an\nadditional twist by a discrete Mellin transform. We first recall the\ndefinition\n$$\n\\Kl_3(n)=\\frac{1}{q}\\sum_{\\substack{x,y,z\\in{\\mathbf{F}^\\times_q}\\\\xyz=n}}\ne\\Bigl(\\frac{x+y+z}{q}\\Bigr)\n$$\nfor a hyper-Kloosterman sum with two variables modulo a prime~$q$.\n\n\\begin{corollary}\\label{cor-average}\n  Let $\\varphi$ be an $\\SL_3(\\mathbf{Z})$-invariant cuspidal Hecke-eigenform\n  with Hecke eigenvalues $(\\lambda(m,n))$.  Let~$q$ be a prime number\n  and let~$\\chi\\mapsto M(\\chi)$ be a function of Dirichlet characters\n  modulo~$q$.\n  \\par\n  Let~$K$ and~$L$ be the $q$-periodic functions defined\n  by~$K(0)=L(0)=0$ and\n  \\begin{align*}\n    K(n)&=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}\\chi(n)M(\\chi)\\\\\n    L(n)&=\\frac{1}{q^{1\/2}} \\sum_{x\\in{\\mathbf{F}_q}}K(x)\\Kl_3(nx)\n  \\end{align*}\n  for~$n$ coprime to~$q$.  We then have\n  $$\n  \\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi, 1\/2) \\ll\n  \\Bigl(\\norm{\\widehat{K}}_{\\infty}+\\norm{\\widehat{L}}_{\\infty}\\Bigr) q^{2\/9\n    + \\varepsilon},\n  $$\n  for any~$\\varepsilon>0$, where the  implied constant depends\n  on~$\\varphi$ and,~$\\varepsilon$.\n \n \n \n \n \n \n \n \n\\end{corollary}\n\n\n\nA further natural application concerns the symmetric square lift, $\\mathrm{sym}_2(\\psi)$, of a\n$\\GL_2$-cusp form of level~$1$. Precisely, let $\\psi$ be a cuspidal\nHecke-eigenform for~$\\SL_2(\\mathbf{Z})$ with Hecke eigenvalues\n$(\\lambda(n))_{n\\geq 1}$. This implies the following\n\\begin{corollary}\\label{cor-gl2}\n  Let $K$ and $V$ be as above and assume that\n  $Z^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}$. Then, for any~$\\varepsilon>0$, we\n  have\n  $$\n  \\sum_{n\\geq 1}\\lambda(n^2)K(n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\n  \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}+\n  \\norm{K}_{\\infty}Z^{1\/3}q^{2\/3}X^{1\/2+\\varepsilon},\n  $$\n  where the implied constant depends only on~$\\varepsilon$, on~$\\psi$, and on\n  the implicit constants in~\\emph{(\\ref{eq:Vprop})}.\n\\end{corollary}\n\\begin{remark}\\label{blomerremark} As pointed out to us by V. Blomer, when $K=\\chi$ is a Dirichlet character a stronger bound should be available: for $\\chi$ quadratic,  one has (see \\cite{Blomer}) the stronger  subconvex bound for the central value\n\\begin{equation}\\label{blomerbound}\n\tL(\\mathrm{sym}_2(\\psi)\\otimes\\chi,s)\\ll_s q^{3\/4-1\/8+o(1)},\\ \\Re s=1\/2.\n\\end{equation}\nThis would amount to a bound of the shape\n$$\n  \\sum_{n\\geq 1}\\lambda(n^2)\\chi(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr) \\ll_Z\n  q^{3\/2-1\/8+\\varepsilon}.\n  $$\nThe bound \\eqref{blomerbound} actually extends to any character $\\chi\\mods q$ by the same method, using the Petrow-Young variant of the Conrey-Iwaniec method \\cite{CI,PY}. However, since this approach uses positivity of central values, it is not entirely clear yet whether this could be extended to general trace functions. \n\\end{remark}\n\n\nFrom this corollary, one can easily derive an estimate for twists of\nthe arithmetic function~$\\lambda(n)^2=|\\lambda(n)|^2$, which is related\nto~$\\lambda(n^2)$ by the convolution identity\n\\begin{equation}\\label{convol}\n  \\lambda(n)^2=\\sum_{ab=n}\\lambda(a^2).\t\n\\end{equation}\n\nHowever, in terms of $L$-functions, a straightforward estimate\nconcerns sums of length close to~$q^2$, and not~$q^{3\/2}$ anymore (it\namounts, when~$K=\\chi$, to a subconvexity estimate\nfor~$L(f\\otimes f\\otimes \\chi,{\\textstyle{\\frac{1}{2}}})$, which results directly from the\nfactorization of this $L$-function of degree~$4$).\n\nOne can however recover a bound for sums of length about~$q^{3\/2}$\nwith more work, and here we require that $K$ be a trace function (more\nprecisely, a \\emph{non-exceptional} trace function, in the sense\nof~\\cite[p. 1686]{FKM2}).\n\n\n\\begin{corollary}\\label{cor2-gl2}\\label{RScor}\n  Let $V$ be as above.  Let $K$ be a the trace function of an\n  $\\ell$-adic sheaf $\\mathcal{F}$ modulo~$q$ which is a geometrically\n  irreducible middle-extension sheaf, pure of weight~$0$, on the\n  affine line over~$\\mathbf{F}_q$. Assume that the sheaf $\\mathcal{F}$ is not\n  geometrically isomorphic to the tensor product\n  $\\mathcal{L}_\\psi\\otimes\\mathcal{L}_\\chi$ of an Artin-Schreier sheaf and a Kummer\n  sheaf.\n  \\par\n  If $Z^{-4\/3}q^{4\/3+8\\gamma\/3} \\leq X \\leq Z^{-2}q^{2}$, then  we have\n  $$\n  \\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\n  X^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{-\\gamma}\n  $$\n  for any~$\\varepsilon>0$, where the implied constant depends only\n  on~$\\psi~$, $\\varepsilon$ and on the conductor~$\\cond(\\mathcal{F})$ of~$\\mathcal{F}$.\n\\end{corollary}\n\n\\begin{remark}\n  (1) Suppose that~$Z$ is fixed. The estimate is then non-trivial as\n  long as $X\\gg q^{4\/3+\\delta}$; for $X=q^{3\/2}$, it saves a factor\n  $q^{1\/48}$ over the trivial bound.\n  \\par\n  (2) The assumption that~$\\mathcal{F}$ is not exceptional means intuitively\n  that $K$ is not proportional to the product of an additive and a\n  multiplicative character modulo~$q$. We then have in particular\n  $$\n  \\|K\\|_{\\infty}+\\|\\widehat{K}\\|_{\\infty}\\ll 1\n  $$\n  where the implied constant depends only on the conductor of~$\\mathcal{F}$.\n\\end{remark}\n\n\\begin{remark}\\label{remcor17}\n  (1) The reader may wonder why this paper is much shorter\n  than~\\cite{FKM1}, and (with the exception of Corollary \\ref{RScor})\n  requires much less input from algebraic geometry in the case of\n  trace functions. One reason is that we are considering (essentially)\n  sums of length~$q^{3\/2}$ whereas the coefficients functions~$K$ are\n  $q$-periodic. This means that periodicity properties of the\n  summand~$K(n)$ have a non-trivial effect, whereas they do not for\n  the sums of length about~$q$ which are considered in~\\cite{FKM1} in\n  the context of~$\\GL_2$.\n  \\par\n  Moreover, observe that an analogue of Theorem~\\ref{thm:main}, with\n  an estimate that depends (in terms of~$K$) only on the size of the\n  Fourier transform~$\\widehat{K}$, is \\emph{false} in the setting\n  of~\\cite{FKM1}, i.e., for sums\n  $$\n  \\sum_{n\\geq 1}\\lambda(n)K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\n  $$\n  with~$X$ of size about~$q$, where $\\lambda(n)$ are the\n  Hecke-eigenvalues of a cusp forms~$\\psi$ for $\\SL_2(\\mathbf{Z})$ (as in\n  Corollary~\\ref{cor-gl2}). Indeed, if we take $X=q$ and define~$K$ to\n  be the $q$-periodic function that coincides with the (real-valued)\n  function $n\\mapsto \\lambda(n)$ for $1\\leq n\\leq q$, then~$K$ has\n  discrete Fourier transform of size~$\\ll \\log q$ by the well-known\n  Wilton estimate (see, e.g.,~\\cite[Th. 5.3]{iwaniec}, when~$\\psi$ is\n  holomorphic), and yet\n  $$\n  \\sum_{n\\leq q}K(n)\\lambda(n)=\\sum_{n\\leq q}|\\lambda(n)|^2\\asymp q\n  $$\n  by the Rankin--Selberg method.\n  \\par\n  On the other hand, the same bound of Wilton combined with discrete\n  Fourier inversion implies quickly that if~$K$ is any $q$-periodic\n  function, then\n  $$\n  \\sum_{n\\leq q^{3\/2}}\\lambda(n) K(n)\\ll\n  q^{1+1\/4+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n  $$\n  for any~$\\varepsilon>0$. However, the natural length for applications\n  is~$q$ in the $\\GL_2$ case.\n  \\par\n  (2) The most obvious function $K$ for which Theorem~\\ref{thm:main}\n  gives trivial results is an additive character $K(n)=e(an\/q)$ for\n  some integer~$a\\in\\mathbf{Z}$, since the Fourier transform takes one value\n  of size~$q^{1\/2}$. However, a useful estimate also exists in this\n  case: Miller~\\cite{Miller} has proved that\n  $$\n  \\sum_{n\\geq 1}\\lambda(1,n)e(\\alpha\n  n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll_{\\varphi,Z} X^{3\/4+\\varepsilon}\n  $$\n  for~$X\\geq 2$, any~$\\alpha\\in\\mathbf{R}$ and any~$\\varepsilon>0$, where the\n  implied constant is independent of~$\\alpha$. This is the\n  generalization to~$\\GL_3$ of the bound of Wilton mentioned in the\n  first remark.\n  \\par\n  (3) Using either the functional equation for the $L$-functions\n  $L(\\varphi\\otimes\\chi,s)$, or the Voronoi summation formula, one can\n  show that the estimate of Miller implies a bound of the shape\n  $$\n  S_V(\\Kl_2(a\\cdot;q),X)\\ll_{\\varphi,Z} (qX)^{\\varepsilon}X^{1\/4}q^{3\/4}\n  $$\n  for any~$\\varepsilon>0$, where\n  $$\n  \\Kl_2(n;q)=\\frac{1}{q^{1\/2}}\\sum_{x\\in{\\mathbf{F}^\\times_q}}e_q(\\ov x+nx)\n  $$ \n  is a normalized Kloosterman sum. This bound is non-trivial as long\n  as $X\\geq q$. Since~$\\Kl_2$ is a trace function that is bounded\n  by~$2$ and has Fourier transform bounded by~$1$, this gives (in a\n  special case) a stronger bound than what follows from\n  Theorem~\\ref{thm:main}.\n  \\par\n  (4) Remark (2) suggests a direct approach by the discrete Fourier\n  inversion formula, which gives\n  $$\n  \\sum_{n\\leq X}\\lambda(1,n)K(n)=\\frac{1}{\\sqrt{q}} \\sum_{0\\leq h<q}\n  \\widehat{K}(h) \\sum_{n\\leq X}\\lambda(1,n)e\\Bigl(\\frac{nh}{q}\\Bigr).\n  $$\n  A non-trivial bound for~$X\\approx q^{3\/2}$ in terms\n  of~$\\norm{\\widehat{K}}_{\\infty}$ would then follow from a bound\n  $$\n  \\sum_{n\\leq X}\\lambda(1,n)e\\Bigl(\\frac{nh}{q}\\Bigr) \\ll X^{\\alpha}\n  $$\n  for additive twists of the Fourier coefficients\n  where~$\\alpha<2\/3$.\n  \\par\n  Unsurprisingly, in the case of~$\\GL_2$, although we have the best\n  possible estimate of Wilton (with the analogue of~$\\alpha$\n  being~$1\/2$), the resulting estimate for a sum of length~$q$ is\n  trivial.\n\\end{remark}\n\nThe plan of the paper is as follows: we will explain the idea and\nsketch the key steps of the proof in\nSection~\\ref{sec-principle}. Section~\\ref{sec-examples} recalls the\nmost important examples of trace functions, for which~$K$ has small\nFourier transform and hence for which Theorem~\\ref{thm:main} is\nnon-trivial. Section~\\ref{sec-reminders} presents a key\nFourier-theoretic estimate and some reminders concerning automorphic\nforms and the Voronoi summation formula for~$\\GL_3$. Then the last\nsections complete the proof of Theorem~\\ref{thm:main} following the\noutline presented previously, and explain how to deduce\nCorollaries~\\ref{cor-average}, \\ref{cor-gl2} and~\\ref{RScor} (the last\nof which requires further ingredients).\n\n\\subsection*{Acknowledgements}\n\nWe are very grateful to R. Holowinsky and P. Nelson for sharing with\nus and explaining their work \\cite{HN} which has directly inspired the\npresent paper. We are also very grateful to V. Blomer for Remark \\ref{blomerremark} and to the referees for their\ncareful reading of the manuscript, comments and suggestions and in particular for pointing out\na serious error in the first version of Corollary~\\ref{RScor}.\n\n\n\\subsection*{Notation}\n\nFor any~$z\\in\\mathbf{C}$, we define $e(z)=\\exp(2\\pi i z)$.  If~$q\\geq 1$,\nthen we denote by $e_q(x)$ the additive character modulo~$q$ defined\nby $e_q(x)=e(x\/q)$ for~$x\\in\\mathbf{Z}$. We often identify a $q$-periodic\nfunction defined on~$\\mathbf{Z}$ with a function on~$\\mathbf{Z}\/q\\mathbf{Z}$.\n\nFor any finite abelian group~$A$, we use the notation~$\\widehat{f}$ for\nthe unitary Fourier transform defined on the character\ngroup~$\\widehat{A}$ of~$A$ by\n$$\n\\widehat{f}(\\psi)=\\frac{1}{\\sqrt{|A|}}\n\\sum_{x\\in A}f(x)\\psi(x).\n$$\nWe have then the Plancherel formula~$\\norm{f}_2=\\norm{\\widehat{f}}_2$,\nwhere\n$$\n\\|f\\|_2=\\sum_{x\\in A}|f(x)|^2,\\quad\\quad \\|\\widehat{f}\\|_2=\\sum_{\\psi\\in\n  \\widehat{A}}|\\widehat{f}(\\psi)|^2.\n$$\n\n\nFor any integrable function on~$\\mathbf{R}$, we denote its Fourier transform\nby\n$$\n\\widehat{V}(y)=\\int_{\\mathbf{R}}V(x)e(-xy)dx.\n$$\nWe recall the Poisson summation formula when performed with a\n$q$-periodic function~$K$ in addition to a smooth function~$V$ with\nfast decay at infinity: for any~$X\\geq 1$, we have\n\\begin{equation}\\label{eq-poisson}\n  \\sum_{n\\in\\mathbf{Z}}K(n)V\\Bigl(\\frac{n}{X}\\Bigr)= \\frac{X}{q^{1\/2}} \\sum_{h\\in\\mathbf{Z}}\n  \\widehat{K}(h)\\widehat{V}\\Bigl(\\frac{hX}{q}\\Bigr).\n\\end{equation}\nThis follows directly from the usual Poisson formula and the\ndefinition of $\\widehat{K}$ after splitting the sum into congruence\nclasses modulo~$q$.\n\n\n\n\\section{Principle of the proof}\\label{sec-principle}\n\nWe use a direct generalisation of the method of Holowinsky and\nNelson~\\cite{HN} that led to Theorem~\\ref{th-hn}. Although it was\nmotivated by Munshi's approach, based on the use of the Petersson\nformula as a tool to express the delta symbol, there is no remaining\ntrace of this point of view; however, we refer to~\\cite[App. B]{HN}\nfor a detailed and insightful description of the origin of this\nstreamlined method, starting from Munshi's.\n\n\\subsection{Amplification}\\label{ssec-amplification}\n\nThe first step is to realize the $q$-periodic function~$K$ within a\none-parameter family of $q$-periodic functions. Precisely, let\n$\\widehat K$ be the Fourier transform of $K$ (see \\eqref{eq:fourierK})\nand define\n\\begin{equation}\\label{whatKzdef}\n\\widehat K(z,h):= \\begin{cases} \\widehat K(z)e_q(-h\\ov z)& q\\nmid z\\\\\n  \\widehat K(0)& q\\mid z\n\\end{cases}\n\\end{equation}\nfor $(z,h)\\in \\mathbf{Z}^2$. Then put\n\\begin{equation}\\label{eq-kn}\nK(n,h)=\\frac{1}{q^{1\/2}}\\sum_{z\\in{\\mathbf{F}^\\times_q}}\\widehat K(z,h)e_q(-nz).\n\\end{equation}\nfor $(n,h)\\in\\mathbf{Z}^2$.  By the discrete Fourier inversion formula, we\nhave\n\\begin{equation}\\label{eq-detector}\nK(n,0)=K(n)-\\frac{\\widehat K(0)}{q^{1\/2}}\n\\end{equation}\nMore generally, for any probability measure $\\varpi$ on~${\\mathbf{F}^\\times_q}$, the\naverage\n$$\nK_{\\varpi}(n,h)=\\sum_{l\\in{\\mathbf{F}^\\times_q}}\\varpi(l)K(n,\\ov lh)\n$$ \nsatisfies $K_{\\varpi}(n,0)=K(n)-\\frac{\\widehat K(0)}{q^{1\/2}}$.  It follows\nthat, for any parameter $H\\geq 1$, we can express the sum $S_V(K,X)$\nas the difference of double sums\n$$\nS_V(K,X)=\\sum_{l\\in\\mathcal{L}}\\varpi(l)\\sum_{|h|\\leq H}S_{V}(K(\\cdot,h\\ov\nl),X)- \\sum_{l\\in\\mathcal{L}}\\varpi(l)\\sum_{0<|h|\\leq H}S_{V}(K(\\cdot,h\\ov\nl),X),\n$$\nup to an error $\\ll X\/q^{1\/2}$.  We write this difference as\n$$\nS_V(K,X)=\\mathcal{F}-\\mathcal{O},\n$$\nsay. One then needs to select a suitable probability measure~$\\varpi$,\nand then the two terms are then handled by different methods. It\nshould be emphasized that no main term arises (which would have to be\ncanceled in the difference between the two terms).\n\n\\begin{remark}\n  The argument is reminiscent of the \\emph{amplification method}, the\n  function $K(n)=K(n,0)$ being ``amplified'' (up to a small error)\n  within the family $(K(n,h))_{|h|\\leq H}$.\n\\end{remark}\n\n\\subsection{Bounding $\\mathcal{F}$}\n\nAs in~\\cite{HN}, we consider a probability measure~$\\varpi$\ncorresponding to a product structure: we average over pairs $(p,l)$ of\nprimes such that $p\\sim P$ and $l\\sim L$, and take $\\varpi(x)$\nproportional to the number of representations $x=\\ov p l\\mods q$,\nwhere $p\\sim P$ and $l\\sim L$ are primes (their sizes being parameters\n$1\\leq P,L<q\/2$ to be chosen later).\n\nThe treatment of $\\mathcal{F}$ is essentially the same as\nin~\\cite{HN, Lin}. By applying the Poisson summation formula to the\n$h$-variable, with dual variable~$r$, we see the function\n$$\n(n,r,p,l)\\mapsto \\widehat K(-p\\ov{l r})\\lambda(1,n)e_q(np\\ov{l r}),\n$$\nappear. We then appeal to the classical ``reciprocity law'' for\nadditive exponentials, namely\n$$\ne_q(np\\ov {l r})\\approx e_{rl}(-np\\ov q),\n$$\ntrading the modulus $q$ for the modulus $rl$, which will be\nsignificantly smaller than $q$. We then apply the Voronoi summation\nformula for the cusp form $\\varphi$ on the $n$-variable (this is the\nonly real automorphic input), which transforms the additive phase\n$e_{rl}(-np\\ov q)$ into Kloosterman sums of modulus $rl$. We then\nobtain further cancellation by smoothing out the resulting variable\n(dual to $n$) by using Cauchy--Schwarz and detecting cancellations on\naverages of products of Kloosterman sums, where the product structure\nof the averaging set is essential.\n\nIn this part of the argument, the coefficient function~$K$ plays very\nlittle role, and we could just more or less quote the\ncorresponding statements in~\\cite{HN, Lin}, if the parameter~$Z$ was\nfixed. Since we wish to keep track of its behavior (for the purpose of\nflexibility for potential applications), we have to go through the\ncomputations anew. This is done in detail in Section~\\ref{sec:Fsum}.\n\n\\subsection{Bounding $\\mathcal{O}$}\n\nIn the sum~$\\mathcal{O}$, with the averaging performed in the same way as\nfor~$\\mathcal{F}$, the key point is that the $n$-variable in the sum\n$S_V(K(\\cdot, h\\bar{l},X)$ is very long compared to $q$. We apply\nCauchy's inequality to smooth it, keeping the other variables $h,p,l$\nin the inside, thus eliminating the automorphic coefficients\n$\\lambda(1,n)$ (for which we only require average bounds, which we\nborrow from the Rankin--Selberg theory, our second important\nautomorphic input). This leads quickly to the problem of estimating\nthe sum\n$$\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\\sum_{n\\sim X}K(n,h_1p_1\\ov\nl_1)\\ov{K(n,h_2p_2\\ov l_2)}.\n$$\nWe apply the Poisson formula in the $n$-variable; since $X$ is\ntypically much larger than $q$, only the zero frequency in the dual\nsum contributes. This yields a key sum of the shape\n$$\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\\sum_{u\\in{\\mathbf{F}^\\times_q}}|\\widehat\nK(u)|^2e_q((h_1p_1\\bar{l}_1-h_2p_2\\bar{l}_2)u^{-1})=\n\\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2}\nK_2(h_1p_1\\bar{l}_1-h_2p_2\\bar{l}_2),\n$$\nsay.\n\\par\nWhen $K$ is a multiplicative character, as in the work of\nHolowinsky--Nelson, the proof is essentially finished then, since\n$\\widehat K(u)$ is a normalized Gauss sum, with a constant modulus,\nhence~$K_2$ is simply a Ramanujan sum, which we can evaluate\nexplicitly.\n\\par\nIn general, we obtain cancellation using a very general\nFourier-theoretic bound for general bilinear forms\n$$\n\\sum_{m\\in{\\mathbf{F}_q}}\\sum_{n\\in{\\mathbf{F}_q}}\\alpha_m\\beta_nK_2(m-n),\n$$\nwhich involves only $L^2$-norm bounds for the coefficients and\n$L^{\\infty}$-norm bounds for the Fourier transform of~$K_2$\n(see~Proposition~\\ref{willprop}). The latter, it turns out, is\nessentially~$|\\widehat{K}|^2$, and we can obtain a good estimate purely\nin terms of~$\\norm{\\widehat{K}}_{\\infty}$.  This part of the argument is\nperformed in Section~\\ref{sec:O}.\n\n\n\\section{Examples of trace functions}\\label{sec-examples}\n\nTheorem~\\ref{thm:main} certainly applies to ``random'' $q$-periodic\nfunctions $K\\colon \\mathbf{Z}\\to\\mathbf{C}$, for all reasonable meanings of the word\n``random'', but the basic motivating examples in number theory are\noften provided by \\emph{trace functions}.  Since there are by now a\nnumber of surveys and discussions of important examples (see,\ne.g.,~\\cite[\\S 10]{FKM1} or~\\cite[\\S 2.2]{short-sums} or~\\cite{pisa}),\nwe only recall some of them for concreteness.\n\n\\begin{itemize}\n\\item If $r\\geq 1$ is a fixed integer and $\\chi_1$, \\ldots, $\\chi_r$\n  are distinct non-trivial Dirichlet characters modulo~$q$, of\n  order~$d_i\\geq 2$, and if $f_1$, \\ldots, $f_r$, $g$ are polynomials\n  in $\\mathbf{Z}[X]$ such that either $\\deg(g\\mods q)\\geq 2$, or one of\n  the~$f_i\\mods q$ is not proportional to a $d_i$-th power\n  in~$\\bar{\\Ff}_q[X]$, then\n  $$\n  K(n)=\\chi_1(f_1(n))\\cdots \\chi_r(f_r(n))e\\Bigl(\\frac{g(n)}{q}\\Bigr)\n  $$\n  has Fourier transform of size bounded only in terms of~$r$ and the\n  degrees of the polynomials~$f_i$ and~$g$. (This is a consequence of\n  the Weil bounds for exponential sums in one variable).\n  \\par\n  Moreover (as is relevant only for Corollary~\\ref{RScor} in this\n  paper), $K$ is a trace function, and it is non-exceptional,\n  unless~$g$ is of degree~$1$, $r=1$ and~$f_1$ is of degree~$\\leq 1$.\n\\item  Let~$r\\geq 2$. Define~$\\Kl_r(0)=0$ and\n  $$\n  \\Kl_r(n)=\\frac{1}{q^{(r-1)-2}}\n  \\sum_{\\substack{x_1,\\ldots,x_r\\in {\\mathbf{F}_q}\\\\\n      x_1\\cdots x_r=n}}e\\Bigl(\\frac{x_1+\\cdots+x_r}{q}\\Bigr)\n  $$\n  for~$n\\in{\\mathbf{F}^\\times_q}$ (these are hyper-Kloosterman sums). Then\n  $\\norm{\\widehat{\\Kl}_r}_{\\infty}\\leq c_r$, where~$c_r$ depends only\n  on~$r$ (this depends on Deligne's general proof of the Riemann\n  Hypothesis over finite fields and on the construction and basic\n  properties of Kloosterman sheaves).\n  \\par\n  For all~$r\\geq 2$, the function~$\\Kl_r$ is a trace function of a\n  non-exceptional sheaf.\n\\end{itemize}\n\nWe also mention one important principle: if~$K$ is the trace function\nof a Fourier sheaf~$\\mathcal{F}$ (in the sense of~\\cite{ESDE}),\nthen~$\\widehat{K}$ is also such a function for a sheaf~$\\ft(\\mathcal{F})$;\nmoreover, if~$\\mathcal{F}$ has conductor~$c$ (in the sense of~\\cite{FKM1}),\nthen $\\ft(\\mathcal{F})$ has conductor $\\leq 10c^2$, and in\nparticular~$\\norm{\\widehat{K}}_{\\infty}\\leq 10c^2$. \n\nFinally, one example that is not usually discussed explicitly\n(formally, because it arises from a skyscraper sheaf) is\n$K(n)=q^{1\/2}\\delta_{n=a\\mods{q}}$, the $L^2$-normalized delta\nfunction at a point~$a\\in\\mathbf{Z}$. In this case, the Fourier transform is\nan additive character, hence is bounded by one, and dividing\nby~$q^{1\/2}$, we obtain the bound\n$$\n\\sum_{\\substack{n\\geq 1\\\\n\\equiv a\\mods{q}}}\n\\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr) \\ll\nZ^{10\/9}q^{-5\/18+\\varepsilon}X^{5\/6},\n$$\nunder the assumptions of Theorem~\\ref{thm:main}; in particular,\nif~$X=q^{3\/2}$ and~$V$ satisfies~(\\ref{eq:Vprop}) for~$Z=1$, we get\n$$\n\\sum_{\\substack{n\\geq 1\\\\n\\equiv a\\mods{q}}}\n\\lambda(1,n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr) \\ll q^{35\/36+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$.  Note that, under the generalized\nRamanujan--Petersson conjecture $\\lambda(1,n)\\ll n^{\\varepsilon}$, we would\nobtain the stronger bound $q^{1\/2+\\varepsilon}$ (and knowing the\napproximation~$\\lambda(1,n)\\ll n^{\\theta}$ for some $\\theta<1\/3$ would\nbe enough to get a non-trivial bound).  We discuss this case in\nfurther details in Remark \\ref{lastremark}, in the context of\nCorollary \\ref{cor-average}.\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec-reminders}\n\n\n\\subsection{A Fourier-theoretic estimate}\n\nA key estimate in Section~\\ref{sec:O} will arise from the following\ngeneral bound (special cases of which have appeared before, e.g. in\nthe case of multiplicative characters for problems concerning sums\nover sumsets).\n \n\\begin{proposition}\\label{willprop}\n  Let $A$ be a a finite abelian group, with group operation denoted\n  additively.  Let $\\alpha$, $\\beta$ and $K$ be functions from~$A$\n  to~$\\mathbf{C}$. We have\n  $$\n  \\Bigl|\\sum_{m,n\\in A}\\alpha(m)\\beta(n)K(m-n)\\Bigr|\\leq\n  |A|^{1\/2}\\|\\widehat K\\|_\\infty \\|\\alpha\\|_2 \\|\\beta \\|_2.\n  $$\n\\end{proposition}\n\n\\begin{proof}\n  Using orthogonality of characters, we write\n  $$\n  \\sum_{m,n\\in A}\\alpha(m)\\beta(n) K(m-n)=\n  \\sum_{m,n,h\\in A}\\alpha(m)\\beta(n) K(h)\\frac{1}{|A|}\n  \\sum_{\\psi\\in\\widehat{A}}\\psi(h-(m-n)).\n  $$\n  Moving the sum over~$\\psi$ to the outside, this is equal to\n  $$\n  |A|^{1\/2}\\sum_{\\psi\\in\\widehat{A}}\n  \\widehat{\\alpha}(\\psi^{-1})\\widehat{\\beta}(\\psi)\\widehat{K}(\\psi),\n  $$\n  whose absolute value is\n  $$\n  \\leq |A|^{1\/2}\\norm{\\widehat{K}}_{\\infty}\n  \\sum_{\\psi\\in\\widehat{A}}|\\widehat{\\alpha}(\\psi^{-1})\\widehat{\\beta}(\\psi)|\n  \\leq |A|^{1\/2}\n  \\|\\widehat K\\|_\\infty \\|\\alpha\\|_2 \\|\\beta \\|_2,\n  $$\n  by the Cauchy--Schwarz inequality and the discrete Plancherel\n  formula.\n\\end{proof}\n\n\n\n\\subsection{Background on $\\GL_3$-cusp forms}\n\nWe refer to \\cite[Chap. 6]{Goldfeld} for notations. Let $\\varphi$ be a\ncusp form on~$\\GL_3$ with level~$1$ and with Langlands parameters\n$\\mu=(\\mu_1,\\mu_2,\\mu_3)\\in\\mathbf{C}^3$.  We denote by\n$(\\lambda(m,n))_{m,n\\not=0}$ its Fourier--Whittaker coefficients, and\nassume that\n$\\varphi$ is an eigenform of the Hecke operators $T_n$ and $T_n^*$,\nnormalized so that $\\lambda(1,1)=1$. The eigenvalue of~$T_n$ is\nthen~$\\lambda(1,n)$ for~$n\\geq 1$.\n\n\nLet $\\theta_3=5\/14$. The archimedean parameters and the Hecke\neigenvalues are bounded individually by\n$$\n|\\Re(\\mu_{i})|\\leq \\theta_3.\\quad\\quad |\\lambda(1,p)|\\leq 3p^{\\theta_3}\n$$\nfor any~$i$ and any prime number~$p$ (see ~\\cite{KimSar}).\n\nAverage estimates follow from the Rankin--Selberg method. We have\n\\begin{equation}\n  \\label{eq:RS}\n  \\sum_{1\\leq n\\leq X}|\\lambda(1,n)|^2\\ll X^{1+\\varepsilon},\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq-RS2}\n  \\sum_{1\\leq m^2n\\leq X}m|\\lambda(m,n)|^2\\ll X^{1+\\varepsilon},\n\\end{equation}\nfor $X\\geq 2$ and any $\\varepsilon>0$, where the implied constant depends\nonly on $\\varphi$ and $\\varepsilon$. (See~\\cite{Molteni} and ~\\cite[Lemma 2]{Munshi1}.)\n\nThe key analytic feature of $\\GL_3$-cusp forms that we use (as in\nprevious works) is the Voronoi summation formula for~$\\varphi$\n(originally due to Miller--Schmid, and Goldfeld--Li\nindependently). Since our use of the ``archimedean'' part of the\nformula is quite mild, we use the same compact formulation as\nin~\\cite[\\S 2.3]{HN}, where references are given.\n\nLet~$q\\geq 1$ be an integer (not necessarily prime). For $n\\in\\mathbf{Z}$, we\ndenote\n$$\n\\Kl_2(n;q)=\\frac{1}{\\sqrt{q}}\n\\sum_{x\\in (\\mathbf{Z}\/q\\mathbf{Z})^{\\times}}\ne\\Bigl(\\frac{nx+\\bar{x}}{q}\\Bigr)\n$$\nwhere~$\\bar{x}$ is the inverse of~$x$ modulo~$q$.\n\n\n\\begin{lemma}[Voronoi summation formula]\\label{Voronoi}\n  For $\\sigma\\in\\{-1,1\\}$, there exist functions $\\mathcal{G}^{\\sigma}$,\n  meromorphic on~$\\mathbf{C}$, holomorphic for~$\\Re(s)>\\theta_3$, with\n  polynomial growth in vertical strips~$\\Re(s)\\geq \\alpha$ for\n  any~$\\alpha>\\theta_3$, such that the following properties hold.\n  \\par\n  Let $a$ and~$q\\geq 1$ be coprime integers, let $X>0$, and let $V$ be\n  a smooth function on~$]0,+\\infty[$ with compact support.\n \n  We have\n  $$\n  \\sum_{n\\geq 1}\\lambda(1,n)e_q(an)V\\Bigl(\\frac{n}{X}\\Bigr) = q^{3\/2}\n  \\sum_{\\sigma\\in\\{-1,1\\}} \\sum_{n\\geq 1} \\sum_{m\\mid q}\n  \\frac{\\lambda(n,m)}{nm^{3\/2}} \\Kl_2\\Bigl(\\sigma n\\ov\n  a;\\frac{q}{m}\\Bigr) \\mathcal{V}_{\\sigma}\\Bigl(\\frac{m^2n}{q^3\/X}\\Bigr),\n  $$\n  where \n  $$\n  \\mathcal{V}_{\\sigma}(x)= \\frac{1}{2\\pi i}\\int_{(1)}x^{-s}\\mathcal{G}^{\\sigma} (s+1)\n  \\Bigl(\\int_{0}^{+\\infty}V(y)y^{-s}\\frac{dy}{y}\\Bigr)ds.\n  $$\n\\end{lemma}\n\nNote that the functions~$\\mathcal{G}^{\\sigma}$ depend (only) on the\narchimedean parameters of~$\\varphi$. We record some properties of the\nfunctions $\\mathcal{V}_{\\sigma}(x)$; for~$Z$ fixed they are already explained\nin~\\cite[\\S 2.3]{HN}.\n\n\\begin{lemma}\\label{bounds-for-V}\n  Let $\\sigma\\in\\{-1,1\\}$. For any $j\\geq 0$, any $A\\geq 1$ and\n  any~$\\varepsilon>0$, we have\n$$\nx^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll \\min\\Bigl( Z^{j+1}x^{1-\\theta_3-\\varepsilon},\nZ^{j+5\/2+\\varepsilon}\\Bigl(\\frac{Z^3}{x}\\Bigr)^A\\Bigr)\n$$\nfor~$x>0$, where the implied constant depends on~$(j,A,\\varepsilon)$.\nMoreover, for $x\\geq 1$, we have\n$$\nx^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll x^{2\/3}\\min(Z^j, x^{j\/3})\n$$\nwhere the implied constant depends on~$j$.\n\\end{lemma}\n\n\\begin{proof}\n  The first inequality in the first bound follows by shifting the\n  contour in $\\mathcal{V}_{\\pm}(x)$ to $\\Re s=\\theta_3-1+\\varepsilon$, while the\n  second one follows by shifting contour to the far right.  The second\n  bound follows from \\cite[Lemma 6]{Blomer}.\n\\end{proof}\n\nIn particular, we see from the lemma that the functions\n$\\mathcal{V}_{\\sigma}(x)$ decay very rapidly as soon as\n$x\\geq X^{\\delta}Z^{3}$ for some~$\\delta>0$.\n\n\\begin{remark}\n  The bound $x^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll Z^{j+1}x^{1-\\theta_3-\\varepsilon}$\n  can be replaced by $x^j\\mathcal{V}_{\\sigma}^{(j)}(x)\\ll Z^{j}x^{1-\\varepsilon}$,\n  under the Ramanujan-Selberg conjecture, i.e., if $\\Re (\\mu_i)=0$ for\n  all~$i$.\n\\end{remark}\n\n\\begin{remark}\n  Let~$N\\geq 1$, and define a congruence\n  subgroup~$\\Gamma_N\\subset \\SL_3(\\mathbf{Z})$ by\n  $$\n  \\Gamma_N=\\Bigl\\{\\gamma\\in\\SL_3(\\mathbf{Z})\\,\\mid\\, \\gamma\\equiv\\begin{pmatrix}\n    *&*&*\\\\ *&*&*\\\\0&0&*\n  \\end{pmatrix}\\mods N\n  \\Bigr\\}.\n  $$\n  Zhou~\\cite{FZ} has established an explicit Voronoi summation\n  formula for $\\GL_3$-cuspforms that are invariant under~$\\Gamma_N$,\n \n \n \n \n \n \n \n \n  for additive twists by~$e_q(an)$ when either $(q,N)=1$ or $N\\mid\n  q$. It should then be possible to use this formula to generalise\n  Theorem~\\ref{thm:main} to such cuspforms by slight adaptations of\n  the argument below.\n \n \n \n \n \n \n\\end{remark}\n\n\n\\section{Amplification of the trace function}\n\nWe now begin the proof of Theorem~\\ref{thm:main}. Let $q$ be a prime\nnumber and $K$ a $q$-periodic function on $\\mathbf{Z}$. Let $\\widehat{K}$ be its\ndiscrete Fourier transform~(\\ref{eq:fourierK}), which is also a\n$q$-periodic function on~$\\mathbf{Z}$.  \n\nLet $P,L\\geq 1$ be two parameters to be chosen later, with $2P<q$ and\n$2L<q$. We define auxiliary sets\n\\begin{align*}\n  \\mathrm{P}&:=\\{p\\in[P,2P[\\,\\mid\\, p\\equiv 1\\mods{4}, \\text{ prime}\\}\\\\\n  \\mathrm{L}&:=\\{l\\in[L,2L[\\,\\mid\\, l\\equiv 3\\mods{4}, \\text{ prime}\\}.\n\\end{align*}\nNote that these sets are disjoint.  We denote\n\\begin{equation}\\label{eq-H}\nH=\\frac{q^2L}{XP}.\n\\end{equation}\nIn the sequel, we assume that $H\\geq 1$, that is\n\\begin{equation}\\label{Hcond}\n  XP\\leq q^2L.\n\\end{equation}\nLet $W$ be a smooth function on $\\mathbf{R}$ that satisfies \\eqref{eq:Vprop}\nwith $Z=1$ and furthermore $\\widehat W(0)=1$.\n\nWe now use the notation~$K(n,h)$ and~$\\widehat{K}(z,h)$ and the basic\namplicatifon idea discussed in Section~\\ref{ssec-amplification}\n(see~\\ref{whatKzdef} and~\\ref{eq-kn}).  We define\n\\begin{align*}\n  \\mathcal{F}\n  &=\n    \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\in \\mathbf{Z}}S_{V}(K(\\cdot,hp\\ov l),X)\n    \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\\\\\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\in\\mathbf{Z}}\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{align*}\n\nSeparating the contribution of $h=0$ and applying~(\\ref{eq-detector}),\nwe can write\n\\begin{equation}\\label{an-identity}\n  \\mathcal{F}=S_{V}(K,X)+\\mathcal{O}+\n  O\\Bigl(\\frac{q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}X}{q^{1\/2}}\\Bigr),\n\\end{equation}\nfor any~$\\varepsilon>0$, where\n\\begin{equation}\\label{eq-m2}\n\\mathcal{O}= \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n\\sum_{h\\neq 0}\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{equation}\nIndeed, the contribution of $h=0$ is\n\\begin{align*}\n  \\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n  S_{V}(K(\\cdot,0),X)\\widehat{W}(0)\n  &=S_{V}(K,X)-\n    \\frac{\\widehat K(0)}{|\\mathrm{P}||\\mathrm{L}|q^{1\/2}}\n    \\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{n\\geq 1}\\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr)\\\\\n  &=S_{V}(K,X)+O\\Bigl(\\frac{\\norm{\\widehat{K}}_{\\infty}X^{1+\\varepsilon}}{q^{1\/2}}\\Bigr),\n\\end{align*}\nfor any $\\varepsilon>0$, by \\eqref{eq:RS}.\n\n\\section{Evaluation of $\\mathcal{F}$}\\label{sec:Fsum}\n\nThe evaluation of~$\\mathcal{F}$ is close to the arguments of~\\cite{HN}\nand~\\cite[\\S 6]{Lin}. In fact, we could extract the desired bounds\nfrom these sources (especially~\\cite{Lin}) in the important special\ncase when the parameter~$Z$ is fixed as~$q$ varies. The reader who is\nfamiliar with one of these references may therefore wish to skip the\nproof of the next proposition in a first reading.\n\n\n\\begin{proposition}\\label{proposition-for-F}\nLet $\\eta>0$. Assume that \n\\begin{equation}\n  \\label{XZlower}\n  X\/Z\\geq q^{1+\\eta}.\n\\end{equation}\nand\n\\begin{equation}\\label{boundsPL}\nL\\leq P^4.\n\\end{equation}\nThen for any $\\varepsilon>0$, we have\n$$\n\\mathcal{F}\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n\\Bigl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\\Bigr),\n$$\nwhere the implied constant depends on $\\varphi$, $\\varepsilon$ and $\\eta$.\n\\end{proposition}\n\nThe remainder of this section is dedicated to the proof of this\nproposition. We fix $\\eta$ satisfying~(\\ref{XZlower}).\n \nWe apply the Poisson summation formula to the sum over $h$ in\n$\\mathcal{F}$, for each $(p,l)$. We obtain\n$$\n\\sum_{h\\in \\mathbf{Z}}K(n,hp\\ov\nl)\\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)=\\frac{H}{q^{1\/2}}\\sum_{(r,q)=1}\\widehat\nK(-p\\ov l\\ov r)e_q(np\\ov l\\ov r)W\\Bigl(\\frac{r}{R}\\Bigr),\n$$\nwhere\n\\begin{equation}\\label{eq-R}\nR=q\/H=\\frac{XP}{qL}.\n\\end{equation}\nHence it follows that\n$$\n\\mathcal{F}= \\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}} \\sum_{(r,q)=1}\\widehat K(-p\\ov l\\ov r)\n\\sum_{n\\geq 1}e_q(n p\\ov l\\ov r) \\lambda(1,n)V\\Bigl(\\frac{n}{X}\\Bigr)\nW\\Bigl(\\frac{r}{R}\\Bigr).\n$$\n\nSince $l\\leq 2L<q$, we have $(q,rl)=1$ in the sums. By reciprocity, we\nhave\n$$\ne_q(n p\\ov l\\ov r)=e_{rl}(-np\\ov q)e_{qrl}(np)\n$$\nfor $n\\geq 1$.\n\n\\begin{remark}\n  Note that for $n\\asymp X$, we have\n  $$\n  \\frac{np}{qrl}\\asymp \\frac{XP}{qLXP\/(qL)}\\asymp 1,\n  $$\n  so that the additive character $e_{qrl}(np)$ doesn't oscillate.\n\\end{remark}\n\nWe define\n$$\nV_1(x)=e\\Bigl(\\frac{xXp}{qrl}\\Bigr)V(x).\n$$\nWe can then rephrase the above as\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov\nq)e_{qrl}(np)V\\Bigl(\\frac{n}{X}\\Bigr)= \\sum_{n\\geq\n  1}\\lambda(1,n)e_{rl}(-np\\ov q)V_1\\Bigl(\\frac{n}{X}\\Bigr),\n$$\nand\n$$\n\\mathcal{F}\n=\\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\\sum_{p,l}\\sum_{r\\geq 1}\n\\widehat K(-p\\ov l\\ov r)W\\Bigl(\\frac{r}{R}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov q)V_1\\Bigl(\\frac{n}{X}\\Bigr).\n$$\n\nLet $\\mathcal{F}'$ be the contribution to the last expression of\nthose $(p,r,l)$ such that $(p,rl)=1$, and let $\\mathcal{F}''$ be the\nremaining contribution.\n\nIn the case $p\\mid r$, we can apply the Voronoi formula with modulus\n$rl\/p$; estimating the resulting expression directly, one obtains an\nestimate for the contribution $\\mathcal{F}''$ to $\\mathcal{F}$ that is bounded by\n$$\n\\mathcal{F}''\\ll \\frac{\\norm{\\widehat{K}}_{\\infty}Z^2X^{3\/2+\\varepsilon}}{qP}\n$$\nfor any $\\varepsilon>0$ (see~\\cite[\\S 6]{Lin} for a similar computation,\nwhere such contribution is denoted $\\mathcal{F}_1^{\\sharp}$).\n\nNow let $p$ be such that $(p,rl)=1$.  By the Voronoi summation\nformula (Lemma \\ref{Voronoi}), we have\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)e_{rl}(-np\\ov q)\nV_1\\Bigl(\\frac{n}{X}\\Bigr)=(rl)^{3\/2} \\sum_{\\sigma\\in\\{-1,1\\}}\n\\sum_{n\\geq 1} \\sum_{m\\mid rl} \\frac{\\lambda(n,m)}{nm^{3\/2}}\n\\Kl_2(\\sigma\\ov\npqn;rl\/m)\\mathcal{V}_{1,\\sigma}\n\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n$$\nTherefore $\\mathcal{F}'=\\mathcal{F}'_{1}+\\mathcal{F}'_{-1}$, where\n\\begin{multline*}\n  \\mathcal{F}'_{\\sigma}\n  =\\frac{q^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n  \\sum_{r\\geq 1}\\widehat K(-p\\ov l\\ov\n  r)\n  W\\Bigl(\\frac{r}{R}\\Bigr)(rl)^{3\/2}\n  \\\\\n  \\sum_{n\\geq 1}\\sum_{m\\mid rl}\n  \\frac{\\lambda(n,m)}{nm^{3\/2}}\\Kl_2(\\sigma \\ov\n  pqn;rl\/m)\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n\\end{multline*}\nWe re-arrange the sums to get\n\\begin{multline*}\n  \\mathcal{F}'_{\\sigma} =\\frac{(qRL)^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n  \\sum_{r\\geq 1}\\Bigl(\\frac{r}{R}\\Bigr)^{3\/2}\n  W\\Bigl(\\frac{r}{R}\\Bigr) \\sum_{n,m}\n  \\frac{\\lambda(n,m)}{\\sqrt{nm}} \\\\\n  \\sum_{p\\in\\mathrm{P}}\\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid rl}}\n  \\frac{(l\/L)^{3\/2}}{\\sqrt{n}m}\\widehat K(-p\\ov l\\ov r)\\Kl_2(\\sigma \\ov\n  pqn;rl\/m)\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l^3\/X}\\Bigr).\n\\end{multline*}\n\nLet~$\\delta>0$ be a small parameter. For fixed $r$ and $l$, using the\nbounds from Lemma \\ref{bounds-for-V} with a suitably large value\nof~$A$, the contribution to the sum over~$m$ and~$n$ of $(m,n)$\nsuch that\n\\begin{equation}\\label{eq-truncate}\n  m^2n\\geq q^{\\delta}\\frac{Z^3(rl)^3}{X}\\asymp \\frac{q^{\\delta}Z^3X^2P^3}{q^3}\n\\end{equation}\nis $\\ll \\norm{\\widehat{K}}_{\\infty}q^{-10}$ (say).\n\\par\nTo handle the remaining part of the sum, we apply the Cauchy--Schwarz\ninequality to the sum over $(m,n)$, and we obtain\n\\begin{equation}\\label{eq-mcfsigma}\n  \\mathcal{F}'_{\\sigma} \\ll \\frac{(qRL)^{3\/2}L}{XP|\\mathrm{P}||\\mathrm{L}|}\n  \\Bigl(\\sum_{r\\sim R} \\sum_{\\substack{n,m\\geq 1\\\\m^2n<\n      q^{\\delta}Z^3X^2P^3\/q^3}}\n  \\frac{|\\lambda(n,m)|^2}{nm}\\Bigr)^{1\/2}\\mathcal{N}_{\\sigma}^{1\/2}\n  +\\norm{\\widehat{K}}_{\\infty}q^{-1},\n\\end{equation}\nwhere\n\\begin{multline*}\n  \\mathcal{N}_{\\sigma}= \\sum_{r,m\\geq 1} W\\Bigl(\\frac{r}{R}\\Bigr)\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l_1,l_2\\\\p_i\\in\\mathrm{P},l_i\\in\\mathrm{L}\\\\m\\mid\n      (rl_1,rl_2)}} \\Bigl(\\frac{l_1l_2}{L^2}\\Bigr)^{3\/2}\n  \\widehat K(-p_1\\ov l_1\\ov r)\\ov{\\widehat K(- p_2\\ov l_2\\ov r)}\\\\\n  \\times\\sum_{n\\geq 1}\\frac{1}{n}\\Kl_2(\\sigma \\ov\n  p_1qn;rl_1\/m)\\ov{\\Kl_2(\\sigma \\ov p_2qn;rl_2\/m)}\n  \\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l_1^3\/X}\\Bigr)\n  \\ov{\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2n}{r^3l_2^3\/X}\\Bigr)}.\n\\end{multline*}\nWe will prove the bound\n\\begin{equation}\\label{Nsigmabound}\n\\mathcal{N}_{\\sigma}\\ll\nq^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\n\\Bigl(\nZ^4RPL\n+\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}\\Bigr).\t\n\\end{equation}\nfor any~$\\varepsilon>0$.\nIf we select~$\\delta>0$ small enough in terms of~$\\varepsilon$, then by \nthe Rankin--Selberg bound~(\\ref{eq-RS2}), we deduce that\n\\begin{equation}\\label{eq-fpsigma}\n\\mathcal{F}'_{\\sigma} \\ll \\frac{q^{3\/2+\\varepsilon}Z^{\\varepsilon}L^{5\/2}R^2}\n{XP|\\mathrm{P}||\\mathrm{L}|}\\mathcal{N}_{\\sigma}^{1\/2}+\n\\norm{\\widehat{K}}_{\\infty}q^{-1}\n\\end{equation}\nfor any $\\varepsilon>0$.\nWe conclude, using~(\\ref{eq-mcfsigma}) and recalling\nthat~$R=XP\/(qL)$, that\n\\begin{align*}\n  \\mathcal{F}'_{\\sigma}\n  &\\ll \n    \\frac{q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n    R^2(qL)^{3\/2}}{XP^2}\\bigg(Z^4RPL+\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}\n    \\bigg)^{1\/2}\n  \\\\\n  &\\ll\n    q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n    \\Bigl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\\Bigr).\n\\end{align*}\nfor any~$\\varepsilon>0$. Assuming \\eqref{Nsigmabound}, this concludes the proof of\nProposition~\\ref{proposition-for-F}.\n\n\\subsection{Proof of \\eqref{Nsigmabound}} We will now investigate the inner sum over~$n$\nin~$\\mathcal{N}_{\\sigma}$, and then perform the remaining summations\n(over $r$, $m$, $p_i$, $l_i$) essentially trivially. We let\n$$\nU=\\frac{q^{\\delta\/2}Z^{3\/2}XP^{3\/2}}{q^{3\/2}},\n$$\nso that the sum over~$m$ has been truncated to~$m\\leq U$.\n\nLet~$F$ be a smooth non-negative function on $\\mathbf{R}$ which is supported\non $[1\/2,3]$ and equal to $1$ on $[1,2]$. Let $Y\\geq 1$ be a parameter\nwith\n\\begin{equation}\\label{eq-boundy}\nY\\leq \\frac{q^{\\delta}Z^3X^2P^3}{m^2q^3},\n\\end{equation}\nand define\n$$\n\\mathcal{W}_{Y}(x)=\\frac{1}{x}\n\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2xY}{r^3l_1^3\/X}\\Bigr)\n\\ov{\\mathcal{V}_{1,\\sigma}\\Bigl(\\frac{m^2xY}{r^3l_2^3\/X}\\Bigr)} F(x).\n$$\nWe study the sums\n$$\n\\mathcal{P}_Y= \\frac{1}{Y}\\sum_{n\\geq 1}\\Kl_2(\\ov\np_1qn;rl_1\/m)\\ov{\\Kl_2(\\ov p_2qn;rl_2\/m)}\n\\mathcal{W}_{Y}\\Bigl(\\frac{n}{Y}\\Bigr),\n$$\nand their combinations\n\\begin{equation}\\label{eq-py}\n  \\mathcal{N}_{Y,\\sigma}= \\sum_{r\\geq 1}\\sum_{1\\leq m\\leq U}\n  W\\Bigl(\\frac{r}{R}\\Bigr)\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l_1,l_2\\\\p_i\\in\\mathrm{P},l_i\\in\\mathrm{L}\\\\m\\mid\n      (rl_1,rl_2)}} \\Bigl(\\frac{l_1l_2}{L^2}\\Bigr)^{3\/2}\n  \\widehat K(-p_1\\ov l_1\\ov r)\\ov{\\widehat K(- p_2\\ov l_2\\ov\n    r)}\\,\\mathcal{P}_Y.\n\\end{equation}\n\nWe will prove the following bound: for any $\\varepsilon>0$, if $\\delta$ is chosen small enough we have\n\\begin{equation}\\label{NYbound}\n\\mathcal{N}_{Y,\\sigma}\\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2RPL+q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}.\n\\end{equation}\nPerforming a dyadic partition of unity on the $n$ variable in $\\mathcal{N}_{\\sigma}$ we deduce \\eqref{Nsigmabound}.\n\\subsection{Bounding $\\mathcal{P}_Y$} \nWe apply the Poisson\nsummation formula~(\\ref{eq-poisson}) with modulus $r[l_1,l_2]\/m$, to\nget\n\\begin{equation}\\label{after-poisson}\n  \\mathcal{P}_Y\n  =\\frac{1}{r[l_1,l_2]\/m}\\sum_{n\\in\\mathbf{Z}}C(n,p_1,p_2, l_1,l_2,r, m)\n  \\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr),\n\\end{equation}\nwhere\n\\begin{eqnarray*}\n  C(n,p_1,p_2,l_1,l_2, r,m)=\n  \\sum_{\\beta \\mods {r[l_1,l_2]\/m}}\n  \\Kl_2(\\ov p_1q\\beta;rl_1\/m)\n  \\ov{\\Kl_2(\\ov p_2q\\beta;rl_2\/m)} \\, e_{r[l_1,l_2]\/m}(\\beta n),\n\\end{eqnarray*}\nwith~$\\ov{p}_i$ denoting the inverse of~$p_i$ modulo~$rl_i\/m$. We write\n$$\n\\mathcal{P}_Y=\\mathcal{P}_0+\\mathcal{P}_1\n$$\nwhere\n$$\n\\mathcal{P}_{0}=\\frac{1}{r[l_1,l_2]\/m} C(0,p_1,p_2,l_1,l_2,r,\nm)\\widehat{\\mathcal{W}}_Y(0)\n$$\nis the contribution of the term~$n=0$ and~$\\mathcal{P}_1$ is the\nremainder in \\eqref{after-poisson}. We show below that for any $\\varepsilon>0$, if $\\delta$ is chosen small enough we have\n\\begin{equation}\\label{P0bound}\n\\mathcal{P}_{0}\\ll \\delta_\\stacksum{l_1=l_2}{p_1=p_2}(qr)^{\\varepsilon}Z^4+\\delta_\\stacksum{l_1=l_2}{p_1\\not=p_2}\\mathcal{P}_0\\ll\nq^{\\varepsilon}Z^4\\frac{m}{rl}\n\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\t\n\\end{equation}\nand that\n\\begin{equation}\\label{P1bound}\\mathcal{P}_{1}\\ll q^{2\\varepsilon}Z^3\\Bigl(\\frac{r[l_1,l_2]1}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}.\\end{equation}\nUsing \\eqref{P0bound} in  the sum~(\\ref{eq-py}), we find that the contribution to~$\\mathcal{N}_{Y,\\sigma}$ of $\\mathcal{P}_0$ is bounded by \n\\begin{gather}\\nonumber\n  \\sum_{r\\geq 1} W\\Bigl(\\frac{r}{R}\\Bigr) \\sum_{1\\leq m\\leq U}\n  \\frac{1}{m^2}\n  \\sumsum_{\\substack{p_1,p_2,l\\\\p_i\\in\\mathrm{P},l\\in\\mathrm{L}\\\\m\\mid rl}}\n  \\Bigl(\\frac{l}{L}\\Bigr)^{3} \\widehat K(-p_1\\ov l\\ov r)\\ov{\\widehat K(-\n    p_2\\ov l\\ov r)}\\,\\mathcal{P}_0\n  \\\\ \\nonumber\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 \\sum_{r\\asymp R}\n  \\sum_{1\\leq m\\leq U}\\frac{1}{m^2} \\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid\n      rl}} \\Bigl(\\sum_{p\\in\\mathrm{P}}1+\n  \\frac{m}{rl}\\sum_{\\substack{p_1,p_2\\in\\mathrm{P}\\\\p_1\\not=p_2}}\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\n  \\Bigr)\\\\ \\nonumber\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 \\Bigl(RPL+ \\sum_{r\\asymp\n    R} \\frac{1}{r}\\sum_{1\\leq m\\leq U}\\frac{1}{m}\n  \\sum_{\\substack{l\\in\\mathrm{L}\\\\m\\mid rl}}\\frac{1}{l} \\sum_{d\\mid rl\/m}\\varphi(d)\n  \\sum_{\\substack{p_1,p_2\\in\\mathrm{P}\\\\p_1\\equiv p_2\\mods{d}}}1\n  \\Bigr)\\\\\n  \\ll q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2 (RPL+P^2)\\ll\n  q^{\\varepsilon}Z^4\\norm{\\widehat{K}}_{\\infty}^2RPL.\\label{NYP0}\n\\end{gather}\nHere $RPL=XP^2\/q\\gg P^2$, since $X$ satisfies \\eqref{XZlower}.\nUsing \\eqref{P1bound} we find that the contribution of~$\\mathcal{P}_1$\nto~$\\mathcal{N}_{\\sigma,Y}$ is bounded by\n\\begin{multline}\\label{NYP1}\n  \\ll q^{\\varepsilon}Z^3\\norm{\\widehat{K}}_{\\infty}^2\\sum_{r\\asymp R}\n  \\sum_{1\\leq m\\leq U}\n  \\frac{1}{m^2}\n  \\sum_{\\substack{p_1,p_2,l_1,l_2\\\\m\\mid (rl_1,rl_2)}}\n  \\Bigl(\\frac{r[l_1,l_2]}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}\\\\\n  \\ll q^{\\varepsilon}Z^3\\norm{\\widehat{K}}_{\\infty}^2\n  R\\frac{q^3}{X^2P^3}(P^2L^2)(RL^2)^{1\/2}\n  \\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2\\frac{Z^3R^{3\/2}q^3L^3}{X^2P}.\n\\end{multline}\n\n\n\nCombining \\eqref{NYP0} and \\eqref{NYP1} we obtain \\eqref{NYbound}. \n\n\\subsection{Proof of \\eqref{P0bound} and \\eqref{P1bound}}\n\nThe next two lemmas evaluate the archimedan and non-archimedan Fourier transforms which occur in \\eqref{after-poisson} :\n\n\\begin{lemma}\\label{lm-trunc}\n  With notation as above, in particular~\\emph{(\\ref{eq-boundy})},\n  let~$j\\geq 0$ be an integer and let~$\\varepsilon>0$.\n  \\par\n  \\emph{(1)} We have\n \n \n \n \n  \\begin{equation}\\label{eq-wyzero}\n    \\widehat{\\mathcal{W}}_Y(0)\\ll q^{\\delta}Z^4.\n  \\end{equation}\n  \\par \n  \\emph{(2)} We have\n  \\begin{equation}\\label{derivatives-of-F}\n    x^j\\mathcal{W}^{(j)}_{Y}(x)\\ll\n    \\begin{cases}\n      Z^{2+j}\\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{2-2\\theta_3-\\varepsilon}\n      & \\quad \\text{if } Y<\\frac{X^2P^3}{m^2q^3}\\\\\n      \\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{4\/3+j\/3} & \\quad \\text{if }\n      Y\\geq \\frac{X^2P^3}{m^2q^3},\n     \n    \\end{cases}\t\n  \\end{equation}\n  where the implied constants depends on~$(\\varphi,j,\\varepsilon)$.\n  \\par\n  \\emph{(3)} If $1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}$\n \n \n  then we have\n  $$\n  \\widehat{\\mathcal{W}}_{Y}\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr)\\ll\n  q^{\\delta}Z^{2}\\frac{m^2Yq^3}{X^2P^3}.\n  $$\n\\end{lemma}\n\n\\begin{proof}\n  Since~$F_Y$ has support in $[1\/2,3]$, part~(1) follows from the\n  bound~$\\mathcal{V}_{\\sigma}(x)\\ll x^{2\/3}$ for~$x\\geq 1$ and the\n  fact that\n  $$\n  \\frac{m^2xY}{r^3l_i^3\/X}\\asymp \\frac{m^2Y}{X^2P^3\/q^3}\\ll\n  q^{\\delta}Z^3.\n  $$\n  \\par\n  Part~(2) is obtained using the estimates\n  \\begin{gather*}\n    x^j\\mathcal{V}_{\\pm}^{(j)}(x)\\ll Z^{j+1}x^{1-\\theta_3-\\varepsilon}\\quad \\text{\n      if }\n    0<x<1,\\\\\n    x^j\\mathcal{V}_{\\pm}^{(j)}(x)\\ll x^{2\/3+j\/3}\\quad \\text{ if } x\\geq 1\n  \\end{gather*}\n  (see Lemma \\ref{bounds-for-V}), noting again that\n  $\\frac{(rl)^3}{X}\\sim \\frac{X^2P^3}{q^3}$.\n  \\par\n \n \n \n \n \n \n \n \n \n \n \n \n  From \\eqref{derivatives-of-F}, for any $n$ such that\n  $1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}$, we get the estimates\n  $$\n  \\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr)\\ll\n   \\begin{cases}\n     Z^{2}\\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{2-2\\theta_3-\\varepsilon},&\n     \\quad\n     \\mathrm{if}\\, Y<\\frac{X^2P^3}{m^2q^3}\\\\\n     \\Bigl(\\frac{m^2Yq^3}{X^2P^3}\\Bigr)^{4\/3},& \\quad \\mathrm{if}\\,\n     Y\\geq \\frac{X^2P^3}{m^2q^3}\n   \\end{cases}\n   $$\n   Since $m^2Yq^3\/(X^2P^3)\\ll q^{\\delta}Z^3$, the second bound is also\n   $$\n   \\ll q^{\\delta\/3}Z\\frac{m^2Yq^3}{X^2P^3}.\n   $$\n   Together with the first bound, this implies part (3) of the lemma.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lm-delta}\n  Let $n\\in\\mathbf{Z}$, $p_1$, $p_2$ primes, $l_1$, $l_2$ primes, $r\\geq 1$\n  and $m\\geq 1$ be integers with $m\\mid rl_i$.\n  \\par\n  \\emph{(1)} We have\n  $$\n  C(0,p_1,p_2,l_1,l_2,r,m)=0\n  $$\n  unless $l_1=l_2$.\n  \\par\n  \\emph{(2)} For $l$ prime with $m\\mid rl$, we have\n  $$\n  |C(0,p_1,p_2,l,l,r,m)|\\leq (rl\/m,p_2-p_1)\n. $$\n  \\par\n  \\emph{(3)} Let\n  $$\n  \\Delta=q\\frac{l_2^2p_2-l_1^2p_1}{(l_1,l_2)^2}.\n  $$\n \n \n  We have\n  $$\n  |C(n,p_1,p_2,l_1,l_2,r,m)|\\leq\n  2^{O(\\omega(r))}\\Bigl(\\frac{r[l_1,l_2]}{m}\\Bigr)^{1\/2}\n  \\frac{(\\Delta,n,m\/rl_1,m\/\/rl_2)}{(n,m\/rl_1,m\/rl_2)^{1\/2}}.\n  $$\n  \\par\n  \\emph{(4)} Suppose that $\\Delta=0$. If $(p_1,p_2)$ are\n  $\\equiv 1\\mods{4}$ and $(l_1,l_2)$ are $\\equiv 3\\mods{4}$, then\n  $p_1=p_2$ and $l_1=l_2$. For $p$ prime and $l$ prime with\n  $m\\mid rl$, we have\n$$\n|C(n,p,p,l,l,r,m)|\\leq 2^{O(\\omega(r))}\n\\Bigl(\\frac{rl}{m}\\Bigr)^{1\/2}\\Bigl(n,\\frac{rl}{m}\\Bigr)^{1\/2},\n$$\nIn particular, $C(0,p,p,l,l,r,m)\\ll r^{\\varepsilon}\\frac{rl}{m}$ for\nany~$\\varepsilon>0$.\n\\end{lemma}\n\n\\begin{proof}\n  Part (1) follows by direct computation (the sum vanishes unless\n  $[l_1,l_2]=l_1$ and $[l_1,l_2]=l_2$). If $n=0$ and $l_1=l_2$, then\n$$\n|C(0,p_1,p_2,l,l,r,m)|=\\bigg|\\sum_{\\substack{x\\bmod rl\/m\\\\(x,rl\/m)=1}}\ne\\Bigl(\\frac{( p_2-p_1)x}{rl\/m}\\Bigr)\\bigg| = \\bigg| \\sum_{d|(rl\/m, p_2-p_1)} d \\mu\\Bigl(\\frac{rl}{md} \\Bigr) \\bigg| \\leq (rl\/m,p_2-p_1)\n$$\nby a classical bound for Ramanujan's sum, which proves (2). Finally, part (3) is a special case of~\\cite[Lemma\nA.2 (A.3)]{HN} (applied with \n$(\\xi,s_1,s_2)=(n,rl_1\/m,rl_2\/m)$, and\n$(a_1,b_1,a_2,b_2)=(q,p_1,q,p_2)$ in the definition of~$\\Delta$).\nIf $\\Delta=0$, then necessarily $p_1=p_2$ and $l_1=l_2$, and we\nobtain~(4) immediately.\n\\end{proof}\n\n\\subsubsection{Estimation of $\\mathcal{P}_0$}\n\nNote that~$\\mathcal{P}_0=0$ unless $l_1=l_2$. If that is the case, we\ndenote~$l=l_1=l_2$. We then have two bounds for~$\\mathcal{P}_{0}$. If\nwe have also~$p_1=p_2$, then the quantity~$\\Delta$ of\nLemma~\\ref{lm-delta} (3) is zero.\nSince~$\\widehat{\\mathcal{W}}_Y(0)\\ll q^{\\varepsilon}Z^4$ for any~$\\varepsilon>0$\n(provided~$\\delta>0$ is chosen small enough) by Lemma~\\ref{lm-trunc}\n(1), we obtain\n$$\n\\mathcal{P}_0\\ll q^{\\varepsilon}Z^4\\frac{m}{rl}|C(0,p_1,p_1,l,l,r,m)|\\ll\n(qr)^{\\varepsilon}Z^4\n$$\nby the last part of Lemma~\\ref{lm-delta} (4).\n\\par\nOn the other hand, if~$p_1\\not=p_2$, we have~$\\Delta\\not=0$ hence\n$$\n\\mathcal{P}_0\\ll\nq^{\\varepsilon}Z^4\\frac{m}{rl}\n\\Bigl(\\frac{rl}{m},p_2-p_1\\Bigr)\n$$\nby Lemma~\\ref{lm-delta} (1) (which shows that the sum\n$C(0,p_1,p_2,l_1,l_2,r,m)$ is zero unless~$l_1=l_2$) and (2).\n\\par\n\n\\subsubsection{Estimation of $\\mathcal{P}_1$}\nUsing Lemma~\\ref{lm-trunc} (2) for a\nsuitable value of~$j$, we obtain first\n$$\n\\mathcal{P}_{1}= \\frac{1}{r[l_1,l_2]\/m}\\sum_{1\\leq |n|\\leq\n  q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}} C(n,p_1,p_2,l_1,l_2,r,m)\n\\widehat{\\mathcal{W}}_Y\\Bigl(\\frac{nY}{r[l_1,l_2]\/m}\\Bigr) +O(q^{-1})\n$$\nfor any~$\\varepsilon>0$ if~$\\delta$ is chosen small enough.  Then, by\nLemma~\\ref{lm-delta} and Lemma~\\ref{lm-trunc} (3), we deduce that\n\\begin{align*}\n  \\mathcal{P}_1\n  &\\ll q^{\\varepsilon+2\\delta}\\frac{1}{(r[l_1,l_2]\/m)^{1\/2}}\n    \\sum_{1\\leq |n|\\leq q^{\\delta}Z\\frac{r[l_1,l_2]}{mY}}\n    \\frac{(\\Delta,n,rl_1\/m,rl_2\/m)}{(n,rl_1\/m,rl_2\/m)^{1\/2}}\n    \\frac{Z^{2}m^2Yq^3}{X^2P^3}\n  \\\\\n  &\\ll q^{2\\varepsilon}Z^3\\Bigl(\\frac{r[l_1,l_2]1}{m}\\Bigr)^{1\/2}\\frac{m^2q^3}{X^2P^3}\n\\end{align*}\nif $\\delta<\\varepsilon\/2$.\n\n\n\\section{Estimate of $\\mathcal{O}$}\\label{sec:O}\n\nIn this section, we bound the sum~$\\mathcal{O}$ defined\nin~(\\ref{eq-m2}). Our goal is:\n\n\\begin{proposition}\\label{corollary-for-O}\n  Let~$\\eta>0$ be a parameter such that \\eqref{XZlower} holds.\n  Let~$\\varepsilon>0$. If~$\\delta$ is a sufficiently small positive real\n  number and if $P,L,X$ satisfy\n  \\begin{equation}\\label{eqboundsPHLX}\n    XP\\leq q^2L,\\quad q^{1+\\delta}L^2<X\/8,\\quad q^{\\delta}PHL<q\/8,\n  \\end{equation}\n  then we have\n  \\begin{equation}\\label{final-bound-for-O}\n    \\mathcal{O}\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{qX^{1\/2}}{P},\n  \\end{equation}\n  where the implied constant depends on~$\\varphi$ and~$\\varepsilon$.\n\\end{proposition}\n\n\n\nWe start by decomposing $\\mathcal{O}$ into\n$$\n\\mathcal{O}=\\mathcal{O}_1+\\mathcal{O}_2\n$$\naccording to whether the prime $l$ divides $h$ or not, in other words\n\\begin{align*}\n  \\mathcal{O}_1\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_{h\\not=0}\\widehat{W}\\Bigl(\\frac{hl}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp)V\\Bigl(\\frac{n}{X}\\Bigr)\\\\\n  \\mathcal{O}_2\n  &=\\frac{1}{|\\mathrm{P}||\\mathrm{L}|}\n    \\sum_{p\\in\\mathrm{P}}\\sum_{l\\in\\mathrm{L}}\n    \\sum_\\stacksum{h\\neq 0}{(h,l)=1}\n    \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n    \\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr).\n\\end{align*}\n\nBoth of these sums will be handled in a similar way in the next two\nsubsections, beginning with the most difficult one.\n\n\\subsection{Bounding $\\mathcal{O}_2$}\n\nIn the sum $\\mathcal{O}_2$, we first use the bound\n$$\n\\widehat{W}(x)\\ll (1+|x|)^{-A}\n$$\nfor any~$A\\geq 1$ and~$x\\in\\mathbf{R}$, and\n$$\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp\\ov l)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{1+\\varepsilon}q^{1\/2}\\norm{\\widehat{K}}_{\\infty}\\ll\nq^{5\/2+2\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n$$\nfor any~$\\varepsilon>0$ (by~(\\ref{eq:RS}) and discrete Fourier inversion) to\ntruncate the sum over~$h$ to~$|h|\\leq q^{\\delta}H$, for\nsome~$\\delta>0$ that may be arbitrarily small.\n\n\nLet~$T\\geq 0$ be a smooth function with compact support such that\n$T(x)=\\norm{V}_{\\infty}$ for $x\\in [1\/2,3]$ and such that~$T$\nsatisfies \\eqref{eq:Vprop} with a fixed value of~$Z$. We then\nhave~$|V|\\leq T$.\n\nIn the sum~$\\mathcal{O}_2$, we split the $h$-sum into $O(\\log q)$ dyadic\nsums. We then apply the Cauchy--Schwarz inequality to smooth the\n$n$-variable, and we obtain\n$$\n\\mathcal{O}_2\\ll \\frac{\\log^3 q}{PL} \\Bigl(\\sum_{n\\sim\n  X}|\\lambda(1,n)|^2\\Bigr)^{1\/2} \\mathop{\\mathrm{Max}}\\limits_{1\\leq H'\\leq\n  q^{\\delta}H}\\mathcal{R}_{H'}^{1\/2}\\ll \\frac{X^{1\/2}\\log^3 q}{PL}\n\\mathop{\\mathrm{Max}}\\limits_{1\\leq H'\\leq q^{\\delta}H}\\mathcal{R}_{H'}^{1\/2},\n$$\nby~(\\ref{eq:RS}) again, where\n$$\n\\mathcal{R}_{H'}= \\sumsum_{p_1,h_1,l_1,p_2,h_2,l_2} \\sum_{n\\geq\n  1}K(n,h_1p_1\\ov l_1)\\ov{K(n,h_2p_2\\ov l_2)}\n\\widehat{W}\\Bigl(\\frac{h_1}{H}\\Bigr)\n\\ov{\\widehat{W}\\Bigl(\\frac{h_2}{H}\\Bigr)}\nT\\Big(\\frac{n}X\\Bigr),\n$$\nwith the variables in the sums constrained by the conditions\n$$\np_i\\in\\mathrm{P},\\quad l_i\\in\\mathrm{L},\\quad\nH'<h_i\\leq 2H',\\quad (l_i,h_i)=1.\n$$\n\nFor $x\\in{\\mathbf{F}_q}$, we define\n\\begin{equation}\\label{nudef}\n  \\nu(x)=\n  \\sum_{\\substack{\n      (p,h,l)\\in \\mathrm{P}\\times[H',2H'[  \\times\\mathrm{L},\\\\\n      (h,l)=1\\\\\n      ph\\ov l\\equiv x\\mods q}}\n  \\widehat{W}\\Bigl(\\frac{h}{H}\\Bigr)\n\\end{equation}\nso that we have\n\\begin{equation}\\label{eq:nsum}\n  \\mathcal{R}_{H'}=\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n  \\nu(x_1)\\nu(x_2)\\sum_{n\\geq 1}K(n,x_1)\\ov{K(n,x_2)}\n  T\\Bigl(\\frac{n}X\\Bigr).\n\\end{equation}\n\nWe apply the Poisson summation formula~(\\ref{eq-poisson}) for the sum\nover~$n$. This results in the formula\n$$\n\\sum_{n\\geq 1}K(n,x_1)\\ov{K(n,x_2)}T\\Bigl(\\frac{n}X\\Bigr)\n=\\frac{X}{\\sqrt{q}}\n\\sum_{h\\in\\mathbf{Z}} \\Bigl(\\frac{1}{\\sqrt{q}}\n\\sum_{n\\mods{q}}K(n,x_1)\\ov{K(n,x_2)}e\\Bigl(\\frac{nh}{q}\\Bigr)\n\\Bigr)\n\\widehat{T}\\Bigl(\\frac{hX}{q}\\Bigr).\n$$\n\nObserve that for any~$h\\in\\mathbf{Z}$, we have\n$$\n\\frac{1}{\\sqrt{q}}\n\\sum_{n\\mods{q}}K(n,x_1)\\ov{K(n,x_2)}e\\Bigl(\\frac{nh}{q}\\Bigr)\n=\\frac{1}{\\sqrt{q}}\\sum_{u\\mods{q}} \\widehat K(u,x_1)\\ov{\\widehat\n  K(u+h,x_2)}\n$$\nwhere $ \\widehat K(u,x)$ is defined as in \\eqref{whatKzdef}. In\nparticular, this quantity is bounded by\n$q^{1\/2}\\norm{\\widehat{K}}_{\\infty}^2$.\n\nNow, for all $h\\not=0$ and all~$A\\geq 1$, we have\n$$\n\\widehat{T}\\Bigl(\\frac{hX}{q}\\Bigr)\\ll_A\n\\Bigl(\\frac{qZ}{hX}\\Bigr)^A\\leq \\Bigl(\\frac{qZ}{X}\\Bigr)^A\\leq\nq^{-A\\eta},\n$$\nby~(\\ref{XZlower}), where the implied constant depends on~$A$. Hence,\ntaking~$A$ large enough in terms of~$\\eta$, the contribution of\nall~$h\\not=0$ to the sum over~$n$ is\n$\\ll \\norm{\\widehat{K}}_{\\infty}^2q^{-5}$, and the total contribution\nto~$\\mathcal{R}_{H'}$ is (using very weak bounds on~$\\nu(x)$)\n$$\n\\ll \\norm{\\widehat{K}}_{\\infty}^2 q^{-3}(PHL)^2\\ll\n\\norm{\\widehat{K}}_{\\infty}^2q^{-1}\n$$\nby~(\\ref{eqboundsPHLX}).\n\n\nThe remaining contribution to~$\\mathcal{R}_{H'}$ from the\nfrequency~$h=0$ is\n$$\n\\frac{X}{\\sqrt{q}}\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n\\nu(x_1)\\nu(x_2)\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}} \\widehat\nK(u,x_1)\\ov{\\widehat K(u,x_2)}\\widehat{T}(0).\n$$\n\n\\begin{lemma}\n  For any~$(x_1,x_2)\\in{\\mathbf{F}_q}\\times{\\mathbf{F}_q}$, we have\n  $$\n  \\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}}\n  \\widehat K(u,x_1)\\ov{\\widehat K(u,x_2)}=\n  L(x_1-x_2)\n  $$\n  where\n  $$\n  L(x)=\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}^\\times_q}}|\\widehat{K}(u)|^2e_q(-\\bar{u}x)\n  +\\frac{1}{\\sqrt{q}}|\\widehat{K}(0)|^2.\n  $$\n  Moreover, we have\n  $$\n  \\widehat{L}(h)=|\\widehat{K}(0)|^2\\delta_{h\\equiv 0\\mods{q}}+\n  |\\widehat{K}(\\bar{h})|^2\\delta_{h\\not\\equiv 0\\mods{q}},\n  $$\n  and in particular~$|\\widehat{L}(h)|\\leq \\norm{\\widehat{K}}_{\\infty}^2$ for\n  all~$h\\in{\\mathbf{F}_q}$.\n\\end{lemma}\n\n\\begin{proof}\n  The first formula is an immediate consequence of the\n  definition~(\\ref{whatKzdef}), and the second results from a\n  straightforward computation.\n\\end{proof}\n\n\\begin{lemma}\n  We have\n  $$\n  \\norm{\\nu}_2^2=\\sum_{x\\in{\\mathbf{F}_q}}\\nu(x)^2\\ll q^{\\varepsilon+\\delta}PHL\n  $$\n  for any~$\\varepsilon>0$.\n\\end{lemma}\n\n\\begin{proof}\nFrom the last condition in~(\\ref{eqboundsPHLX}), we have the\nimplications\n\\begin{equation}\\label{PHLbound}\n  h_2p_2\\ov l_2=h_1p_1\\ov l_1\\mods q\\Longleftrightarrow\n  l_1h_2p_2\\equiv l_2h_1p_1\\mods q\\Longleftrightarrow l_1h_2p_2=l_2h_1p_1.\n\\end{equation}\nTherefore, if $(p_1,h_1,l_2)$ are given, the number of possibilities\nfor $(p_2,h_2,l_1)$ is $\\ll q^{\\varepsilon}$ for any~$\\varepsilon>0$.\nThe bound\n$$\n\\sum_{x\\in{\\mathbf{F}^\\times_q}}\\nu(x)^2\\ll q^{\\varepsilon}PH'L\\leq q^{\\varepsilon+\\delta}PHL\n$$\nfollows immediately.\n\\end{proof}\n\nWe can now combine these two lemmas with Proposition~\\ref{willprop} to\ndeduce that\n\\begin{align*}\n  \\frac{X}{\\sqrt{q}}\\Bigl|\\sumsum_{x_1,x_2\\in{\\mathbf{F}_q}}\n  \\nu(x_1)\\nu(x_2)\\frac{1}{\\sqrt{q}}\\sum_{u\\in{\\mathbf{F}_q}}\n  \\widehat K(u,x_1)\\ov{\\widehat K(u,x_2)}\\widehat{T}(0)\\Bigr|\n  &\\leq X\\norm{\\widehat{L}}_{\\infty}\\norm{\\nu}_2^2|\\widehat{T}(0)|\\\\\n  &\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}^2XPHL\n\\end{align*}\nfor any~$\\varepsilon>0$, by taking~$\\delta$ small enough in terms of~$\\varepsilon$.\nHence we obtain\n\\begin{equation}\\label{diagonal1total}\n  \\mathcal{O}_2\\ll q^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}X\\Bigl(\\frac{H}{PL}\\Bigr)^{1\/2}\\ll\n  q^{1+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{X^{1\/2}}{P}.\n\\end{equation}\n\n\\subsection{Bounding $\\mathcal{O}_1$ and end of the proof of Proposition~\\ref{corollary-for-O}}\n\nThe treatment of $\\mathcal{O}_1$ is similar to that of~$\\mathcal{O}_2$, but simpler,\nso we will be brief. We have\n$$\n\\mathcal{O}_1=\\frac{1}{|\\mathrm{L}|}\\sum_{l\\in\\mathrm{L}}\\frac{1}{|\\mathrm{P}|}\n\\sum_{p\\in\\mathrm{P}}\\sum_{h\\not=0} \\widehat{W}\\Bigl(\\frac{h}{H\/l}\\Bigr)\n\\sum_{n\\geq 1}\\lambda(1,n)K(n,hp)V\\Bigl(\\frac{n}{X}\\Bigr).\n$$\nWe bound the sum over~$p$ for each individual~$l\\in\\mathrm{L}$, with\n$h\\ll H\/l\\asymp H\/L$, by repeating the arguments of the previous\nsection with $H$ replaced by $H\/l$ and $L$ replaced by $1$. We\nobtain\n\\begin{equation}\\label{O1bound}\n  \\mathcal{O}_1\\ll \\norm{\\widehat{K}}_{\\infty}\n  q^{\\varepsilon}X\\Bigl(\\frac{H}{PL}\\Bigr)^{1\/2}\n  \\ll q^{1+\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\\frac{X^{1\/2}}{P}\n\\end{equation}\nfor any~$\\varepsilon>0$, as in the previous case.\n\nFinally, since~$\\mathcal{O}=\\mathcal{O}_1+\\mathcal{O}_2$, this bound combined\nwith~(\\ref{diagonal1total}) implies Proposition~\\ref{corollary-for-O}.\n\n\n\\section{End of the proof}\n\nWe can now finish the proof of our main theorem.  We recall that $X,Z$\nare such that\n  \\begin{equation}\n    \\label{eqcondZXq}\nZ^{2\/3}q^{4\/3}\\leq X \\leq Z^{-2}q^{2}.\n\\end{equation}\nIn particular $Z\\leq q^{1\/4}$ and\n$$X\\geq Z^{2\/3}q^{4\/3}\\geq Zq^{1+1\/4}$$\ntherefore \\eqref{XZlower} holds for~$\\eta=1\/4$.\n\nAssuming that the conditions \\eqref{eqboundsPHLX} hold,\ncombining~\\eqref{an-identity}, Proposition~\\ref{proposition-for-F} and\nProposition~\\ref{corollary-for-O}, we deduce the estimate\n$$\nS_V(K,X) \\ll\nq^{\\varepsilon}\\norm{\\widehat{K}}_{\\infty}\n\\Biggl(\\frac{Z^{2}X^{3\/2}P}{qL^{1\/2}}+Z^{3\/2}X^{3\/4}(qPL)^{1\/4}\n+\\frac{qX^{1\/2}}{P}\\biggr)\n$$\nfor any~$\\varepsilon>0$.  When $L=Z^{2\/3}XP\/q^{5\/3}$, the first two terms are\nequal to $Z^{5\/3}XP^{1\/2}\/q^{1\/6}$. For $P=q^{7\/9}\/(X^{1\/3}Z^{10\/9})$,\nthey are also equal to the third term which is\n$Z^{10\/9}q^{2\/9}X^{5\/6}$.  Moreover, the conditions \\eqref{eqcondZXq}\nand~$Z\\leq q^{1\/4}$ imply then by simple computations that\n$$\n1\\leq P,\\ 1\\leq L,\\  L\\leq P^4,\\ XP\\leq q^2L\n$$\n(for instance, $X^3Z^{10}\\leq Z^{10}(q^2\/Z^2)^3=Z^4q^6\\leq q^7$\ngives~$P\\geq 1$), and then we get\n$$\nq^{1+\\delta}L^2<\\frac{X}{8}\n$$\nfor~$\\delta=1\/18$ provided~$q$ is large enough (since\n$qL^2=q^{-7\/9}Z^{-8\/9}X^{4\/3}\\leq X(X^{1\/3}q^{-7\/9})\\leq Xq^{-1\/9}$\nusing~$X\\leq q^2$). By~(\\ref{eq-H}), this also implies\nthat~$q^{\\delta}PHL<q\/8$. Hence this choice of the parameters\nsatisfies~(\\ref{eqboundsPHLX}). We finally conclude that\n$$\nS_V(K,X) \\ll \\norm{\\widehat{K}}_{\\infty}\nZ^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}\n$$\nfor any~$\\varepsilon>0$.\n\n\n\\section{Applications}\n\nIn this section, we prove \nCorollaries~\\ref{cor-average}, \\ref{cor-gl2}.\n\n\\subsection{ Proof of Corollary~\\ref{cor-average}} Applying the approximate functional\nequation for~$L(\\varphi\\otimes\\chi,s)$ in balanced form, we\nimmediately express the first moment\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi, 1\/2)\n$$\nin terms of the sums\n$$\n\\frac{1}{\\sqrt{q}}\n\\sum_{n\\geq 1} \\frac{\\lambda(1,n)}{\\sqrt{n}}\nK(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr)\n$$\nand\n$$\n\\frac{1}{\\sqrt{q}}\\sum_{n\\geq 1}\n\\frac{\\overline{\\lambda(1,n)}}{\\sqrt{n}}\nL(n)V\\Bigl(\\frac{n}{q^{3\/2}}\\Bigr),\n$$\nfor suitable test functions satisfying~(\\ref{eq:Vprop}) for~$Z=1$,\nwhere\n$$\nK(n)=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}M(\\chi){\\chi(n)},\\quad\nL(n)=\\frac{q^{1\/2}}{q-1}\\sum_{\\chi\\mods{q}}\\tau(\\chi)^3M(\\chi)\\ov{\\chi(n)},\n$$\nin terms of the normalized Gauss sum~$\\tau(\\chi)$. An elementary\ncomputation shows that this function~$L$ coincides with the function\nin the statement of Corollary~\\ref{cor-average}. Since moreover\nthe~$\\overline{\\lambda(1,n)}$ are the Hecke-eigenvalues of the dual\ncusp form~$\\widetilde{\\varphi}$, the corollary follows from\nTheorem~\\ref{thm:main} applied to~$K$ and~$L$.\n\n\\begin{remark}\\label{lastremark}\n  (1) If\n$$\nM(\\chi)=\\frac{1}{\\sqrt{q}}\\sum_{x\\in{\\mathbf{F}^\\times_q}}K(x)\\ov{\\chi(x)}\n$$\nis the discrete Mellin transform of the trace function~$K$ of a\nFourier sheaf~$\\mathcal{F}$ that is a middle-extension sheaf on~$\\mathbf{G}_m$ of\nweight~$0$, and if no sheaf of the\nform~$[x\\mapsto x^{-1}]^*\\dual(\\mathcal{K}\\ell_3)$ is among the geometrically\nirreducible components of~$\\mathcal{F}$, then \nboth~$\\norm{\\widehat{K}}_{\\infty}$ and~$\\norm{\\widehat{L}}_{\\infty}$ are\nbounded in terms of the conductor of~$\\mathcal{F}$ only and we obtain\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}M(\\chi)L(\\varphi\\otimes\\chi,\n1\/2)\\ll q^{2\/9+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, where the implied constant depends only\non~$\\varepsilon$,~$\\varphi$ and the conductor of~$\\mathcal{F}$.\n\\par\n(2) Applying the approximate functional equation in a balanced form\nmay not always be the best move. For instance, consider the important\nspecial case where $M(\\chi)=1$. We are then evaluating the first\nmoment\n\\begin{equation}\\label{eqfirstmoment}\n\\frac{1}{q-1}\\sum_{\\chi\\mods q}L(\\varphi\\otimes\\chi,1\/2)\n\\end{equation}\nof the central values of the twisted $L$-functions. Then we are\nworking with the functions\n$$\nK(n)=q^{1\/2}\\delta_{n\\equiv 1\\mods q},\\quad L(n)=\\Kl_3( n;q),\n$$\nwhose Fourier transforms are bounded by absolute constants. Hence the\nabove leads to\n$$\n\\frac{1}{q-1} \\sum_{\\chi\\mods{q}}L(\\varphi\\otimes\\chi, 1\/2)\n\\ll q^{2\/9 + \\varepsilon}\n$$\nfor any~$\\varepsilon>0$, where the implied constant depends on~$\\varphi$\nand~$\\varepsilon$.\n\\par\nOn the other hand, the approximate functional equation in unbalanced\nform yields sums of the shape\n$$\n\\sum_{n\\equiv 1\\mods q} \\frac{\\lambda(1,n)}{\\sqrt{n}}\nV\\Bigl(\\frac{n}{Yq^{3\/2}}\\Bigr)\n\\quad\\text{ and }\\quad\n\\frac{1}{\\sqrt{q}}\\sum_{n\\geq 1}\n\\frac{\\overline{\\lambda(1,n)}}{\\sqrt{n}}\n\\Kl_3(n;q)V\\Bigl(\\frac{nY}{q^{3\/2}}\\Bigr),\n$$\nfor some parameter $Y>0$ at our disposal. Assuming the\nRamanujan--Petersson conjecture for~$\\varphi$\nand~$\\widetilde{\\varphi}$, and using Deligne's bound\n$|\\Kl_3(n;q)|\\leq 3$ for $(n,q)=1$, we obtain the much stronger bound\n$$\n\\frac{1}{q-1}\\sum_{\\chi\\mods\n  q}L(\\varphi\\otimes\\chi,1\/2)=1+(qY)^{\\varepsilon}\n\\bigl({Y^{1\/2}}\/{q^{1\/4}}+q^{1\/4}\/Y^{1\/2}\\bigr)\\ll\nq^{\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, on choosing $Y=q^{1\/2}$.\n\\par\nNote that, again under the Ramanujan--Petersson conjecture\nfor~$\\varphi$ and its dual, we would obtain an \\emph{asymptotic\n  formula} for the first moment \\eqref{eqfirstmoment} \\emph{provided}\nwe could obtain an estimate for $S_V(\\Kl_3,X)$ with a power-saving in\nterms of~$q$, when $X$ is a bit smaller than $q$. Results of this type\nare however currently only known if $\\varphi$ is an Eisenstein series\n(starting from the work~\\cite{FI} of Friedlander and Iwaniec for the\nternary divisor function; see also the papers of Fouvry, Kowalski and\nMichel~\\cite{FKMd3}, of Kowalski, Michel and Sawin~\\cite{KMS} and of\nZacharias~\\cite{Zac}).\n\nThis illustrates the importance of the problem of obtaining\nnon-trivial bounds for short sums in Theorem \\ref{thm:main}. However,\nwe expect that much more refined properties of trace functions and\ntheir associated sheaves will be necessary for such a purpose (as\nindicated by Remark~\\ref{remcor17}).\n\\end{remark}\n\n\\subsection{Proof of Corollary~\\ref{cor-gl2}}\n\nThe symmetric square~$\\varphi$ of~$\\psi$ has Hecke eigenvalues\n\\begin{equation}\\label{lambda(n2)}\n\\lambda(1,n)=\\sum_{d^2\\mid n}\\lambda\\Bigl(\\frac{n^2}{d^2}\\Bigr),\n\\end{equation}\nand hence, by M\\\"obius inversion, we have\n$$\n\\lambda(n^2)=\\sum_{d^2\\mid n}\\mu(d)\\lambda\\Bigl(1,\\frac{n}{d^2}\\Bigr).\n$$\nWe deduce that\n$$\n\\sum_{n\\geq 1}\\lambda(n^2)K(n)V\\Bigl(\\frac{n}{X}\\Bigr)= \\sum_{d\\geq\n  1}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr).\n$$\nFor\n$$\n1\\leq d\\leq \\frac{X^{1\/2}}{Z^{1\/3}q^{2\/3}},\n$$\nwe can apply Theorem~\\ref{thm:main} to the sum over~$n$ and the\n$q$-periodic function $L(n)=K(nd^2)$, with~$X$ replaced\nby~$X\/d^2$. Since~$q\\nmid d$, we have\n$\\widehat{L}(x)=\\widehat{K}(\\bar{d}^2x)$ for any~$x\\in \\mathbf{Z}$, so\nthat~$\\norm{\\widehat{L}}_{\\infty}=\\norm{\\widehat{K}}_{\\infty}$, and we get\n\\begin{align*}\n  \\sum_{d\\leq X^{1\/2}\/(Z^{1\/3}q^{2\/3})}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr)\n  &\\ll\n    \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon} \\sum_{d\\geq\n    1}\\frac{X^{5\/6}}{d^{5\/3}}\n  \\\\\n  &\\ll\n    \\norm{\\widehat{K}}_{\\infty}Z^{10\/9}q^{2\/9+\\varepsilon}X^{5\/6}\n\\end{align*}\nfor any~$\\varepsilon>0$.\n\\par\nSince~$V$ has compact support in $[1\/2,3]$, the sum over~$n$ is empty\nif $d\\geq \\sqrt{3X}$. Since\n$$\n\\sum_{n\\geq 1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr) \\ll\n\\norm{K}_{\\infty}\\Bigl(\\frac{X}{d^2}\\Bigr)^{1+\\varepsilon}\n$$\nfor any~$\\varepsilon>0$, by the Rankin--Selberg bound~(\\ref{eq:RS}), we can\nestimate the remaining part of the sum as follows:\n\\begin{align*}\n  \\sum_{X^{1\/2}\/(Z^{1\/3}q^{2\/3})<d\\leq \\sqrt{3X}}\\mu(d) \\sum_{n\\geq\n  1}K(nd^2)\\lambda(1,n)V\\Bigl(\\frac{nd^2}{X}\\Bigr)\n  &\\ll\n    \\norm{K}_{\\infty}X^{1+\\varepsilon}\n    \\sum_{X^{1\/2}\/(Z^{1\/3}q^{2\/3})<d\\leq\n    \\sqrt{3X}}\\frac{1}{d^{2+2\\varepsilon}}\n  \\\\\n  &\\ll\n    \\norm{K}_{\\infty}X^{1\/2+\\varepsilon}Z^{1\/3}q^{2\/3}\n\\end{align*}\nfor any~$\\varepsilon>0$.\n\n\n\\begin{remark}\n  The additional dependency on~$\\norm{K}_{\\infty}$ seems to be\n  unavoidable in Corollary~\\ref{cor-gl2}.\n\\end{remark}\n\n\\section{Proof of Corollary~\\ref{RScor}}\n\nThe proof of Corollary~\\ref{RScor} requires additional ingredients\nbesides Theorem~\\ref{thm:main}.  We will be somewhat brief in handling\nthese additional arguments (especially standard analytic arguments),\nsince similar computations have been performed in a number of other\npapers (e.g.~\\cite{FKM2}).\n\nFirst, in terms of the\nHecke-eigenvalues $\\lambda(m,n)$ of the symmetric square~$\\psi$\nof~$\\varphi$, we have the identity\n$$\n\\lambda(n)^2=\\sum_{d^2bc=n}\\mu(d)\\lambda(1,c)\n$$\n(see~(\\ref{lambda(n2)}) and~(\\ref{convol})).  Thus we have\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)=\n\\sum_{d,m,n\\geq\n  1}\\mu(d)\\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr).\n$$\n\\par\nWe bound the contribution of the integers~$n$ divisible by~$q$ using\nthe Kim--Sarnak bound~\\cite{KimSar} for~$\\lambda(1,n)$, which shows\nthat it is\n$$\n\\ll\n\\|K\\|_\\infty X^{1+\\varepsilon}q^{-1+7\/32},\n$$\nfor any~$\\varepsilon>0$, and hence is negligigle. We may therefore restrict\nthe sum on the right-hand side to integers such that $(dmn,q)=1$.\n\nFor a fixed value of $d\\leq D$, coprime to~$q$, we consider the sum\n\\begin{equation}\\label{eq-Td}\n  T_{d,x}=\\sum_{m,n\\geq\n    1}\\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr).\n\\end{equation}\nWe need to estimate the sum of~$T_{d,x}$ over~$d\\geq 1$.\n\nLet $D\\geq 1$ be some small parameter to be fixed later. The\ncontribution of the integers $d> D$ is bounded trivially and is\n$$\n\\sum_{d>D}T_{d,X}\\ll \\frac{\\|K\\|_\\infty X^{1+\\varepsilon}}{D}\n$$\nfor any~$\\varepsilon>0$.\n\nWe now fix $d\\leq D$, coprime to~$q$. We handle the sum~$T_{d,X}$ by a\nsmooth dyadic partition of unity on the $m$-variable. This reduces the\nproblem to estimates of $O(\\log X)$ sums of the form\n\\begin{equation}\\label{DMsum}\n  S_{d,M}=  \\sum_{\\substack{m,n\\geq 1\\\\(mn,q)=1}}\n  \\lambda(1,n)K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr)\n  W\\Bigl(\\frac{m}M\\Bigr)\n\\end{equation}\nwhere $W$ is smooth and compactly supported in $[1\/2,5\/2]$. We\nset\n$$\nN=\\frac{X}{d^2M},\n$$\nso that $n\\sim N$ in the sum.\n\nThe estimate for~(\\ref{DMsum}) splits in three cases, depending on the\nsize of~$M$.\n\n\\subsection{When $M$ is small}\n\nWe assume that $\\frac{X}{d^2m}\\geq \\frac{X}{D^2M}\\geq\nZ^{2\/3}q^{4\/3}$, or in another words, that\n\\begin{equation}\\label{condition-on-M}\nD^2M\\leq \\frac{X}{Z^{2\/3}q^{4\/3}}.\n\\end{equation}\nWe can then apply Theorem~\\ref{thm:main}, and we derive\n\\begin{align}\\nonumber\n  S_{d,M}&\\ll \\|K\\|_\\infty q^{7\/32-1}\\frac{X^{1+\\varepsilon}}{d^2}\n           +\\|\\widehat K\\|_\\infty Z^{10\/9}q^{2\/9+\\varepsilon}\n           \\sum_{m\\sim M}\\Bigl(\\frac{X}{d^2m}\\Bigr)^{5\/6}\\\\\n         &\\ll \\|K\\|_\\infty X^{\\varepsilon}q^{7\/32-1}\\frac{X}{d^2}\n           +\\|\\widehat K\\|_\\infty X^{\\varepsilon}Z^{10\/9}\n           \\Bigl(\\frac{X}{d^2}\\Bigr)^{5\/6}q^{2\/9}{M^{1\/6}}\n\\label{bound1}\n\\end{align}\nfor any~$\\varepsilon>0$ (the first term corresponds to removing the\nconstraint $(n,q)=1$).\n\n\\subsection{When $M$ is in the Fourier range}\n\nIf $M\\geq q^{1\/2}$, then it is\nbeneficial to apply the Poisson summation formula to the\n$m$-variable. As in the previous case, the cost of removing the\ncondition $(m,q)=1$ is $\\ll \\|K\\|_\\infty q^{7\/32-1}X^{1+\\varepsilon}\/d^2$\nfor~$\\varepsilon>0$. The Poisson summation formula implies that\n$$\n\\sum_{m\\geq\n  1}K(d^2mn)V\\Bigl(\\frac{d^2mn}{X}\\Bigr)W\\Bigl(\\frac{m}M\\Bigr)\\ll\n\\|\\widehat K\\|_\\infty \\Bigl(\\frac{M}{q^{1\/2}}+q^{1\/2}\\Bigr)\n$$\nand therefore\n\\eqref{DMsum} is bounded by\n\\begin{equation}\\label{bound2}\n  S_{d,M}\\ll \\|K\\|_\\infty X^{\\varepsilon}q^{7\/32-1}\\frac{X}{d^2}\n  +\\|\\widehat K\\|_\\infty X^{\\varepsilon}\\frac{X}{d^2}\n  \\Bigl(\\frac{1}{q^{1\/2}}+\\frac{q^{1\/2}}{M}\\Bigr)\n\\end{equation}\nfor any~$\\varepsilon>0$.\n\n\\subsection{When $M$ is large but not in Fourier range}\n\nIf~$M\\leq q^{1\/2}$, thinking of the prototypical case when\n$X\\sim q^{3\/2}$ and $D$ is close to one, the $n$-sum is of length\nclose to $q$, so the natural move is to smooth the $n$-sum, and then\nuse the Poisson summation formula on the resulting sums.\n\nThus we apply the Cauchy--Schwarz inequality to \\eqref{DMsum}, leaving\nthe $n$ variable outside, namely\n\\begin{equation}\\label{eqCS}\n  |S_{d,M}|^2\n  \\ll\\sum_{n\\sim X\/d^2M}|\\lambda(1,n)|^2\\times\n  \\sumsum_\\stacksum{m_i\\sim\n    M}{(m_i,q)=1}\\sum_{n\\geq 1}K(d^2m_1n)\\ov{K(d^2m_2n)}\n  V\\Bigl(\\frac{d^2m_1n}X\\Bigr)\\ov{V}\\Bigl(\\frac{d^2m_2n}X\\Bigr).\n\\end{equation}\n\nHere, we have dropped the constraint $(n,q)=1$ on the right-hand side\nby positivity, and replaced the expressions $W\\Bigl(\\frac{m_i}M\\Bigr)$\nby the summation conditions $ m_i\\sim M$.\n\nBy the Poisson summation formula, we have\n\\begin{equation}\\label{applypoisson}\n  \\sum_{n\\geq 1}\n  K(d^2m_1n)\\ov{K(d^2m_2n)}\n  V\\Bigl(\\frac{d^2m_1n}X\\Bigr)\\ov{V}\\Bigl(\\frac{d^2m_2n}X\\Bigr)\n  =\\frac{N}{q^{1\/2}}\\sum_{h\\in\\mathbf{Z}}\n  \\widehat{K}_{(2)}(h)\\mathcal{W}\\Bigl(\\frac{h}{q\/(X\/d^2M)}\\Bigr),\n\\end{equation}\nwhere $\\mathcal{W}(y)$ is a smooth function depending on $d,m_1,m_2$,\nrapidly decaying as $y\\rightarrow\\infty$, and\n$$\n\\widehat{K}_{(2)}(h)=\\frac{1}{\\sqrt{q}}\\sum_{n\\in\\mathbf{F}_q}\nK(d^2m_1n)\\ov{K(d^2m_2n)}e\\Bigl(\\frac{nh}{q}\\Bigr).\n$$\n\\par\nTo go further, we use the assumption of Corollary~\\ref{RScor} that $K$\nis the trace function of a middle-extension $\\ell$-adic sheaf~$\\mathcal{F}$\nthat is not exceptional. Indeed, from \\cite[Theorem 6.3]{FKM2}, we can\ndeduce that there exists a set $B\\subset {\\mathbf{F}^\\times_q}$ such\nthat $|B|$ is bounded in terms of the conductor of~$\\mathcal{F}$ only, and\nsuch that whenever\n\\begin{equation}\\label{eqdiag}\n  m_1\/m_2\\mods q\\not\\in B,\n\\end{equation}\nthen we have\n$$\n\\|\\widehat{K}_{(2)}\\|_\\infty\\ll 1\n$$\nwhere the implied constant depends on the conductor of~$\\mathcal{F}$ only.\n\nReturning to \\eqref{eqCS}, we apply the bound \\eqref{applypoisson} to\nthe pairs pairs $(m_1,m_2)$ which satisfy \\eqref{eqdiag}, and apply\nthe trivial bound otherwise.\n\nWe see then that the contribution to the second factor of \\eqref{eqCS}\nof the ``diagonal'' pairs not satisfying \\eqref{eqdiag} is bounded by\n$$\n\\ll X^{\\varepsilon}M\\Bigl(\\frac{M}q+1\\Bigr)\\frac{X\/M}{d^2}\n$$\nfor any~$\\varepsilon>0$, while the contribution of the pairs $(m_1,m_2)$\nsatisfying \\eqref{eqdiag} is bounded by\n$$\n\\ll X^{\\varepsilon}M^2\\Bigl(\\frac{X\/M}{d^2q^{1\/2}}+q^{1\/2}\\Bigr),\n$$\nfor any~$\\varepsilon>0$, where in both cases the implied constant depends\nonly on~$\\varepsilon$ and on the conductor of~$\\mathcal{F}$.\n\n\nCollecting these bounds, we obtain from \\eqref{eqCS} the bound\n\\begin{equation}\\label{bound3}\n  S_{d,M} \\ll\n  \\frac{X^{1+\\varepsilon}}{d^2}\n  \\Bigl(\\frac{1}{M^{1\/2}}+\\frac{1}{q^{1\/4}}+q^{1\/4}M^{1\/2}\n  \\frac{d}{X^{1\/2}}\\Bigr),\n\\end{equation}\nfor any~$\\varepsilon>0$, where the implied constant depends only on~$\\varepsilon$\nand on the conductor of~$\\mathcal{F}$.\n\n\n\\subsection{End of the proof}\n\nNow we can combine the previous bounds.  Let~$\\eta>0$ and~$\\delta$\nwith~$0<\\delta<1\/4$ be parameters to be determined later.\n\n\\subsubsection*{-- If $M\\leq q^{2\\delta}$,} we then apply the bound\n\\eqref{bound1} (and the dyadic decomposition of~$T_{d,x}$ in a\ncombination of sums~$S_{d,M}$) to derive\n\\begin{equation}\\label{bound-c}\n  \\sum_{d\\leq D}T_{d,X}\\ll X^{1+\\varepsilon}q^{7\/32-1}\n  +Z^{10\/9}X^{5\/6+\\varepsilon}q^{2\/9+\\delta\/3},\n\\end{equation}\nunder the condition that\n\\begin{equation}\\label{condition-X}\nX\\geq Z^{2\/3}D^2q^{4\/3+2\\delta}\n\\end{equation}\n(see \\eqref{condition-on-M}).\n\n\\subsubsection*{-- If $M\\geq q^{1\/2+\\eta}$,} we apply the bound\n\\eqref{bound2} and sum over $d\\leq D$, to find that\n\\begin{equation}\\label{bound-a}\n  \\sum_{d\\leq D} T_{d,X}\n  \\ll X^{1+\\varepsilon}\\Bigl(\\frac{1}{q^{1\/2}}+\\frac{q^{1\/2}}{M}\\Bigr)\n  \\ll X^{1+\\varepsilon}q^{-\\eta}\n\\end{equation}\nin that case.\n\n\\subsubsection*{-- If $q^{2\\delta}\\leq M< q^{1\/2+\\eta}$,} we then\napply the bound \\eqref{bound3} and sum over $d\\leq D$, obtaining\n\\begin{equation}\\label{bound-b}\n  \\sum_{d\\leq D}T_{d,X}\\ll\n  X^{1+\\varepsilon}\\Bigl(\\frac{1}{q^{\\delta}}\n  +\\frac{1}{q^{1\/4}}+\\frac{q^{1\/2+\\eta\/2}}{X^{1\/2}}\\Bigr)\n  \\ll X^{1+\\varepsilon}\\Bigl(q^{-\\delta}+\\frac{q^{1\/2+\\eta\/2}}{X^{1\/2}}\\Bigr).\n\\end{equation}\n\nThis covers all of the ranges for $M$. We now choose $\\eta, \\delta>0$\nsuch that the bound in \\eqref{bound-a} is equal to the second term in\n\\eqref{bound-b}, and the first term in \\eqref{bound-b} is consistent\nwith the second term in \\eqref{bound-c}. That is, we choose\n$q^{\\eta}=(X\/q)^{1\/3}$ and\n$q^{\\delta}=\\frac{X^{1\/8}}{Z^{5\/6}q^{1\/6}}$. Therefore we have in all\ncases the estimate\n$$\n\\sum_{d\\leq D}T_{d,X} \\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{7\/32-1},\n$$\nfor any~$\\varepsilon>0$, under the assumption that\n$$\nX\\gg D^{8\/3}q^{4\/3}Z^{-4\/3},\n$$\nand the implied constant depends only on~$\\varepsilon$ and the conductor\nof~$\\mathcal{F}$.\n\nFinally we combine this with the previously noted estimate\n$$\n\\sum_{d>D}T_{d,X}\\ll\n\\frac{\\|K\\|_\\infty X^{1+\\varepsilon}}{D}\n$$\n(recall that for a non-exceptional trace function, we\nhave~$\\|\\widehat{K}\\|_{\\infty}\\ll 1$ where the implied constant depends\nonly on the conductor of~$\\mathcal{F}$), to conclude that\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}\/D.\n$$\nWe take $D=q^{\\gamma}$ for some small~$\\gamma>0$, and then we have\n$$\n\\sum_{n\\geq 1}\\lambda(n)^2K(n)V\\Bigl(\\frac{n}{X}\\Bigr)\\ll\nX^{2\/3+\\varepsilon}q^{1\/3}+Z^{5\/6}X^{7\/8+\\varepsilon}q^{1\/6}+X^{1+\\varepsilon}q^{-\\gamma},\n$$\nprovided that\n$$\nX\\gg q^{4\/3+8\\gamma\/3}\/Z^{4\/3}, \n$$\nwhere the implied constant depends only on~$\\varepsilon$ and the conductor\nof~$\\mathcal{F}$.\n\nThis concludes the proof of Corollary \\ref{cor2-gl2}.\n\n\n\n\\begin{bibdiv}\n\n\\begin{biblist}\n\n\\bib{AHLS}{article}{\n  author={Aggarwal, K.},\n  author={Holowinsky, R.},\n  author={Lin, Y.},\n  author={Sun, Q.},\n  title={The Burgess bound via a trivial delta method},\n  note={\\url{arXiv:1803.00542v1}},\n  date={2018},\n}   \n\n\\bib{Blomer}{article}{\n   author={Blomer, V.},\n   title={Subconvexity for twisted $L$-functions on ${\\rm GL}(3)$},\n   journal={Amer. J. Math.},\n   volume={134},\n   date={2012},\n   number={5},\n   pages={1385--1421},\n\n}\n \\bib{CI}{article}{\n   author={Conrey, J. B.},\n   author={Iwaniec, H.},\n   title={The cubic moment of central values of automorphic $L$-functions},\n   journal={Ann. of Math. 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Journal},\n   date={2014},\n   volume={163},\n   pages={1683-1736},\n   number={9},\n }\n\n\n \\bib{pisa}{article}{\n   author={Fouvry, {\\'E}.},\n   author={Kowalski, E.},\n   author={Michel, Ph.},\n   title={Trace functions over finite fields and their applications},\n   book={\n     series={Colloquio de Giorgi},\n     publisher={Springer},\n   },\n   date={2014},\n }\n \n\n \\bib{short-sums}{article}{\n   author={Fouvry, {\\'E}.},\n   author={Kowalski, E.},\n   author={Michel, Ph.},\n   author={Raju, C.},\n   author={Rivat, J.},\n   author={Soundararajan, K.},\n   title={On short sums of trace functions},\n   journal={Ann. Inst. Fourier},\n   date={2017},\n   volume={67},\n   pages={423--449},\n }\n \n \\bib{FI}{article}{\n  author={Friedlander, J.B.},\n  author={Iwaniec, H.},\n  title={Incomplete Kloosterman sums and a divisor problem},\n  note={(with an appendix by\n  B. J. Birch and E. Bombieri)},\n  journal={Ann. of Math. 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{"text":"\n\\section{Shape Priors for Object SLAM}\n\nIn this section, we explain in more detail the benefit of using learnt shape priors in object SLAM. We evaluate the related object-oriented SLAM methods under three important properties: \\textit{1. Can the system reconstruct previously unseen objects? 2. Is the reconstruction complete? 3. Does the reconstruction preserve detailed shape?} \n\nThe first line of works~\\cite{SLAM++, Tateno2016When2.5D} relies on a pre-scanned CAD model database to perform online detection and registration. The reconstructed object shapes are complete and detailed, but the system cannot handle new objects. The second category of works~\\cite{maskfusion, fusion++} leverages 2D segmentation masks, treat each segmented object as individual identities and perform reconstruction on-the-fly. This reconstruction-by-segmentation strategy can reconstruct arbitrary unseen object shapes in great detail, but the reconstructed shape is incomplete due to partial observation and missing depth. \nThe last line of research represents objects as simple geometric shapes, such as ellipsoids~\\cite{quadricslam} or cuboids~\\cite{cubeslam}. \nThis kind of geometric representation can be applied to arbitrary unseen shapes and preserve some important properties like scale and orientation, but we've lost the level of details in the shapes.\n\nWith learnt shape priors, all the three properties can be achieved at the same time. Its generative property means it can generalize to unseen shapes. Object shape decoded from latent code is guaranteed to be detailed and complete, even from a single partial view. The compact representation also benefits the optimization process. Table \\ref{tab:related_work} provides an overview of the related works under the different properties. Only NodeSLAM~\\cite{node-slam} and DSP-SLAM can achieve all the three important properties. Unlike NodeSLAM, DSP-SLAM can also deal with large scale environments and monocular RGB inputs.\n\n\\section{Full Derivation of Jacobians}\n\nAs mentioned in the main paper, one of our contribution is the fast Gauss-Newton optimizer, which is crucial to achieve real-time performance. This section provides full derivation of Jacobians of each individual residual term.\n\n\\subsection{Jacobian of Surface Term Residual \\label{sec:J_surf}}\n\nAs shown in Eq. 1 in the main paper, the surface term residual at pixel $\\matr{u} \\in \\mathbf{\\Omega}_s$ is simply the SDF value of the back-projected point under object coordinate:\n\n\\begin{equation}\n    e_s(\\matr{u}) = G(\\matr{T}_{oc} \\pi^{-1}(\\matr{u}, {\\mathcal{D}}), \\matr{z}) = G(^o\\matr{x}, \\matr{z})\n\\end{equation}\n\nIts Jacobian with respect to object pose and shape $[\\boldsymbol{\\xi}_{oc};\\matr{z}]$ is defined as:\n\n\\begin{equation} \\label{eq:J_surf}\n    \\matr{J}_s = \\frac{\\partial e_s(\\matr{u})}{\\partial [\\boldsymbol{\\xi}_{oc};\\matr{z}]}\n\\end{equation}\n\nwhere $\\boldsymbol{\\xi}_{oc} \\in \\mathbb{R}^7$ is the Cartesian representation (twist coordinate) of the corresponding element in Lie Algebra $\\mathfrak{sim}(3)$. The Jacobian with respect to the shape code $\\frac{\\partial e_s(\\matr{u})}{\\partial \\matr{z}}$ could be obtained directly via back-propagation. To derive the Jacobian with respect to object pose $\\boldsymbol{\\xi}_{oc}$, we first factorize it using chain rule:\n\n\\begin{table}[tbp]\n  \\center\n  \\setlength\\tabcolsep{4pt}\n    \\begin{tabular}{c||l|l|l||l|l}\n    Method       & \\specialcell[b]{Unseen\\\\Objects}  & \\specialcell[b]{Full\\\\Shape}  & \\specialcell[b]{Detailed\\\\Shape} & \\specialcell[b]{RGB\\\\Input} & \\specialcell[b]{Large\\\\Scene}  \\\\ \\hline \\hline\n    \\specialcell[b]{2.5D is not\\\\enough\\cite{Tateno2016When2.5D}} &  ~ & \\checkmark & \\checkmark & ~ & ~ \\\\ \\hline\n    \\specialcell[b]{SLAM++\\\\\\cite{SLAM++}} & ~ & \\checkmark & \\checkmark & ~ & ~\\\\ \\hline\\hline\n    \\specialcell[b]{Mask-\\\\Fusion \\cite{maskfusion}} & \\checkmark & ~ & \\checkmark & ~ & ~ \\\\ \\hline\n    \\specialcell[b]{Fusion++\\\\\\cite{fusion++}} & \\checkmark & ~ & \\checkmark & ~ & ~\\\\ \\hline\\hline\n     \\specialcell[b]{Quadric-\\\\SLAM \\cite{quadricslam}} & \\checkmark & \\checkmark & ~ & \\checkmark &\\checkmark \\\\ \\hline\n    \\specialcell[b]{Cube-\\\\SLAM \\cite{cubeslam}} & \\checkmark & \\checkmark & ~ & \\checkmark & \\checkmark\\\\ \\hline\\hline\n    \\specialcell[b]{Node-\\\\SLAM \\cite{node-slam}} & \\checkmark & \\checkmark & \\checkmark & ~ & ~ \\\\ \\hline\n    \\textbf{\\specialcell[b]{DSP-\\\\SLAM}} & \\checkmark & \\checkmark & \\checkmark & \\checkmark & \\checkmark\n    \\end{tabular}\n    \\caption{Comparison of the properties of DSP-SLAM with respect\n      to other object-oriented SLAM systems. \n      \\vspace{-1.0cm}}\n    \\label{tab:related_work}\n\\end{table}\n\n\\begin{equation} \\label{eq:J_surf_pose_1}\n    \\frac{\\partial e_s(\\matr{u})}{\\partial \\boldsymbol{\\xi}_{oc}} = \\frac{\\partial G(^o\\matr{x}, \\matr{z})}{\\partial ^o\\matr{x}} \\frac{\\partial ^o\\matr{x}}{\\partial \\boldsymbol{\\xi}_{oc}}\n\\end{equation}\n\nThen the first term $\\frac{\\partial G(^o\\matr{x}, \\matr{z})}{\\partial ^o\\matr{x}}$ can also be obtained via back-propagation. The second Jacobian term could be computed by applying a left perturbation to the pose $\\matr{T}_{oc}$:\n\n\\begin{align}\n    \\frac{\\partial ^o\\matr{x}}{\\partial \\boldsymbol{\\xi}_{oc}} & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\exp{\\big(\\delta \\boldsymbol{\\xi}^\\wedge}\\big) \\matr{T}_{oc} {^c\\matr{x}} - \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\big(\\matr{I} + \\delta \\boldsymbol{\\xi}^\\wedge\\big) \\matr{T}_{oc} {^c\\matr{x}} - \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge \\matr{T}_{oc} {^c\\matr{x}}}{\\delta \\boldsymbol{\\xi}} = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge {^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}}  \\label{eq:J_surf_pose_2}\n\\end{align}\n\nwhere $\\exp(\\cdot)$ is the exponential map from Lie Algebra $\\mathfrak{sim}(3)$ to the corresponding Lie Group $\\mathbf{Sim}(3)$, and $\\cdot ^\\wedge$ is the hat-operator that maps a twist coordinate in $\\mathbb{R}^7$ to $\\mathfrak{sim}(3)$:\n\n\\begin{equation} \\label{eq:hat}\n    \\boldsymbol{\\xi}^ \\wedge = \\begin{bmatrix} \\boldsymbol{\\nu} \\\\ \\boldsymbol{\\phi} \\\\ s \\end{bmatrix}^ \\wedge = \\begin{bmatrix} \\boldsymbol{\\phi}_{\\times} + s\\matr{I} & \\boldsymbol{\\nu} \\\\ \\matr{0} & 0\\end{bmatrix}\n\\end{equation}\n\n$\\boldsymbol{\\nu} \\in \\mathbb{R}^3$, $\\boldsymbol{\\phi} \\in \\mathbb{R}^3$ and $s \\in \\mathbb{R}$ represent the translation, rotation and scale part of the twist coordinate respectively. $(\\cdot)_\\times$ maps from $\\mathbb{R}^3$ to $\\mathfrak{so}(3)$ (skew-symmetric matrices). With the equation above, the Jacobian term in Eq. 6 can be computed as: \\vspace{-0.3cm}\n\n\\begin{align}\n    \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\xi}^\\wedge {^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}} & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{\\delta \\boldsymbol{\\phi}_\\times {^o\\matr{x}} + \\delta s{^o\\matr{x}} + \\delta \\boldsymbol{\\nu}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\lim_{\\delta \\boldsymbol{\\xi} = \\matr{0}} \\frac{ \\delta \\boldsymbol{\\nu} - {^o\\matr{x}}_\\times \\delta \\boldsymbol{\\phi}+ \\delta s{^o\\matr{x}}}{\\delta \\boldsymbol{\\xi}} \\\\\n    & = \\begin{bmatrix} \\matr{I} & -^o\\matr{x}_{\\times} & ^o\\matr{x} \\end{bmatrix}  \\label{eq:J_surf_pose_3}\n\\end{align}\n\nThe full Jacobian of the surface consistency term residual with respect to the object pose can be obtained by combining Eq. \\ref{eq:J_surf_pose_1}, \\ref{eq:J_surf_pose_2}, \\ref{eq:J_surf_pose_3}. Note that here we derive the Jacobian with slight abuse of notations and neglected the notation of homogeneous representation. For a more detailed explanation of Lie Theory, please refer to ~\\cite{barfoot_2017} or \\cite{sola2018_micro_lie}. \n\n\\subsection{Jacobian of Rendering Term Residual}\n\nAs stated in the main paper, the rendering term residual of pixel $\\matr{u}$ is\n\n\\begin{equation}\n    e_r(\\matr{u}) = d_\\matr{u} - \\hat{d}_{\\matr{u}}\n\\end{equation}\n\nTo compute the Jacobian of the rendering terms $\\matr{J}_{r}$, we can expand it using chain rule:\n\n\\begin{align} \n    \\matr{J}_{r} & = \\frac{\\partial e_{r}}{\\partial \\left[\\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]} \\\\\n    & = \\sum_{k=1}^M \\frac{\\partial e_{r}}{\\partial o_k} \\frac{\\partial o_k}{\\partial s_k} \\frac{\\partial{G({^o\\matr{x}_k}, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]} \\label{eq:J_render}\n\\end{align}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/rendering_values.png}\n\\caption{Demonstration of the depth rendering process. Only very few ray points contribute to the overall Jacobian term.\n\\vspace{-0.5cm}}\n\\label{fig:render_values}\n\\end{figure}\n\nwhere $\\{^o\\matr{x}_k\\}_{k=1}^M$ is the depth-ordered set of sampled points along the ray of back-projecting pixel $\\matr{u}$. The third term $\\frac{\\partial{G( ^o\\matr{x}_k, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]}$ has exactly the same form as the surface term Jacobian in Eq.~\\ref{eq:J_surf}, thus it can be computed following Sec.~\\ref{sec:J_surf} . The second term $\\frac{\\partial o_k}{\\partial s_k}$ is either a constant value or $0$, as the sdf-occupancy conversion is linear inside the cut-off threshold and constant elsewhere. \n\n\\begin{equation} \\label{eq:conversion_grad}\n    \\frac{\\partial o_k}{\\partial s_k} =   \n    \\begin{cases}\n        -\\frac{1}{2\\sigma} & |s_k| < \\sigma\\\\\n        0 & \\text{elsewhere}\n    \\end{cases}\n\\end{equation}\n\nTo compute the first term, we expand the residual term using Eq. 3 and Eq. 4 in the main paper following~\\cite{multi-view_supervision}: \\vspace{-0.3cm}\n\n\\begin{equation*}\n    \\begin{aligned}\n        e_r & = d_\\matr{u} - \\bigg( \\sum_{i=1}^M d_i \\ o_i \\prod_{j=1}^{i-1} (1-o_j) + d_{M+1} \\prod_{j=1}^M (1 - o_j) \\bigg) \\\\\n        & = \\sum_{i=1}^M \\psi(i) \\ o_i \\prod_{j=1}^{i-1} (1-o_j) + \\psi(M+1) \\prod_{j=1}^M (1 - o_j) \\\\\n       \n        & = \\sum_{i=1}^{M+1} \\psi(i) \\prod_{j=1}^{i-1} (1-o_j) - \\sum_{i=1}^M \\psi(i) \\prod_{j=1}^i (1-o_j) \\\\\n        & = \\psi(1) + \\sum_{i=1}^M (\\psi(i+1) - \\psi(i)) \\prod_{j=1}^i (1-o_j)\n    \\end{aligned}\n\\end{equation*}\n\nwhere $\\psi(i) = d_{\\matr{u}} - d_i$ is the depth residual for each of the sampled points along the ray. Differentiating with respect to the sample point, we reach: \\vspace{-0.2cm}\n\n\\begin{align} \n    \\frac{\\partial e_{r}}{\\partial o_k} \n    & = \\sum_{i=1}^M (\\psi(i+1) - \\psi(i)) \\frac{\\partial}{\\partial o_k}\\prod_{j=1}^i (1-o_j) \\\\\n    & = \\sum_{i=k}^M (\\psi(i) - \\psi(i+1)) \\prod_{j=1, j \\neq k}^i (1-o_j) \\\\\n    & = \\Delta d\\sum_{i=k}^M \\prod_{j=1, j \\neq k}^i (1-o_j) \\label{eq:dr_do}\n\\end{align}\n\nAs we are sampling uniformly along the ray, the coefficient $\\psi(i) - \\psi(i+1) = d_{i+1} - d_i$ reduces to $\\Delta d$. To understand this expression we can multiply $(1-o_k)$ on both sides, of Eq.~\\ref{eq:dr_do}, which gives us:\n\n\\begin{equation}\n        \\frac{\\partial e_{r}}{\\partial o_k}(1 - o_k)  = \\Delta d\\sum_{i=k}^M t_i\n\\end{equation}\n\nwhere $t_i = \\prod_{j=1}^i (1 - o_j)$ represents the accumulated transmittance at point $^o\\matr{x}_i$ along the ray. Before hitting the surface, $o_k$ is zero, so the term $(1-o_k)$ has no effect, and the Jacobian term becomes the sum of transmittance of all the points starting from point k. And after the ray entering the solid body, this Jacobian term will be zero. This means only points before entering the solid body contribute to the rendered depth value.\n\nNow from Eq.~\\ref{eq:conversion_grad} and Eq.~\\ref{eq:dr_do}, we already know that many Jacobian terms that make up the final Jacobian in Eq.~\\ref{eq:J_render} should be zero. Therefore, we could further speed up the optimization by not evaluating the third term $\\frac{\\partial{G( ^o\\matr{x}_k, \\matr{z})}}{\\partial \\left[ \\boldsymbol{\\xi}_{oc}; \\matr{z} \\right]}$ at those points. Figure~\\ref{fig:render_values} is a demonstration of the rendering process and the sparse nature of the resulting Jacobians.\n\n\\subsection{Jacobian of Prior Terms}\n\nBased on the shape regularisation energy defined in Eq. 6 in the main paper. The residual of this energy term is simply the shape code vector itself:\n\n\\begin{equation} \\label{eq:E_code}\n    e_c = \\matr{z}\n\\end{equation}\n\nBased on Eq.~\\ref{eq:E_code}, we have\n\n\\begin{equation}\n    \\frac{\\partial e_c}{\\partial \\boldsymbol{\\xi}_{oc}} = \\matr{0}, \\frac{\\partial e_c}{\\partial \\matr{z}} = \\matr{I}\n\\end{equation}\n\nOptionally, we can also apply structural priors such as constraining the object orientation to be aligned with the ground plane, as in~\\cite{wang2020directshape}. We can define the rotation prior residual as\n\n\\begin{align}\n    e_{rot} & = 1 - \\matr{R}_{co}(0:2,1) \\cdot \\matr{n}_g \\\\\n    & = 1 - \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix}\\matr{R}_{oc}\\matr{n}_g\n\\end{align}\n\n\\begin{figure}[tpb]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{figures\/redwood\/gt_together.png}\n\\includegraphics[width=0.48\\linewidth]{figures\/redwood\/no_gt_together.png}\n\\caption{Results with GT ($\\leftarrow$) vs. MaskRCNN ($\\rightarrow$) masks \\vspace{-0.6cm}}\n\\label{fig:gt_mask}\n\\end{figure}\n\n\nwhere $\\matr{R} \\in \\mathbf{SO}(3)$ denotes the rotation part of the transformation matrix, and $(0:2, 1)$ represents the operation of getting the second column of the matrix. $\\matr{n}_g$ is the normal direction to the ground plane under camera coordinate frame. The Jacobian with respect to pose can be easily obtained following Eq.~\\ref{eq:J_surf_pose_3}:\n\n\\begin{align}\n    \\frac{\\partial e_{rot}}{\\partial \\boldsymbol{\\xi}_{oc}} &= - \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix}\\frac{\\partial \\matr{R}_{oc}\\matr{n}_g}{\\partial \\boldsymbol{\\xi}_{oc}} \\\\\n    & = \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix} \\begin{bmatrix} \\matr{0} & (\\matr{R}_{oc}\\matr{n}_g)_{\\times} & \\matr{0} \\end{bmatrix} \\\\\n    & = \\begin{bmatrix} 0 & 0 & 0 & (\\matr{R}_{oc}\\matr{n}_g)_{\\times}(1,0:2) & 0 \\end{bmatrix}\n\\end{align}\n\nwhere $(0:2, 1)$ represents the operation of getting the second row of the matrix. As this residual has no effect on object shape, we have:\n\n\\begin{equation}\n    \\frac{\\partial e_{rot}}{\\partial \\matr{z}} = \\matr{0}\n\\end{equation}\n\n\\section{Experiment Details and Run-time Analysis}\n\nWe evaluate the run-time performance of our full SLAM system (stereo+LiDAR) on a Linux system with Intel Core i9 9900K CPU at 3.60GHz, and an nVidia GeForce RTX2080 GPU with 8GB of memory. The 2D detection boxes and masks are obtained using MaskRCNN~\\cite{maskrcnn},\\footnote{\\url{https:\/\/github.com\/facebookresearch\/detectron2}} throughout all the experiments. Initial object poses are inferred using the LiDAR-based 3D b-box detector SECOND~\\cite{second}.\\footnote{\\url{https:\/\/github.com\/traveller59\/second.pytorch}} Our object reconstruction pipeline is implemented in Python with PyTorch. The back-end object-SLAM framework is implemented in C++ as an extension of the original ORB-SLAM2 implementation. The Python part is embedded into C++ using pybind11. \n\nTable~\\ref{tab:timing} lists the breakdown of the run-time performance of different major components. Note that all those components are performed only at key-frames. \nFor KITTI sequences, we adopted two design choices to achieve real-time performance. Firstly, we noticed shape estimation did not improve much over time in KITTI, thus we perform full optimization only for new objects and only update poses for existing objects (as stated in the main paper). Secondly, we abort new object optimization whenever necessary to ensure new key-frame is inserted in time. These make our system able to operate at $10$Hz on KITTI. In sequences such as Freiburg and Redwood-OS, it is beneficial to update the shape more frequently, with fewer GN iterations to guarantee real-time performance. Object meshes can be extracted optionally using marching cube with an extra GPU, for visualization only.\n\n\\begin{table}[tbp]\n\\centering\n\\begin{tabular}{l|c}\n\\textbf{Component} & \\textbf{Time (ms)} \\\\ \\hline\nMask R-CNN Detector & 70 \/ frame \\\\\n3D LiDAR Detector & 60 \/ frame \\\\\nPose-shape Optimization & 20$\\times$10 \/ new object  \\\\\nPose-only Optimization & 4$\\times$5 \/ vis. object  \\\\ \\hline\n\\end{tabular}\n\\caption{Run-time analysis with system components \\vspace{-0.4cm}}\n\\label{tab:timing}\n\\end{table}\n\n\n\\section{Ablation with GT masks}\nWe have shown promising reconstruction results on cars on both KITTI and Freiburg dataset. Chairs have thin structures and a complex topology and are more challenging to reconstruct than cars. We noticed the thin legs of the last chair in Fig.~10 in the main paper were not properly reconstructed. This is because our reconstruction makes use of Mask-RCNN segmentation masks, which are noisy and affected by shadows. This can result in background points being associated to the chair, leading to incorrect results. We conducted ablation study using ground-truth masks as input. Fig.~\\ref{fig:gt_mask} illustrates an upper bound quality that could be achieved with clean GT masks.\n\n\\section{More Qualitative Results}\n\nWe show more qualitative reconstruction results in Fig.~\\ref{fig:recon_results}. For each scene the RGB image is shown on the top and the reconstruction results are shown underneath. We also show an example of reconstructed map in Fig.~\\ref{fig:map_result}. \n\n\n\\begin{figure*}[tpb]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-000117.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-001190.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-000117.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-001190.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-001848.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-003264.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-001848.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-003264.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/00-003516.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/07-000195.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-00-003516.png}\n\\includegraphics[width=0.48\\textwidth]{figures\/supp\/recon-07-000195.png}\n\\caption{More qualitative results on object reconstruction from a single view-point \\vspace{-1cm}}\n\\label{fig:recon_results}\n\\end{figure*}\n\n\\begin{figure*}[tpb]\n\\centering\n\\includegraphics[width=0.90\\textwidth]{figures\/supp\/kitti-07-map.png}\n\\caption{Reconstructed map of KITTI-07. Note the dense reconstruction of the objects and the shape variations. \\vspace{-1cm}}\n\\label{fig:map_result}\n\\end{figure*}\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n\n\\section{Introduction}\nSimultaneous Localization and Mapping (SLAM) is the process of estimating the trajectory of a moving camera while reconstructing its surrounding environment. From a purely geometric perspective, SLAM is often regarded as a well-understood or even solved problem. Many state-of-the-art dense SLAM algorithms can achieve accurate trajectory estimation and create high-quality geometric reconstructions that can be used in obstacle avoidance or path planning for mobile robots. However, when it comes to more complex tasks that require scene understanding, geometry-only scene representations fall short of providing key semantic information. \nTaking advantage of recent deep learning breakthroughs in semantic segmentation and object detection algorithms \\cite{maskrcnn, faster-rcnn, yolo} semantic SLAM systems augment geometric low-level map primitives by fusing semantic labels into the 3D reconstruction~\\cite{semantic-fusion, semantic-stereo-fusion,Chen2019SuMa++}. However, the resulting scene maps merely consist of a set of labelled 3D points where reasoning about the scene at the level of objects to infer meaningful information such as the number of objects of each category, their size, shape or relative pose remains a challenging task. Better use of the semantic information is required in the form of an object-centric map representation that allows detailed shape estimation and meaningful instantiation of scene objects. \n\n\nOur proposed approach forms part of a more recent family of \\emph{object-aware} SLAM methods that reconstruct object-centric maps grouping all the low-level geometric primitives (voxels, points ...) that make up the same object into a single instance. Front-end camera tracking and back-end optimization are both performed at the level of object instances. While the first object-level methods, such as SLAM++~\\cite{SLAM++}, mapped previously known object instances, more recent systems have taken advantage of instance level semantic segmentation masks \\cite{maskrcnn} to achieve object level reconstruction for unknown objects \\cite{fusion++} even in the presence of dynamic objects  \\cite{maskfusion, midfusion}.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.235\\textwidth]{figures\/08-2758.png}\n\\includegraphics[width=0.235\\textwidth]{figures\/recon-08-2758.png}\n\\caption{Qualitative shape and pose results on a stereo+LiDAR KITTI sequence. A very sparse set of LiDAR points was used to reconstruct each car. LiDAR points on the road are only shown for illustration.\\vspace{-0.5cm}}\n\\label{fig:single-view}\n\\end{figure}\n\nHowever, these early object level SLAM systems exhibit major drawbacks: They either require a pre-known database of object instances \\cite{SLAM++}; or reconstruct objects from scratch without exploiting shape priors \\cite{maskfusion, midfusion, fusion++}, which results in partial or incomplete object reconstructions. We improve this by exploiting the regularity of shapes within an object category in the form of learned shape priors, defined as a latent code $\\matr{z}$ and a generative model $G(\\matr{z})$ that decodes it into its full geometry. This brings us several advantages; object shapes decoded from latent codes are guaranteed to be detailed and complete, regardless of partial observations or changes in view-points, they provide a compact representation and they can be optimized using the gradients obtained through back-propagation. \n\nUsing ORB-SLAM2~\\cite{orbslam2} as a sparse camera tracking and mapping backbone, DSP-SLAM takes the reconstructed 3D point-cloud as input and fits a latent code to each detected object instance, using a combination of 3D surface consistency and rendered depth losses. Foreground objects, background features and camera poses are further refined via bundle adjustment using a joint factor graph. We show DSP-SLAM  operating in $3$ different modes: monocular, stereo, and stereo+LiDAR. The monocular and stereo systems use the respective ORB-SLAM2 modalities as the SLAM backbone and the reconstructed 3D point-clouds to reconstruct the detected objects. The stereo+LiDAR system uses stereo ORB-SLAM2 as the SLAM backbone but in addition it incorporates a sparse set of LiDAR measurements (as few as 50 per object) for object reconstruction and pose-only optimization. \n\n\\noindent\\textbf{Contributions:} While DSP-SLAM is not the first approach to leverage shape priors for 3D reconstruction~\\cite{frodo,node-slam} from image sequences, it innovates in various ways. \nFirstly, unlike~\\cite{frodo,node-slam}, our map does not only represent objects, but also reconstructs the background as sparse feature points, optimizing them together in a joint factor graph, marrying the best properties of feature-based~\\cite{orbslam2} (highly accurate camera tracking) and object-aware SLAM (high level semantic map). Secondly, although Node-SLAM \\cite{node-slam} also incorporates shape priors within a real-time SLAM system~\\cite{node-slam}, it uses dense depth images for shape optimization, while DSP-SLAM can operate with RGB-only monocular streams and requires as few as $50$ 3D points per object to obtain accurate shape estimates. Finally, although both FroDO~\\cite{frodo} and DSP-SLAM can operate in a monocular RGB setting, FroDO is a slow batch approach that requires all frames to be acquired in advance and associated with their camera poses, while  \nDSP-SLAM is an online, sequential method that can operate at $10$ frames per second. \n\nIn terms of object shape and pose estimation, we improve quantitative and qualitatively over auto-labelling~\\cite{Zakharov2020Autolabeling3D}, a state-of-the-art prior-based object reconstruction method. Experiments on the KITTI odometry \\cite{kitti_odom} dataset show that, with stereo+LiDAR input our joint bundle adjustment offers improvements in trajectory estimation over the feature-only stereo system ORB-SLAM2~\\cite{orbslam2}, used as our backbone. Moreover, DSP-SLAM offers comparable tracking performance to state-of-the-art stereo~\\cite{Wang2017StereoDSO}, LiDAR-only~\\cite{Chen2019SuMa++} and dynamic~\\cite{Bescos2018DynaSLAM} SLAM systems, while providing rich dense object reconstructions. DSP-SLAM also achieves promising qualitative reconstruction results with monocular input on Freiburg Cars \\cite{freiburg-cars} and Redwood-OS \\cite{choi2016redwood} dataset.\n\n\n\n\n\n\\section{\\label{system-overview}System Overview}\n\nDSP-SLAM is a sequential localisation and mapping method that reconstructs the complete detailed shape of detected objects while representing the background coarsely as a sparse set of feature points. Each object is represented as a compact and optimizable code vector $\\matr{z}$. We employ DeepSDF \\cite{deepsdf} as the shape embedding, that takes as input a shape code $\\matr{z} \\in \\mathbb{R}^{64}$ and a 3D query location $\\matr{x} \\in \\mathbb{R}^{3}$, and outputs the signed distance function (SDF) value $s = G(\\matr{x}, \\matr{z})$ at the given point. An overview of DSP-SLAM is shown in Fig.~\\ref{fig:system}. DSP-SLAM runs at almost real time ($10$ frames per second) and can operate on different modalities: monocular, stereo or stereo with LiDAR; depending on the available input data. \n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000140.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000168.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000428.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/000655.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/140r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.1115\\textwidth]{figures\/single_view_shape\/168r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/428r.png} \\hspace{-0.6em}\n\\includegraphics[width=0.115\\textwidth]{figures\/single_view_shape\/655r.png} \\hspace{-0.6em}\n\\caption{Shape reconstruction: qualitative results. \\vspace{-0.5cm}}\n\\label{fig:single-view-shape}\n\\end{figure}\n\n\\noindent\\textbf{Sparse SLAM backbone:} ORB-SLAM2~\\cite{orbslam2} is used as the tracking and mapping backbone, a feature-based SLAM framework that can operate on monocular or stereo sequences. While the tracking thread estimates camera pose at frame-rate from correspondences, the mapping thread builds a sparse map by reconstructing 3D landmarks.\n\n\\noindent\\textbf{Detections:}  We perform object detection at each key-frame, to jointly infer 2D bounding boxes and segmentation masks. In addition, an initial estimate for the object pose estimation is obtained via 3D bounding box detection~\\cite{smoke,second}.\n\n\\noindent\\textbf{Data association:} New detections will either be associated to existing map objects, or instantiated as a \\textit{new} object via object-level data association. \nEach detected object instance $I$ consists of a 2D bounding box ${\\mathcal{B}}$, a 2D mask ${\\mathcal{M}}$, the dpeth observation of sparse 3D point cloud ${\\mathcal{D}}$, and the initial object pose $\\matr{T}_{co, 0}$. \n\n\\noindent\\textbf{Prior-based object reconstruction:} Newly instantiated objects will be reconstructed following the object reconstruction pipeline described in Sec.~\\ref{object-recon}. DSP-SLAM takes the set of sparse 3D point observations ${\\mathcal{D}}$, which can come from reconstructed SLAM points (in monocular and stereo modes) or LiDAR input (in stereo+LiDAR mode) and optimises the shape code and object pose to minimise surface consistency and depth rendering losses. Objects already present in the map will only have their 6-dof pose updated via pose-only optimization. \n\n\n\\noindent\\textbf{Joint map optimisation:} A joint factor graph of point features (from SLAM), objects and camera poses is optimised via bundle adjustment to maintain a consistent map and incorporate loop closure. New objects are added as nodes to the joint factor graph and their relative pose estimates $\\matr{T}_{co}$ as camera-object edges. Object-level data association and joint bundle adjustment will be discussed in Sec.~\\ref{object-slam}.\n\n\n\n\n\n\n\\section{\\label{object-recon}Object Reconstruction with  Shape Priors}\n\nWe aim to estimate the full dense shape $\\matr{z}$ and 7-DoF  pose $\\matr{T}_{co}$, represented as a homogeneous transformation matrix $\\matr{T}_{co} = [s\\matr{R}_{co}, \\matr{t}_{co}; \\matr{0}, 1] \\in \\matr{Sim}(3)$, for an object with detections $I = \\{{\\mathcal{B}}, {\\mathcal{M}}, {\\mathcal{D}}, \\matr{T}_{co, 0}\\}$. We formulate this as a joint optimization problem, which iteratively refines the shape code and object pose from an initial estimate. We propose two energy terms $E_{surf}$ and $E_{rend}$ and formulate a Gauss-Newton solver with analytical Jacobians.\n\n\\subsection{Surface Consistency Term}\nThis term measures the alignment between observed 3D points and the reconstructed object surface. \n\\begin{equation} \\label{E_surf}\n    E_{surf} = \\frac{1}{\\left| \\mathbf{\\Omega}_s \\right|}\\sum_{\\matr{u} \\in \\mathbf{\\Omega}_s} G^2(\\matr{T}_{oc} \\pi^{-1}(\\matr{u}, {\\mathcal{D}}), \\matr{z})\n\\end{equation}\nwhere $\\mathbf{\\Omega}_s$ denotes the pixel coordinates of the set of sparse 3D points ${\\mathcal{D}}$, which can come from reconstructed SLAM points (in monocular and stereo modes) or LiDAR input (in stereo+LiDAR mode). Ideally, the back-projected point at pixel $\\matr{u}$ should perfectly align with the object surface resulting in zero SDF value, giving a zero error residual. In practice, we observed that the surface consistency term alone is not sufficient for correct shape and pose estimation in the case of partial observations. Fig.~\\ref{fig:shape-ablation} illustrates a case where only points on the back and the right side of a car are observed (shown in green). Using  the surface consistency alone term leads to incorrect shape estimation -- much larger than its actual size. To address this issue, we propose a rendering loss, that provides point-to-point depth supervision and enforces silhouette consistency to penalize shapes that grow outside of the segmentation mask. \n\n\\begin{figure}[tbp]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/shape_ablation_success.png}\n    \\caption{Illustration of the effectiveness of the rendering term in the presence of partial observations. \\textbf{Left:} Detected object and partial surface point observations (green). \\textbf{Middle:} Optimisation result with $E_{surf}$ only. The loss is minimised but the shape grows larger than its actual size. \\textbf{Right:} Optimisation result with the rendering term. Enforcing the silhouette constraint results in the correct scale. \\vspace{-0.6cm}}\n    \\label{fig:shape-ablation}\n\\end{figure}\n\n\\subsection{Differentiable SDF Renderer}\n\nFollowing \\cite{multi-view_supervision, node-slam}, we build our SDF renderer via differentiable ray-tracing. For each pixel $\\matr{u}$, we back-project a ray $^c\\matr{x} = \\matr{o} + d \\matr{K}^{-1} \\dot{\\matr{u}}$ parameterized by the depth value $d$ under camera coordinate space, with $\\matr{o}$ being the camera optical centre and $\\matr{K}$ being camera intrinsic matrix. We sample $M$ discrete depth values $\\{d_i\\}$ along each ray within the range $[d_{min}, d_{max}]$, with $d_i = d_{min} + (i-1) \\Delta d$, and $\\Delta d = (d_{max} - d_{min}) \/ (M-1)$. The bounds of the depth range are determined by the current estimation of object translation and scale, and are re-computed at each iteration.\n\n\\noindent\\textbf{Occupancy Probabilities} \nThe SDF value $s_i$ at each sampled point can be obtained by transforming sampled points to the object coordinate frame and passing through the DeepSDF decoder. The SDF value encodes the probability that a given point is occupied by the object or belongs to free space. We apply a piecewise linear function to the predicted SDF values to indicate the occupancy probability $o_i$, defined in Eq.~\\ref{sdf2occ}, where $\\sigma$ represents the cut-off threshold which controls the smoothness of the transition. We fix $\\sigma = 0.01$ throughout our experiments. \\vspace{-0.4cm}\n\n\\begin{equation} \\label{sdf2occ}\n    s_i = G(\\matr{T}_{oc} {^c\\matr{x}},\\, \\matr{z}) \\quad \\textrm{and} \\quad \n    o_i = \n    \\begin{cases}\n        1 & s_i < -\\sigma\\\\\n        -\\frac{s_i}{2 \\sigma} &  \\left|s_i\\right| \\leq \\sigma \\\\    \n        0 & s_i > \\sigma\n    \\end{cases}\n\\end{equation}\n\n\n\\noindent\\textbf{Event Probabilities} When tracing points along the ray, the ray either terminates or escapes without hitting other points. These $M+1$ event probabilities can be defined as: \n\\begin{eqnarray} \\label{term_prob}\n    & \\phi_i = o_i \\prod_{j=1}^{i-1} (1- o_j), i = 1, \\dots, M \\nonumber \\\\\n    & \\phi_{M + 1} = \\prod_{j=1}^{M} (1- o_j)\n\\end{eqnarray}\n\n\\noindent\\textbf{Rendered Depth and Rendering Term} With the probabilities defined above, the rendered depth value at each pixel $\\matr{u}$ can be computed as the expected depth value of the terminating point as in Eq.~\\ref{render_depth}. To make it consistent, we set $d_{M+1}$, the depth value associated with escape probability, to a constant value $1.1 d_{max}$, as in \\cite{node-slam}.\n\\begin{equation} \\label{render_depth}\n    \\hat{d}_{\\matr{u}} = \\sum_{i=1}^{M+1} \\phi_i d_i\n\\end{equation}\nSince the rendering is fully differentiable, it can be integrated in our optimization. Unlike~\\cite{node-slam, multi-view_supervision}, we perform  ray-tracing in continuous space and do not require to discretize the object model. The final rendering term is as follows:\n\\begin{equation}\n    E_{rend} = \\frac{1}{\\left| \\mathbf{\\Omega}_r \\right|} \\sum_{\\matr{u} \\in \\mathbf{\\Omega}_r} (d_\\matr{u} - \\hat{d}_{\\matr{u}})^2\n\\end{equation}\nwhere $\\mathbf{\\Omega}_r = \\mathbf{\\Omega}_s \\cup \\mathbf{\\Omega}_b$ is the union of surface pixels and pixels not on object surface but inside the 2D bounding box ${\\mathcal{B}}$. Surface pixels $\\mathbf{\\Omega}_s$ are the same set of pixels used in Eq.~\\ref{E_surf}, obtained by projecting the 3D reconstucted SLAM points onto the image masks as discussed in Sec.~\\ref{system-overview}. The pixels in $\\mathbf{\\Omega}_b$ are assigned the same depth value as $d_{M+1} = 1.1 d_{max}$ and provide important silhouette supervision for our optimization since they penalize renderings that lie outside the object boundary, forcing empty space. As the pixels in $\\mathbf{\\Omega}_b$ do not require a depth measurement, we perform uniform sampling inside the 2D bounding box and filter out those inside the segmentation mask.\n\n\\subsection{Optimization details}\n\nOur final energy is the weighted sum of the surface and rendering terms and a shape code regularization term:\n\\begin{equation}\n    E = \\lambda_s E_{surf} + \\lambda_r E_{rend} + \\lambda_c \\norm{\\matr{z}}^2\n\\end{equation}\nThe hyperparameter values used for optimization $\\lambda_s = 100$, $\\lambda_r = 2.5$ and $\\lambda_c = 0.25$ were tuned such that the Hessian matrices of the energy terms are of the same order of magnitude. Since all terms are quadratic, we adopt a  Gauss-Newton optimisation approach with analytical Jacobians (Please refer to supplemental material for detail), initialized from a zero shape code $\\matr{z} = \\matr{0}$. The initialisation for the object pose \n$\\matr{T}_{co, 0}$  is given by a LiDAR 3D detector~\\cite{second} when LiDAR is available. In the monocular\/stereo case, it is given by an image-based 3D detector~\\cite{smoke} or by performing PCA on the sparse object point cloud. \n\n\n\n\\section{\\label{object-slam}Object SLAM}\n\nAs an object-based SLAM system, DSP-SLAM builds a joint factor graph of camera poses, 3D feature points and object locations and poses. As Fig.~\\ref{system-overview} shows, the factor graph introduces object nodes and camera-object edges. \n\n\n\\subsection{\\label{data-association}Object Data Association}\n\nData association between new detections and reconstructed objects is an important step in object-level SLAM. We aim to associate each detection $I$ to its \\textit{nearest} object $o$ in the map, adopting different strategies depending on the different input modalities. When LiDAR input is available we compare the distance between 3D bounding box and reconstructed object. When only stereo or monocular images are used as input, we count the number of matched feature points between the detection and object. If multiple detections are associated with the same object, we  keep the \\textit{nearest} one and reject others. Detections not associated with any existing objects are initialised as new objects and their shape and pose  optimised following Sec.~\\ref{object-recon}. For stereo and monocular input modes, reconstruction only happens when enough surface points are observed. For detections associated with existing objects, only the pose is optimised by running pose-only optimization and a new camera-object edge added to the joint factor-graph.\n\n\\begin{table*}\n\\renewcommand{\\arraystretch}{1.0}\n\\centering\n\\begin{tabular}{c|c c c c|c c c c}\n\\hline\n\\multirow{2}{*}{Diff.} & \\multicolumn{4}{c|}{Auto-labelling\\cite{Zakharov2020Autolabeling3D}} & \\multicolumn{4}{c}{Ours} \\\\\n\\cline{2-9}\n & BEV@0.5 & 3D@0.5 & NS@0.5 & NS@1.0 & BEV@0.5 & 3D@0.5 & NS@0.5 & NS@1.0 \\\\\n\\hline\nE & 80.70 & \\textbf{63.96} & 86.52 & 94.31 & \\textbf{83.31} & 62.58 & \\textbf{88.01} & \\textbf{96.86} \\\\ \nM & 63.36 & 44.79 & 64.44 & 85.24 & \\textbf{75.28} & \\textbf{47.76} & \\textbf{76.15} & \\textbf{89.97} \\\\\n\\hline\n\\end{tabular}\n\\caption{Quantitative comparison of object cuboid prediction quality with Auto-labelling on KITTI3D on Easy and Moderate samples. Results of Auto-labelling are taken from their paper. Best results are shown as bold numbers.\n\\vspace{-0.1cm}}\n\\label{tab:object_detection}\n\\end{table*}\n\n\n\\begin{figure*}[tbp]\n\\centering\n\\includegraphics[width=1.00\\textwidth]{figures\/labelling_ours\/ours-labelling-cropped.png}\n\\caption{A qualitative comparison of shape reconstruction and pose estimation against Auto-labelling~\\cite{Zakharov2020Autolabeling3D}. \\textbf{Left:} input RGB image. \\textbf{Middle:} result with DSP-SLAM \\textbf{Right:} result with auto-labelling~\\cite{Zakharov2020Autolabeling3D}\\vspace{-0.5cm}}\n\\label{fig:comparison-auto-labelling}\n\\end{figure*}\n\n\\subsection{\\label{joint_ba}Joint Bundle Adjustment}\n\nOur joint map consists of a set of camera poses $C = \\{\\matr{T}_{wc_i}\\}_{i=1}^{M}$, object poses $O = \\{\\matr{T}_{wo_j}\\}_{j=1}^{N}$ and map points $P = \\{^w\\matr{p}_k\\}_{k=1}^{K}$. Our joint BA can be formulated as a non-linear least squares optimization problem:\n\\begin{eqnarray}\n    C^*, O^*, P^* &{} = {}& \\mathop{\\arg\\min}_{\\{C, O, P\\}} \\sum_{i, j} \\norm{\\matr{e}_{co}(\\matr{T}_{wc_i}, \\matr{T}_{wo_j})}_{\\Sigma_{i,j}} \\nonumber \\\\\n    && {+}\\:\\sum_{i, k} \\norm{\\matr{e}_{cp}(\\matr{T}_{wc_i}, ^w\\matr{p}_k)}_{\\Sigma_{i,k}}\n\\label{eq:ba}\n\\end{eqnarray}\nwhere $\\matr{e}_{co}$ and $\\matr{e}_{cp}$ represent the residuals for camera-object and camera-point measurements and $\\Sigma$ is the co-variance matrix of measurement residuals. Objects act as additional landmarks, which results in improvements in tracking performance as shown in our evaluation on KITTI. The optimization is solved with Levenberg-Marquardt in g2o \\cite{g2o}.\n\n\\noindent\\textbf{Camera-Object Measurements:} Object-camera poses $\\matr{T}_{co}$ are evaluated by minimising the surface alignment term in Eq. \\ref{E_surf} while keeping  the shape code and scale fixed. New pose observations serve as edges between camera pose $\\matr{T}_{wc}$ and object pose $\\matr{T}_{wo}$, and the residual is defined as:\n $   \\matr{e}_{co} = \\log (\\matr{T}^{-1}_{co} \\cdot \\matr{T}^{-1}_{wc} \\cdot \\matr{T}_{wo})$\nwhere $\\log$ is the logarithm mapping from $\\matr{SE}(3)$ to $\\mathfrak{se}(3)$. Poses in the factor graph are 6-DoF, as object scale is only optimised when first detected.\n\n\\noindent\\textbf{Camera-Point Measurements:} We use the standard formulation of re-projection error used in ORB-SLAM2~\\cite{orbslam2}:\n$    \\matr{e}_{cp} = \\pi(\\matr{T}_{wc}^{-1} {^w\\matr{p}}) - \\matr{\\Tilde{u}}$, \nwhere $\\matr{\\Tilde{u}}$ is the measured pixel coordinate of map point $\\matr{p}$. We follow a similar strategy as ORB-SLAM2 to tune $\\Sigma_{ij}$ such that the two energy terms contribute roughly the same to the overall optimization. \n\\section{Related work} \n\n\\noindent\\textbf{Object-aware SLAM:}\nSLAM++ \\cite{SLAM++} pioneered object-aware RGB-D SLAM, representing the scene at the level of objects using a joint pose-graph for camera and object poses. A database of pre-scanned objects was created in advance and object instances were detected and mapped using a pre-trained 3D detector, ICP losses and pose-graph optimization. In later work, Tateno \\emph{et al.}~\\cite{Tateno2016When2I} aligned object instances from a pre-trained database to volumetric maps while Stuckler \\emph{et al.}~\\cite{stuckler2012model} performed online exploration, learning object models on the fly and tracking them in real time. An important drawback of instance-based approaches is their inability to scale to a large number of objects and their need for object models to be known in advance. More recent object-aware RGB-D SLAM systems have dropped the requirement for known models and instead take advantage of state-of-the art 2D instance-level semantic segmentation masks~\\cite{maskrcnn} to obtain object-level scene graphs~\\cite{semantic-fusion} and per-object reconstructions via depth fusion, even in the case of dynamic scenes~\\cite{maskfusion,midfusion}.\n\nExtensions of object-aware SLAM to the case of monocular video input deal with the additional challenge of relying only on RGB-only information~\\cite{mono-object-slam, mono-object-sparse-slam, quadricslam, cubeslam, category-specific-models} which results in the use of simplistic shape representations. In QuadricSLAM \\cite{quadricslam} objects are represented as ellipsoids and fit to monocular observations while in CubeSLAM \\cite{cubeslam} cuboid proposals generated from single-view detections are optimized in a joint bundle adjustment optimization. \n\nWhile the above SLAM systems represent an important step forward towards equipping robots with the capability of building semantically meaningful object-oriented maps, they fall short of exploiting semantic priors for object reconstruction. In this paper we take this direction of using a category-specific learnt shape prior and embed this within an object-aware SLAM system. \n\n\\noindent\\textbf{3D Reconstruction with Shape Priors:}\nThe use of learnt compact shape embeddings as priors for 3D reconstruction has a long tradition in computer vision. From 3D morphable models for the reconstruction of faces or bodies~\\cite{blanz1999morphable,loper2015smpl}, to PCA models to represent category specific object shape priors~\\cite{directshape}.\nOther examples of the use of embedding spaces for single or multi-view shape reconstruction include GPLVMs \\cite{dame2013dense, prisacariu2011shared, prisacariu2012simultaneous} or neural representations~\\cite{hu2018dense-embedding, zhu2018object} such as a variational autoencoder~\\cite{node-slam},  AltlasNet~\\cite{AtlasNet,2019-lin} or DeepSDF~\\cite{deepsdf,frodo,2019-xu,Zakharov2020Autolabeling3D}. DeepSDF \\cite{deepsdf} provides a powerful implicit learnt shape model that encapsulates the variations in shape across an object category, in the form of an auto-decoder network that regresses the signed distance function (SDF) values of a given object and has been used as a shape prior for single-view~\\cite{2019-xu} and multi-view~\\cite{frodo} reconstruction. Similarly to~\\cite{Zakharov2020Autolabeling3D} DSP-SLAM adopts DeepSDF as the shape prior and takes sparse LiDAR and images as input, however~\\cite{Zakharov2020Autolabeling3D} takes single frames and is not a SLAM method. DOPS~\\cite{Najibi2020DOPSLT} is a single-pass 3D object detection architecture for LiDAR that estimates both 3D bounding boxes and shape.  \n\nOur approach is most closely related to those that build consistent multi-object maps over an entire sequence such as FroDO~\\cite{frodo} and Node-SLAM~\\cite{node-slam}. Unlike FroDO ~\\cite{frodo} ours is a sequential SLAM system and not a batch method. Unlike Node-SLAM \\cite{node-slam}, in our system low-level point features and high-level objects are jointly optimized to bring the best of both worlds: accurate tracking and rich semantic shape information. DeepSLAM++~\\cite{hu2019deep-slam++} leverages shape priors in a SLAM pipeline by selecting 3D shapes predicted by Pix3D~\\cite{Sun_2018_CVPR}, but forward shape generation is often unstable and lead to poor results on real data. \n\\section{Experimental Results}\n\n\\begin{table*}\n\\centering\n\\setlength{\\arrayrulewidth}{0.5mm}\n\\resizebox{1.00\\textwidth}{!}{\n\\begin{tabular}{c c c c c c c c c c c c c}\n\\hline\n\\multirow{3}{*}{Approach} & \\multicolumn{11}{c}{Sequence} & \\multirow{3}{*}{Average} \\\\\n& 00* & 01 & 02* & 03 & 04 & 05* & 06* & 07* & 08* & 09* & 10 & \\\\\n& rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & rpe\/rre & \\\\\n\\hline\nSuMa++ \\cite{Chen2019SuMa++} & \\textbf{0.64}\/\\textbf{0.22} & 1.60\/0.46 & 1.00\/0.37 & 0.67\/0.46 & \\textbf{0.37}\/0.26 & 0.40\/0.20 & 0.46\/0.21 & \\textbf{0.34}\/\\textbf{0.19} & 1.10\/0.35 & \\textbf{0.47}\/\\textbf{0.23} & \\textbf{0.66}\/0.28 & \\textbf{0.70}\/0.29 \\\\\nOurs St+LiDAR (250pts) & 0.75\/\\textbf{0.22} & \\textbf{1.49}\/0.20 & \\textbf{0.79}\/\\textbf{0.23} & \\textbf{0.60}\/\\textbf{0.18} & 0.47\/0.11 & \\textbf{0.32}\/\\textbf{0.15} & 0.39\/0.21 & 0.52\/0.28 & \\textbf{0.94}\/\\textbf{0.27} & 0.79\/0.28 & 0.69\/\\textbf{0.26} & \\textbf{0.70}\/\\textbf{0.22} \\\\\nOurs St+LiDAR (50pts) & 0.80\/0.24 & 1.50\/\\textbf{0.15} & 0.84\/0.26 & 0.61\/0.18 & 0.44\/\\textbf{0.10} & 0.32\/0.16 & \\textbf{0.35}\/\\textbf{0.15} & 0.57\/0.24 & 1.03\/0.30 & 0.78\/0.27 & 0.67\/0.30 & 0.72\/\\textbf{0.22} \\\\\n\\hline\nORB-SLAM2 \\cite{orbslam2} & 0.70\/0.25 & \\textbf{1.38}\/0.20 & 0.76\/0.23 & 0.71\/0.17 & 0.45\/0.18 & \\textbf{0.40}\/\\textbf{0.16} & 0.51\/0.15 & \\textbf{0.50}\/\\textbf{0.28} & 1.07\/0.31 & \\textbf{0.82}\/0.25 & 0.58\/0.28 & \\textbf{0.72}\/0.22 \\\\\nSt DSO \\cite{Wang2017StereoDSO} & 0.84\/0.26 & 1.43\/\\textbf{0.09} & 0.78\/\\textbf{0.21} & 0.92\/\\textbf{0.16} & 0.65\/0.15 & 0.68\/0.19 & 0.67\/0.20 & 0.83\/0.36 & \\textbf{0.98}\/\\textbf{0.25} & 0.98\/\\textbf{0.18} & \\textbf{0.49}\/\\textbf{0.18} & 0.84\/\\textbf{0.20} \\\\\nSt LSD-SLAM \\cite{engel2015stereo-lsd} & \\textbf{0.63}\/0.26 & 2.36\/0.36 & 0.79\/0.23 & 1.01\/0.28 & \\textbf{0.38}\/0.31 & 0.64\/0.18 & 0.71\/0.18 & 0.56\/0.29 & 1.11\/0.31 & 1.14\/0.25 & 0.72\/0.33 & 0.91\/0.27 \\\\\nDynaSLAM \\cite{Bescos2018DynaSLAM} & 0.74\/0.26 & 1.57\/0.22 & 0.80\/0.24 & 0.69\/0.18 & 0.45\/\\textbf{0.09} & \\textbf{0.40}\/\\textbf{0.16} & 0.50\/0.17 & 0.52\/0.29 & 1.05\/0.32 & 0.93\/0.29 & 0.67\/0.32 & 0.76\/0.23 \\\\\nOurs St only & 0.71\/\\textbf{0.24} & 1.45\/0.30 & \\textbf{0.75}\/0.23 & 0.73\/0.19 & 0.47\/0.11 & 0.57\/0.23 & 0.57\/0.22 & 0.51\/0.29 & 1.02\/0.32 & 0.87\/0.26 & 0.65\/0.31 & 0.75\/0.25 \\\\\nOurs St only (5Hz) & 0.71\/0.26 & 1.43\/0.23 & 0.78\/0.24 & \\textbf{0.67}\/0.18 & 0.46\/\\textbf{0.09} & \\textbf{0.40}\/\\textbf{0.16} & \\textbf{0.47}\/\\textbf{0.14} & 0.52\/0.29 & 0.99\/0.31 & 0.90\/0.28 & 0.63\/0.31 & \\textbf{0.72}\/0.22 \\\\\n\\hline\n\\end{tabular}}\n\\caption{Comparison of camera tracking accuracy - average $t_{rel}$ [\\%] and $r_{rel}$ [\\si{\\degree}\/100m] against state-of-the-art stereo and LiDAR SLAM systems. Sequences marked with * contain loops. Note that Stereo-DSO is a purely visual odometry system, so their result is without loop closing. We keep it in the table for completeness. \\vspace{-0.5cm}}\n\\label{tab:traj-lp}\n\\end{table*}\n\nWe perform a quantitative evaluation of our novel prior-based object reconstruction optimisation, using LiDAR input on the KITTI3D Dataset \\cite{kitti3d}, comparing with auto-labelling~\\cite{Zakharov2020Autolabeling3D}, the most related approach. In addition, we evaluate the camera trajectory errors of our full DSP-SLAM system on both stereo+LiDAR and stereo-only input on the  KITTI Odometry~\\cite{kitti_odom} benchmark, comparing with state-of-the-art approaches.  We also provide qualitative results of our full SLAM system on pure monocular input on Freiburg Cars \\cite{freiburg-cars} and Redwood-OS \\cite{choi2016redwood} Chairs dataset. \n\n\\subsection{\\label{det3d}3D Object Reconstruction}\n\nWe conduct a quantitative comparison of our object pose estimation on the KITTI3D benchmark,  against auto-labeling~\\cite{Zakharov2020Autolabeling3D}, a recent approach to prior-based object shape and pose reconstruction based on image and LiDAR inputs, and using the same shape prior embedding (DeepSDF~\\cite{deepsdf}) and similar level of supervision (object masks and sparse depth from the LiDAR measurements).\n\n\n\\noindent\\textbf{Experimental Setting:} For a fair comparison, we evaluate our approach using a single image and LiDAR input and take the 2D segmentation masks and initial pose estimates from the  auto-labelling code release~\\cite{Zakharov2020Autolabeling3D} as initialization for our own optimization approach. We evaluate the results of pose estimation on the trainval split of KITTI3D which consists of 7481 frames, using  the same metrics  proposed in~\\cite{Zakharov2020Autolabeling3D}: BEV AP @ 0.50, 3D AP @ 0.50, and the distance threshold metric (NS) from the nuscenes dataset~\\cite{caesar2020nuscenes}.\n\n\\noindent\\textbf{Results:} We report quantatitive results in Tab.~\\ref{tab:object_detection}. Our method achieves better performance under almost all metrics, especially on harder samples. We also visualize the comparison of reconstructed shapes and pose in Fig.~\\ref{fig:comparison-auto-labelling}. Auto-labelling~\\cite{Zakharov2020Autolabeling3D} does not capture shape accurately for several vehicles: The first two cars on the left side are sedans, but auto-labelling~\\cite{Zakharov2020Autolabeling3D} reconstructs them as \"beetle\"-shaped. In addition, some of the cars on the right side are reconstructed with incorrect poses which do not align with the image. In contrast, DSP-SLAM obtains accurate shape and pose.\n\n\n\n\\noindent\\textbf{Timing Analysis:} To achieve close to real-time performance, we employ a Gauss-Newton solver with faster convergence than first-order methods during our optimization, leading to significant speed-ups.\nTab.~\\ref{tab:timing} shows a run-time comparison between a first-order optimizer and our Gauss-Newton solver with analytical gradients. Our method is approximately one order of magnitude faster to complete a single iteration, and requires fewer iterations to converge.\n\\begin{table}[tbp]\n\\centering\n\\begin{tabular}{c|l|c|c}\nMethod & Energy Terms & msec. \/ iter & \\# of iter \\\\ \\hline\n1st order & $E_{surf} + E_{rend}$ & 183 & 50 \\\\\n1st order & $E_{surf}$ & 88 & 50 \\\\ \\hline\nOurs GN & $E_{surf} + E_{rend}$ & 20 & 10 \\\\\nOurs GN & $E_{surf}$ & 4 & 10 \\\\ \\hline\n\\end{tabular}\n\\caption{Speed comparison between first-order optimization and our Gauss-Newton method with analytical Jacobians\\vspace{-0.5cm}}\n\\label{tab:timing}\n\\end{table}\n\n\\noindent\\textbf{Ablation Study:} We conducted an ablation study for DSP-SLAM with stereo+LiDAR input to analyse the effect of the number of LiDAR points used for shape optimization  on the reconstruction error.  Fig.~\\ref{fig:recon-num-pts} shows that there is no significant difference when reducing the number of LiDAR points from 250 to 50. The reconstruction quality starts to degrade when the number of points is further reduced to 10. \n\n\\subsection{\\label{full_slam}KITTI Odometry Benchmark}\n\nWe evaluate the camera trajectory error for our full DSP-SLAM system on the KITTI odometry benchmark with both stereo+LiDAR and stereo-only input. We evaluate on the 11 training sequences and compare with state-of-the-art SLAM systems of different input modalities using relative translation error $t_{rel}$ (\\%) and relative rotation error $r_{rel}$ (degree per 100m). Quantitative results are shown in Table~\\ref{tab:traj-lp}.\n\n\\noindent\\textbf{Stereo+LiDAR input:} \nThe upper part of Tab.~\\ref{tab:traj-lp} shows trajectory errors of our system with stereo+LiDAR input. Results suggest our method achieves comparable results with SuMa++, a state-of-the-art LiDAR-based semantic SLAM system~\\cite{Chen2019SuMa++}. Note however, that our method only takes very few LiDAR points (several hundred per frame) while SuMa++ uses a full LiDAR point-cloud. It is interesting to see the comparison between our stereo+LiDAR system and stereo ORB-SLAM2, which is used as our backbone system. With our LiDAR-enhanced object reconstruction and joint BA, tracking accuracy improves on most sequences, especially 03, 05, 06, 08 where adequate number of static objects are observed throughout the sequence.\nHowever, our system performs slightly worse on some sequences which contain only moving objects (01, 04) or long trajectory segments where no static objects are observed (02, 10).\nThe table also shows the effect on the  camera trajectory error when using 250 vs 50 points for object reconstruction. The results suggest that the impact of reducing the number of points on camera tracking accuracy is minimal.\n\n\\noindent\\textbf{Stereo-only input:} The lower part of Tab.~\\ref{tab:traj-lp} contains the results of our stereo-only system. It can be seen that our stereo-only system performs slightly worse than stereo ORB-SLAM2, which means dense shape reconstruction and joint BA does not help improve tracking accuracy with stereo-only input. We argue that the reason is two-fold. Firstly, 3D measurements based on stereo images are noisier than LiDAR-based measurements, giving rise to lower accuracy in object pose estimates.\nSecondly, in the stereo-only case, the surface points are obtained from the SLAM system, where the same features are repeatedly measured and not from multiple (LiDAR) measurements.\nWe also noticed that, to guarantee timings, we were performing bundle-adjustment less frequently than ORB-SLAM2. We re-ran DSP-SLAM, at a slightly reduced frame-rate (5Hz), performing BA after every key-frame (as ORB-SLAM2) and the average performance increased, matching ORB-SLAM2 at $0.72\/0.22$. \nA comparison with state-of-the-art stereo SLAM systems is also included in Tab.~\\ref{tab:traj-lp}. \n\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=1.00\\textwidth]{figures\/reconstruction_number_pts_single_row.png}\n\\caption{Object reconstruction results when using different number of LiDAR points per object (N=250, 50, 10). There is no noticeable difference when the number of points is reduced from 250 to 50. The reconstruction quality starts to degrade when further reducing to 10. The degraded parts are marked with a red circle.\n\\vspace{-0.1cm}}\n\\label{fig:recon-num-pts}\n\\end{figure*}\n\n\\subsection{Freiburg Cars \\& Redwood-OS Dataset}\n\n\\begin{figure*}[tbph!]\n\\centering\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-001.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-002.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-003.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/Frame-010.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_001.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_002.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_003.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/freiburg\/render_010.png}\n\\caption{Qualitative results on Freiburg Cars dataset\n\\vspace{-0.1cm}}\n\\label{fig:freiburg}\n\\end{figure*}\n\n\\begin{figure*}[htbp!]\n\\centering\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-01053.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-02484.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-09374.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/Frame-09647.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/01053-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/02484-new-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/09374-front-side.png}\n\\includegraphics[width=0.246\\textwidth]{figures\/redwood\/09647-front-side.png}\n\\caption{Qualitative results on Redwood-OS Chairs dataset\n\\vspace{-0.4cm}\n}\n\\label{fig:redwood}\n\\end{figure*}\n\nFinally, we evaluate our SLAM system with monocular input on the Freiburg Cars dataset \\cite{freiburg-cars} and Redwood-OS Chairs dataset. Both datasets consist of object-centric sequences with the camera moving around the object. Demonstrations can be seen on Fig.~\\ref{fig:freiburg} and~\\ref{fig:redwood} and in the supplementary video.\n\n\\noindent\\textbf{Experimental Setting:}\n3D Bounding boxes are estimated using PCA on the reconstructed surface points. Note that this approach cannot differentiate between the front and back side of the car. To address this issue, we initialize with two flipped hypothesis and keep the orientation that yields a smaller loss.\n\n\\noindent\\textbf{Results:} Fig.~\\ref{fig:freiburg} provides qualitative reconstruction results on 4 Freiburg Cars sequences. Our system is capable of reconstructing dense, accurate and high-quality shapes for cars solely from monocular input at 10-20 fps. Fig.~\\ref{fig:redwood} illustrates results on chairs from the Redwood-OS \\cite{choi2016redwood} dataset. Reconstruction accuracy is slightly worse than on cars as chairs have more complex shape variations. Results are promising nonetheless -- our method still produces dense meshes that capture the overall object shape from monocular RGB-only sequences, in quasi-real time. \n\n\n\n\n\n\n\\section{Conclusions}\n\nWe have presented DSP-SLAM, a new object-aware real-time SLAM system that exploits deep  shape priors for object reconstruction, produces a joint map of sparse point features for the background and dense shapes for detected objects. We show almost real-time performance on challenging real-world datasets such as KITTI (stereo and stereo+LiDAR), and even on monocular setting Freiburg cars and Redwood-OS. Our quantitative comparisons with competing approaches on camera trajectory estimation and shape\/pose reconstruction show comparable or superior performance to state of the art methods.\n\n\\vspace{-0.1cm}\n\n\\section*{Acknowledgements}\n\n\\vspace{-0.1cm}\n\nResearch presented here has been supported by the UCL Centre for Doctoral Training in Foundational AI under UKRI grant number EP\/S021566\/1. We thank Wonbong Jang and Adam Sherwood for fruitful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{%\n  \\@startsection\n    {section}%\n    {1}%\n    {\\z@}%\n    {0.8cm \\@plus1ex \\@minus .2ex}%\n    {0.5cm}%\n    {%\n      \\normalfont\\small\\bfseries\n      \\centering\n    }%\n}%\n\\def\\@hangfrom@section#1#2#3{\\@hangfrom{#1#2}\\MakeTextUppercase{#3}}%\n\\def\\subsection{%\n  \\@startsection\n    {subsection}%\n    {2}%\n    {\\z@}%\n    {.8cm \\@plus1ex \\@minus .2ex}%\n    {.5cm}%\n    {%\n     \\normalfont\\small\\bfseries\n     \\centering\n    }%\n}%\n\\def\\subsubsection{%\n  \\@startsection\n    {subsubsection}%\n    {3}%\n    {\\z@}%\n    {.8cm \\@plus1ex \\@minus .2ex}%\n    {.5cm}%\n    {%\n     \\normalfont\\small\\itshape\n     \\centering\n    }%\n}%\n\\def\\paragraph{%\n  \\@startsection\n    {paragraph}%\n    {4}%\n    {\\parindent}%\n    {\\z@}%\n    {-1em}%\n    {\\normalfont\\normalsize\\itshape}%\n}%\n\\def\\subparagraph{%\n  \\@startsection\n    {subparagraph}%\n    {5}%\n    {\\parindent}%\n    {3.25ex \\@plus1ex \\@minus .2ex}%\n    {-1em}%\n    {\\normalfont\\normalsize\\bfseries}%\n}%\n\\def\\section@preprintsty{%\n  \\@startsection\n    {section}%\n    {1}%\n    {\\z@}%\n    {0.8cm \\@plus1ex \\@minus .2ex}%\n    {0.5cm}%\n    {%\n      \\normalfont\\small\\bfseries\n    }%\n}%\n\\def\\subsection@preprintsty{%\n  \\@startsection\n    {subsection}%\n    {2}%\n    {\\z@}%\n    {.8cm \\@plus1ex \\@minus .2ex}%\n    {.5cm}%\n    {%\n     \\normalfont\\small\\bfseries\n    }%\n}%\n\\def\\subsubsection@preprintsty{%\n  \\@startsection\n    {subsubsection}%\n    {3}%\n    {\\z@}%\n    {.8cm \\@plus1ex \\@minus .2ex}%\n    {.5cm}%\n    {%\n     \\normalfont\\small\\itshape\n    }%\n}%\n \\@ifxundefined\\frontmatter@footnote@produce{%\n  \\let\\frontmatter@footnote@produce\\frontmatter@footnote@produce@endnote\n }{}%\n\\def\\@pnumwidth{1.55em}\n\\def\\@tocrmarg {2.55em}\n\\def\\@dotsep{4.5pt}\n\\setcounter{tocdepth}{3}\n\\def\\tableofcontents{%\n \\addtocontents{toc}{\\string\\tocdepth@munge}%\n \\print@toc{toc}%\n \\addtocontents{toc}{\\string\\tocdepth@restore}%\n}%\n\\def\\tocdepth@munge{%\n  \\let\\l@section@saved\\l@section\n  \\let\\l@section\\@gobble@tw@\n}%\n\\def\\@gobble@tw@#1#2{}%\n\\def\\tocdepth@restore{%\n  \\let\\l@section\\l@section@saved\n}%\n\\def\\l@part#1#2{\\addpenalty{\\@secpenalty}%\n \\begingroup\n  \\set@tocdim@pagenum{#2}%\n  \\parindent \\z@\n  \\rightskip\\tocleft@pagenum plus 1fil\\relax\n  \\skip@\\parfillskip\\parfillskip\\z@\n  \\addvspace{2.25em plus\\p@}%\n  \\large \\bf %\n  \\leavevmode\\ignorespaces#1\\unskip\\nobreak\\hskip\\skip@\n  \\hb@xt@\\rightskip{\\hfil\\unhbox\\z@}\\hskip-\\rightskip\\hskip\\z@skip\n  \\par\n  \\nobreak %\n \\endgroup\n}%\n\\def\\tocleft@{\\z@}%\n\\def\\tocdim@min{5\\p@}%\n\\def\\l@section{%\n \\l@@sections{}{section\n}%\n\\def\\l@f@section{%\n \\addpenalty{\\@secpenalty}%\n \\addvspace{1.0em plus\\p@}%\n \\bf\n}%\n\\def\\l@subsection{%\n \\l@@sections{section}{subsection\n}%\n\\def\\l@subsubsection{%\n \\l@@sections{subsection}{subsubsection\n}%\n\\def\\l@paragraph#1#2{}%\n\\def\\l@subparagraph#1#2{}%\n\\let\\toc@pre\\toc@pre@auto\n\\let\\toc@post\\toc@post@auto\n\\def\\listoffigures{\\print@toc{lof}}%\n\\def\\l@figure{\\@dottedtocline{1}{1.5em}{2.3em}}\n\\def\\listoftables{\\print@toc{lot}}%\n\\let\\l@table\\l@figure\n\\appdef\\class@documenthook{%\n \\@ifxundefined\\raggedcolumn@sw{\\@booleantrue\\raggedcolumn@sw}{}%\n \\raggedcolumn@sw{\\raggedbottom}{\\flushbottom}%\n}%\n\\def\\tableft@skip@float{\\z@ plus\\hsize}%\n\\def\\tabmid@skip@float{\\@flushglue}%\n\\def\\tabright@skip@float{\\z@ plus\\hsize}%\n\\def\\array@row@pre@float{\\hline\\hline\\noalign{\\vskip\\doublerulesep}}%\n\\def\\array@row@pst@float{\\noalign{\\vskip\\doublerulesep}\\hline\\hline}%\n\\def\\@makefntext#1{%\n \\def\\baselinestretch{1}%\n \\reset@font\n \\footnotesize\n \\leftskip1em\n \\parindent1em\n \\noindent\\nobreak\\hskip-\\leftskip\n \\hb@xt@\\leftskip{%\n  \\Hy@raisedlink{\\hyper@anchorstart{footnote@\\the\\c@footnote}\\hyper@anchorend}%\n  \\hss\\@makefnmark\\\n }%\n #1%\n \\par\n}%\n\\prepdef\n\\section{Introduction}\n\\label{sec::intro}\n\nWe observe our Universe primarily though thermal and non-thermal radiation as well as by means of atomic line transitions. The thermal component probes gas (including ionized and neutral components) \nin thermal equilibrium, while the non-thermal emission is emitted by particles such as leptons (e.g. electrons and positrons) \nor hadrons (e.g., protons and ions of heavier elements) that are not in thermal equilibrium.\nThanks to multi-wavelength observations of various astrophysical objects such as supernova\nremnants \\citep[SNRs,][]{2012Helder}, active galactic nuclei \\citep{2010Abdo},\ngamma-ray bursts \\citep{2009Abdo}, galaxy clusters \\citep{Brunetti2014,2019vanWeeren}, and galaxies \\citep{Beck2015}, observational insights\ninto the emission processes and the radiating particle\ndistributions in collisionless astrophysical plasmas is now possible.\nObserved non-thermal emission in our Universe spans a vast range of energies: from radio to TeV gamma-ray energies, and occurs in \nvarious astrophysical objects spanning many length scales: from sub-AU to Mpc scales.\nThe observed non-thermal power-law emission indicates that these plasmas are far from thermal equilibrium \nand suggests an efficient particle acceleration process giving rise to a population of cosmic rays\n(CRs) composed of electrons, protons, and heavier ions \\citep[][]{1987Blandford,Marcowith2016}. In our Galaxy, the acceleration of \nparticles near SNR shocks is thought to be the main mechanism for producing CRs up to particle energies of $10^{16}$eV \\citep[e.g.,][]{Gaisser2016}.\n\n\n\nThe interaction of plasmas on both sides of the shock wave produces electromagnetic perturbations.\nProtons and heavier ions, with speeds larger than a few times the shock speed, interact resonantly with these electromagnetic perturbations and, as a result, are transported diffusively in the vicinity of shock waves.\nBecause these magnetic perturbations are converging at the shock front, this leads to proton and ion acceleration through a process known as diffusive shock acceleration \\citep[DSA,][]{Krymskii1977,Axford1977,Blandford1978,Bell1978a,Bell1978b}. \nThe level of magnetic fluctuations determines the acceleration efficiency of particles, and vice versa, \naccelerated particles excite plasma instabilities \\citep{Bell2004} which further amplify the magnetic field near shock fronts.\n\nObservational evidences for electron acceleration at shocks are numerous. These include puzzling giant radio shocks that are observed at the outskirts of merging galaxy clusters.\nThese diffuse radio-emitting structures are often coincident with weak merger-driven shocks \\citep{2019vanWeeren}, and interpreted as synchrotron emitting regions powered by relativistic CR electrons that underwent the DSA process \\citep{Ensslin1998,Pfrommer2008,Kang2012,Pinzke2013}.\nMoreover, X-ray observations show that the non-thermal emission component at the edges of SNRs such as SN 1006 is due to accelerated CR electrons emitting synchrotron radiation \\citep{Willingale+1996}.\n\nThe particular morphology of SN 1006 in radio and X-rays has raised the question about the impact of the orientation of the upstream magnetic field on the acceleration mechanism of CR electrons. Radio polarization observations suggest a large-scale magnetic field that is aligned along the north-east to south-west direction \\citep{Reynoso2013}. Hence, the magnetic field seems to be aligned with the polar caps visible in radio \\citep{Dyer2009}, X-rays \\citep{Cassam-Chenaie2008} and TeV gamma rays \\citep{Acero2010}, suggesting preferentially quasi-parallel acceleration \\citep{Voelk2003}. Azimuthal variations of X-ray cutoff energies \\citep{Rothenflug2004, Katsuda2010} and the radius ratio of the forward shock to the tangential discontinuity \\citep{Cassam-Chenaie2008} reinforce this picture. Three-dimensional magneto-hydrodynamics (MHD) simulations support the interpretation of preferred quasi-parallel shock acceleration of electrons \\citep{Bocchino2011,Schneiter2015,Winner+2020} and protons \\citep{Winner+2020,Pais2020a,Pais2020b}.\n\n\n\n\nHowever, we still lack a detailed understanding of how electrons are accelerated to high energies at these shocks: for electrons and protons traveling with the same non-relativistic speed, the kinetic energy of electrons is much smaller than that of protons owing to their substantially smaller mass in comparison to protons. Hence, unlike protons, these electrons can not directly scatter off of the magnetic perturbations near shock waves and DSA is not possible for electrons without some process(es) that can provide and sustain pre-acceleration of incoming upstream electrons.\nThis is known as the electron injection problem in shocks \\citep{Amano+2010,Guo+2015}.\nThe importance of understanding the acceleration mechanism of electrons is also reinforced by their much higher radiative yield in comparison to that of protons at the same energy because the Thompson cross section scales inversely with particle mass squared.\n\nIn general, electron pre-acceleration proceeds in two steps: 1.\\ electrons are heated in the downstream region to energies much higher than their initial kinetic energy, and 2.\\ an acceleration phase where electrons scatter off of electromagnetic waves with a wavelength shorter than the shock width, which further increases the electron energy and enables them to participate in the DSA process.\nThe nature of these waves, which accelerate electrons in step 2, depends on the relative angle, $\\theta_{B}$, between the direction of the large-scale background magnetic field and the direction of the propagation of the shock \\citep[see, e.g.,][]{Guo+2015}.\nMoreover, recent global MHD modeling of the multi-wavelength emission from the supernova remnant SN 1006 shows that efficiency of electron acceleration at quasi-parallel ($0^{\\circ}<\\theta_{B} < 45^{\\circ}$) shocks has to be at least an order of magnitude higher in comparison to quasi-perpendicular ($45^{\\circ}<\\theta_{B} < 90^{\\circ}$) shock configurations \\citep{Winner+2020}.\n\n\nIn quasi-perpendicular shocks, electron pre-acceleration is thought to occur via shock drift acceleration (SDA) where electrons are\nreflected and accelerated via a combination of the cross shock electric potential and the shock rest frame motional electric field \\citep{Wu1984,Krauss+Wu1989}\nand\/or shock surfing acceleration (SSA) where electron acceleration occurs when they  interact with electrostatic waves at the shock front \\citep{Shimada+2000,Xu2020,KumarReville2021}.\nTheoretically, the SDA mechanism is shown to be inefficient for accelerating electrons at planar shocks in the so-called scattering-free limit \\citep{Ball+Melrose_2001}. However, in the presence of strong pitch-angle scattering, this mechanism is modified to become stochastic SDA, which could result in efficient acceleration \\citep{Katou+2019}. \nOn the other hand, particle-in-cell (PIC) simulations have shown that the SSA mechanism mediated by waves that are generated due to the Buneman instability is efficient in pre-accelerating electrons \\citep[see, e.g.,][]{Bohdan2019}. While these simulations used unrealistically low ion-to-electron mass ratios and (too) high {\\alf} speeds, it remains to be shown how these results can carry over to more realistic astrophysical conditions.\n\n\nFor quasi-parallel shocks, it is usually assumed that hot electrons generate whistler waves that lead to pitch-angle scattering and acceleration of these electrons. Efficient wave production in this scenario requires high values of {\\alf}ic Mach number, $\\mathcal{M}_{\\rm A}$. However, when tested with PIC simulations, this scenario did not result in any significant electron acceleration \\citep{Riquelme+2011,Niemiec+2012}. \nRecently, \\citet{sharp2} discovered a new instability (called the intermediate-scale instability) that drives comoving ion-cyclotron waves unstable in the vicinity of the shock front. This presents {\\it a new mechanism for generating waves} that can scatter electrons and potentially enables an efficient DSA mechanism for electrons. Unlike the whistler-mediated case, this mechanism requires low values of $\\mathcal{M}_{\\rm A}$ which is a condition for the new instability to operate. \n\nIn this paper we test this mechanism using 1D3V PIC simulations (i.e., one spatial and three velocity-space dimensions) of parallel electron-ion shocks and show that it indeed leads to very efficient electron acceleration.\nThe paper is organized as follows.\nIn Section~\\ref{sec::setup}, we present the setup for our simulations, and compute the linear growth rates for the expected unstable wavemodes at the shock front region.\nThe growth of these wavemodes is responsible for the formation of the shock and for creating non-thermal populations of electrons and ions.\nIn Section~\\ref{sec:Bamp}, we present the evolution of density and magnetic field amplification  in our simulations.\nWe study the impact of the intermediate-scale instability driven wavemodes on the acceleration of electrons in the downstream region of the shock in Section~\\ref{sec:nth1}.\nIn Section~\\ref{sec:thermal}, we discuss the heating of the shocked plasmas in the downstream and the shock front region. We also compare this heating to analytical heating models at shocks.\nThe fraction of energy that is channeled into non-thermal  electrons and ions is quantified and its evolution is presented in Section~\\ref{sec:nothermal}. We discuss our findings and their implications in Section~\\ref{sec:dis}, and conclude the paper in Section~\\ref{sec:concl}.\nThroughout this paper, we assume the SI system of units.\n\n\n\n\\section{Non-relativistic shock simulations}\n\\label{sec::setup}\n\n\\begin{figure}\n\\includegraphics[width=8.8cm]{Plots\/Fig01.pdf}\n\\caption{The formation of a strong parallel shock in the contact discontinuity rest frame (simulation frame).\nLeft-hand side: the reflection of an incident electron-ion plasma drifting to the left side of the simulation with speed $\\varv_u$ leads to the formation of the shock.\nRight-hand side: after shock formation, the shock speed relative to the upstream plasma is $\\varv_{\\rm sh} \\sim 4 \\varv_u\/3$ (assuming a strong shock and adopting the Rankine-Hugoniot jump conditions), and the density jump at the shock front is on average $\\sim 4 n_0$, where $n_0$ is the number density of the far upstream plasma.\nThat is, at the shock front the {\\alf} speed of ions $\\varv_{\\rm A}$ is $0.5$ of that in the far upstream, which means that the {\\alf}ic Mach number, $\\mathcal{M}_{\\rm A}$, in this region is twice of that in the far upstream.\n\\label{fig:ShockSketch}}\n\\end{figure}\n\nHere we discuss the setup for our shock simulations and compute the scales and the growth rates for the expected unstable linear wavemodes at the shock front regions.\n\n\\subsection{Simulation setup}\n\nWe perform 1D3V particle-in-cell (PIC) simulations using the SHARP code \\citep{sharp,resolution-paper,sharp2}, where the shock is formed by reflecting the upstream electron-ion plasma (initially moving towards the left, i.e., negative $\\bs{\\hat{\\vec{x}}}$ direction) at $x=0$.\nThat is, the shock is formed by the interaction of two exact replica of the same plasma moving in opposite directions, and the simulations are performed in the rest-frame of the contact discontinuity. A sketch for the initial configuration and the resulting shock formation is shown in Figure~\\ref{fig:ShockSketch}.\n\nIn all simulations, the upstream plasma skin-depth is resolved by 10 computation cells, and the CFL stability condition on the computational time step, $\\Delta t$, is such that $c \\Delta t = 0.45   ~ \\Delta x$, where $c$ is the speed of light in vacuum.\nWe use 200 particles per species in each computational cell.\nThe boundary of the computational domain expands with time on the right-hand side to save computational cost while containing all energetic particles in the precursor of the shock.\nThe upstream  plasma is initially moving with velocity $\\varv_u = -0.1 c$, and both electrons (with mass $m_e$) and ions (with mass $m_i$) have the same physical temperature  $T_e = T_p = 4 \\times 10^{-8} m_i c^2\/k_{\\rm B}$, where $k_{\\rm B}$ is Boltzmann's constant.\nInitially, both species have the same uniform number density $n_i=n_e=n_0$. \n\n\nFor such shocks, the expected shock speed (in the rest-frame of upstream plasma) is \n$\\varv_{\\rm sh} = 4 |\\varv_u| \/3 \\sim 0.133 c$.\nTherefore, the sonic Mach number $ \\mathcal{M}_s = \\varv_{\\rm sh} \/ \\varv_s \\sim \n365$, where the sonic speed is $\\varv_s = \\sqrt{\\Gamma_{\\rm ad} k_{\\rm B} (T_e+T_p)\/m_i \n}$, and  $\\Gamma_{\\rm ad} = 5\/3$ is the adiabatic index. That is, all shock simulations presented here have high sonic Mach number.\nThe initial (large-scale) background magnetic field $\\vec{ B}_0 = B_0  \\bs{\\hat{\\vec{x}}}$ is such that the {\\alf}ic Mach number $\\mathcal{M}_{\\rm A} = \\varv_{\\rm sh}\/\\varv_{\\rm A}$, where $\\varv_{\\rm A} = B_0\/\\sqrt{ \\mu_0 n_i m_i}$ is the {\\alf} speed of ions.\n\n\nIt is important to note here that collisionless shocks found in the intracluster medium have $\\mathcal{M}_s \\sim 1$ to 3 \\citep{Ryu+2003, Pfrommer+2006,Vazza+2009,Schaal+2016}, and for such low values of the sonic Mach number, the intermediate-scale growth rates are much larger in comparison to that at the gyro-scale \\citep{sharp2}. This implies a stronger impact of the intermediate-scale unstable modes on the acceleration of electrons in the intracluster medium.\nHowever, we leave a demonstration of this point in simulations to future works.\n\n\n\\subsection{Unstable linear wavemodes at the shock front}\n\\label{sec::theory}\n\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig02.pdf}\n\\caption{Top: growth rates ($\\Gamma$) normalized to the non-relativistic ion gyro-frequency, $\\Omega_i$.\nBottom: the phase velocity ($\\varv_{\\rm ph}$) of the fastest growing parallel (unstable) electromagnetic wavemodes created by the penetration of upstream cold plasma drifting  into the denser plasma at the shock front (in the simulation frame).\nThe solutions are shown for $\\mathcal{M}_{\\rm A} = 21.3$, with a realistic ion-to-electron mass ratio, $m_r=m_i\/m_e$ (blue curves).\nIn the case of $\\mathcal{M}_{\\rm A} = 5.3$, solutions are shown for a realistic value of $m_r$ (red curves) and a reduced $m_r$ (black curves).\nOnly the case with $\\mathcal{M}_{\\rm A}=5.3$ and a realistic mass ratio is expected to excite the intermediate-scale instability~\\citep{sharp2} and thus the small scale ion-cyclotron wave modes that are comoving with the upstream plasma with velocity $\\varv_u = -0.1 c$ are amplified, i.e., come from solutions of $D^{+}=0$ (see Equation~\\ref{eq:disp}).\nSmall scale unstable wavemodes in the other (blue and black) cases are whistler waves, i.e., they derive from solutions of $D^{-}=0$ (see Equation~\\ref{eq:disp}).\n\\label{fig:GrowthRate}}\n\\end{figure}\n\n\nTo study the impact of the intermediate-scale instability on the heating and the acceleration of electrons at parallel non-relativistic shocks, we present simulations that differ in their ability to excite the intermediate-scale instability, which manifests itself by exciting ion-cyclotron waves that are comoving with the upstream plasma~\\citep{sharp2}.\nThe upstream drifting plasma drives wavemodes unstable at the shock front\\footnote{In the linear dispersion calculation we present here, the background number density is assumed to be uniform.\nNumber density non-uniformity could change the growth rates if the non-uniformity scale is close to the wavelengths  of the unstable modes.\nHowever studying this theoretically, even in the linear regime, is tedious \\citep[see, e.g.,][]{sim_inho_18, th_inho_20}.}. \nAssuming gyrotropic momentum distributions for various species, the \ndispersion relation, in the simulation frame, for parallel (w.r.t.\\ the background magnetic field $B_0$) propagating electromagnetic wavemodes with wavenumber $k$ and complex frequency \n$\\omega$ can be written as~\\citep{Schlickeiser+2002}\n\\begin{eqnarray}\nD^{\\pm}\n&=& 1- \\frac{k^2c^2}{\\omega ^2} \n\\nonumber \\\\\n&&+\n\\sum_{s=1}^{4}\n\\frac{ \\omega_s^2 }{ \\gamma_s  \\omega ^2 }\n\\left[\n\\frac{ \\omega -k \\varv_{\\parallel,s} }\n{  k \\varv_{\\parallel,s}  - \\omega  \\pm \\Omega_s}\n-\n\\frac{ \\varv_{\\perp,s}^2  \\left(k^2c^2-\\omega ^2\\right) \/c^2 }\n{2 \\left(k \\varv_{\\parallel,s}  -\\omega \\pm \\Omega_s \\right)^2}\n\\right] ~\\hspace{-0.16cm} .\n\\label{eq:disp}\n\\end{eqnarray} \nHere, $s=\\{1,2\\}$ denote the shocked electron and ion plasmas at the shock front, respectively, and $s=\\{3,4\\}$ denote the incoming (upstream) cold electron and ion plasmas, respectively. The drift speeds are $\\varv_{\\parallel,s} = \\{ - \\varv_{u}\/3, - \\varv_{u}\/3, \\varv_{u}, \\varv_{u} \\}$, \nand the relativistic gyro-frequencies $\\Omega_s = \\{- m_r \\Omega_0, \n\\Omega_0, -m_r \\Omega_0, \\Omega_0\\}\/\\gamma_s$, where \n$\\Omega_0 = e B_0\/m_i $ is the non-relativistic ion gyro-frequency, and \nthe Lorentz factor is $\\gamma_s = (1 - \\varv_{\\parallel,s}^2 - \\varv_{\\perp,s}^2)^{-1\/2}$.\n\nSolutions to $D^{+}=0$ ($D^{-}=0$) are solutions for the left (right) handed polarized electromagnetic wavemodes.  When solving for the maximum growth rates at some $k$, we solve the dispersion relation with both signs, and find the fastest growing mode regardless of its polarization.\nFor various species $s$, the perpendicular speeds are $\\varv_{\\perp,s} = \\varv_s\/\\sqrt{2} ~ \\forall s $,\nthe plasma frequencies are $\\omega_s = \\{ \\sqrt{3 m_r} ~\\omega_i, \n\\sqrt{3}~ \\omega_i, \\sqrt{m_r} ~\\omega_i, \\omega_i \\}$, where $\\omega^2_i= e^2 n_i\/(m_i \\epsilon_0)$ is the square of the ion plasma frequency in the far upstream of the shock, $e$ is the elementary charge, and $\\epsilon_0$ is the permittivity of free space. \n\nWe present a simulation that can excite the intermediate-scale instability at the shock front region. It has an upstream {\\alf}ic Mach number $\\mathcal{M}_{\\rm A} = 5.3$ and uses a realistic mass ratio $m_r = m_i\/m_e=1836.$\nThat is, the shock front {\\alf}ic Mach number \n\\begin{eqnarray}\n\\mathcal{M}^f_{\\rm A} \\sim 10.6 < \\sqrt{m_r}\/2 \\sim  21.4 .\n\\end{eqnarray}\nThis condition represents the necessary criteria for driving such comoving ion-cyclotron wave modes \\citep{sharp2}.\nThe solution of the dispersion relation (Equation~\\ref{eq:disp}) for \nthis case is shown in Figure~\\ref{fig:GrowthRate} (red curves).\nThe lower panel shows that the phase velocity of the unstable wave modes (at $kc\/\\omega_i \\sim 5$) are comoving with the upstream drifting plasmas ($\\varv_{\\rm ph} \\sim -0.1 c$) for the simulation with $\\mathcal{M}_{\\rm A} = 5.3$ and $m_r=1836$ (red curve).\nTo demonstrate the importance of this instability in electron-heating, we present two more simulations where the condition of this instability is violated.\n\n\\begin{deluxetable}{ cccccc }\n\\tablewidth{8.7cm}\n\\tabletypesize{\\footnotesize}\n\\tablecolumns{5} \n\\tablecaption{\nParameters of our electron-ion parallel shock simulations \\vspace{-0.2cm}\n\\label{table:Sims}\n}\n\\tablehead{ Name & $\\varv_u\/c$\\tablenotemark{a} &  $\\mathcal{M}_{\\rm A}$\\tablenotemark{b}\n&   $\\mathcal{M}_s$\\tablenotemark{c}\n&    $m_i\/m_e$ \n& Condition\\tablenotemark{d} \n}\n\\startdata\n\\rule{0pt}{12pt}  Ma5Mr1836  & -0.1 & 5.3 & 365 &  1836 & \\checkmark \n\\\\\n\\rule{0pt}{12pt}  Ma5Mr100\t & -0.1 & 5.3 & 365 &  100\t  & $\\times$ \n\\\\\n\\rule{0pt}{12pt}  Ma21Mr1836 & -0.1 & 21.3 & 365 & 1836 & $\\times$\n\\enddata\n\\tablenotetext{a}{Upstream plasma velocity in the contact-discontinuity rest frame.}\n\\tablenotetext{b}{Upstream {\\alf}ic Mach number.}\n\\tablenotetext{c}{Upstream sonic Mach number.}\n\\tablenotetext{d}{This shows whether the condition ($\\varv_{\\rm sh}\/\\varv_{\\rm A} < \\sqrt{m_r}\/2 $) for exciting the intermediate-scale instability at the shock front is satisfied.}\n\\end{deluxetable}\n\nThe first simulation has $\\mathcal{M}_{\\rm A} = 21.3$, with a realistic mass ratio, $m_r=1836$.\nIn this case, $\\mathcal{M}_{\\rm A}^f \\sim 40.6 > \\sqrt{m_r}\/2 \\sim  21.4$. \nIndeed, the solutions of the dispersion relation for this case (blue curves in Figure~\\ref{fig:GrowthRate}) show that sub ion-skin-depth unstable whistler wavemodes are  not comoving with the upstream plasma and thus can be easily quenched.\nFor this simulation, we increase $\\mathcal{M}_{\\rm A}$ by decreasing $B_0$, i.e., by lowering $\\varv_{\\rm A}$. \n\n\n\n\n\nThe second simulation has $\\mathcal{M}_{\\rm A} = 5.3$, but with a reduced mass ratio, $m_r=100$. In this case, $\\mathcal{M}_{\\rm A}^f \\sim 10.3 > \\sqrt{m_r}\/2 =  5 $.\nAgain, the solutions of the dispersion relation for this case (black curves in Figure~\\ref{fig:GrowthRate}) show that the sub ion-skin-depth unstable whistler wavemodes are not comoving with the upstream plasma and thus can also be easily quenched.\nFor this simulation, we increase $T_e=T_i$ such that $T_i\/m_i$ and thus the sonic speed $\\varv_s$ is unchanged. \nConsequently, the sonic Mach number $\\mathcal{M}_s$ is also unchanged.\n\n\n\n\nA summary of parameters of the three simulation that we present here is given in Table~\\ref{table:Sims}.\nIt is important to note that larger values of the number density at the shock front, i.e., $n > 4n_0$, increases $\\mathcal{M}_{\\rm A}^f$ in simulations. Hence, we have added a safety margin to our chosen values of $\\mathcal{M}_{\\rm A}^f$ that can in principle account for on additional factor of four enhancement in density over its MHD prediction without changing our conclusion on the condition of whether the intermediate-scale instability is excited in our simulations, as given in Table~\\ref{table:Sims}.\n\n\n\n\\section{Shock formation and magnetic field amplification; self-driven scattering centers}\n\\label{sec:Bamp}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig03.png}\n\\caption{Left: evolution of the number density normalized to the far upstream number density ($n_0$).\nRight: evolution of $B_y$ normalized to the far upstream parallel magnetic field ($B_0$). \nThese are the evolution in the simulation Ma5Mr1836 (top),  Ma5Mr100 (middle), and Ma20Mr1836 (bottom). \n\\label{fig:XT-evol}\n}\n\\end{figure*}\n\n\n\nThe interaction of the drifting and reflected plasma results in instabilities on scales both longer and shorter than the ion-skin depth.\nSuch unstable modes slow down the reflected plasma and thus create an over-density ($n> 2 n_0$) behind the shock transition region\\footnote{Here, $n$ is the number density of both ions and electrons together.}, where $n_0$ is the far upstream number density.\nAfter the formation of the shock, the interaction of the incoming upstream plasma with the denser plasma behind the shock also forms an unstable plasma configuration.\nThis leads to particle scattering and a constant heating and acceleration of the upstream plasma as it is swept over by the shock.\nThe left-hand side of Figure~\\ref{fig:XT-evol} shows the time evolution of the number density, $n$, normalized to $n_0$, in all the simulations.\nThe right-hand side shows the time evolution of $B_y$ normalized to the initial background (parallel) magnetic fields $B_0$ (the $B_z$ evolution is very similar to that of $B_y$, albeit shifted slightly in space).\nFigure~\\ref{fig:XT-evol} shows the formation of the shock via the excitation of unstable magnetic field wavemodes in various simulations.\nAfter the formation of the shock, wavemodes on small and large scales are unstable as seen in Figure~\\ref{fig:GrowthRate}. These modes are continuously driven at the shock front region as the shock swipes through the upstream plasma.\n\nIn Figure~\\ref{fig:XT-evol}, the white lines that separate the shock front and the upstream regions indicates the location of the shock in various simulations. While the exact position of the white line is determined visually, we note that its slope is approximately given by \n\\begin{eqnarray} \n\\frac{ \\Delta t \\Omega_i }{  \\Delta x \\omega_i\/c }  =\n\\frac{ \\Omega_i\/\\omega_i }{ \\varv_{\\rm sh}  \/c }   \n= \\frac{ 1}{\\mathcal{M}_{\\rm A} }.\n\\end{eqnarray}\nThat is, the slope for the Ma20Mr1836 simulation is smaller by a factor of $\\sim 4$ in comparison to the other simulations, and thus the range in  $x$-direction is larger by the same factor.\nWe define the shock front (transition) to be a region that is 200 $c\/\\omega_i$ wide behind the location of the shock, i.e., it is the region between the two white lines in various panels of Figure~\\ref{fig:XT-evol}.\n\nAs particles (especially ions) scatter back and forth across the shock front (transition) region, they are accelerated to higher energies, and thus some particles escape from this region toward the upstream plasma. \nThese excites unstable wavemodes, which  in turn scatter most of these counter-streaming energetic particles back toward the shock front region \\citep{Bell1978a}.\nThis process generates the so called shock precursor region ahead of the shock in the upstream plasma as seen in the right-hand panels of Figure~\\ref{fig:XT-evol}.\nIn the next sections, we closely look at the nature of these driven modes and their impact on particle acceleration and heating for both ions and electrons.\n\nThe overall density jump between the far upstream and far downstream region ($n\/n_0$) is notably larger than expected from the MHD jump condition that predicts $n\/n_0= 4$ (for a strong parallel shock). Instead, our simulations with $\\mathcal{M}_{\\rm A}=5.3$ show an overall density jump of $n\/n_0 \\sim 6$.\nThis has already been seen in a number of PIC simulation before as summarized by \\citet{Bret2020}.\nIn hybrid-PIC simulations, where the jump in the density is not constrained by using a very stiff electron equation of state, \n\\citet{Colby+2019} shows that for a simulation with $\\mathcal{M}_{\\rm A}=20$ the density jump grows to $n\/n_0  \\gtrsim 5.5$.\n\n\n\n\\section{Impact of intermediate-scale instability driven wavemodes on acceleration}\n\\label{sec:nth1}\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig04.pdf}\n\\caption{Top: downstream electron (solid) and ion (dashed) momentum spectra at $t ~\\Omega_i= 260$ from Ma5Mr1836 (red),  Ma5Mr100 (black),  Ma20Mr1836 (blue) simulations.\nThe downstream is defined to be at a distance that is larger than 200 $c\/\\omega_i$ from the shock front; see Figure~\\ref{fig:XT-evol}.\nBottom: downstream perpendicular magnetic energy in Fourier space at $t ~\\Omega_i= 260$ .\nThis shows that the co-moving (traveling towards the downstream) unstable waves that are driven at the shock front in the Ma5Mr1836 simulation generate a much higher level of small-scale magnetic fields. These increased magnetic fluctuations imply a much stronger scattering and thus, more efficient acceleration of electrons in comparison to the other simulations.\n\\label{fig:spectra300}}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.5cm]{Plots\/Fig05.png}\n\\caption{Time evolution of the average perpendicular magnetic field power, $\\langle \\delta B^k_{\\perp} \\rangle$, in the downstream (left) and shock front (right) regions.\nThe dashed lines indicate the small-scale ($k c\/\\omega_i > 2$) time evolution while solid lines show the large-scale ($k c\/\\omega_i< 2$) time evolution. \nDifferent simulations are indicated with different colors; Ma5Mr1836 with red,  Ma5Mr100 with black, and   Ma20Mr1836 with blue.\n\\label{fig:KLS}\n}\n\\end{figure*}\n\nThe importance of the intermediate-scale instability on the particle acceleration process can be quantified by comparing simulations which differ in their ability to excite comoving ion-cyclotron modes via the intermediate-scale instability.\nAs seen in Figure~\\ref{fig:GrowthRate}, all simulations are predicted to have unstable wavemodes on scales smaller than the ion skin depth at the shock front (transition) region.\nHowever, only in the simulation Ma5Mr1836, these  modes are comoving with the incoming flow (red, see the bottom panel of Figure~\\ref{fig:GrowthRate}), and hence their mode power can be transferred to the downstream region where they can scatter electrons.\n\nThe top panel of Figure~\\ref{fig:spectra300} shows the electron and proton spectra as a function of dimensionless velocity $u_s\/c$. To obtain these spectra, we first transform to the frame in which the plasma is at rest before we compute the momentum spectra from which we estimate the plasma temperature as laid out in Appendix~\\ref{app:temp}. \nAs shown in the top panel of Figure~\\ref{fig:spectra300}, the simulation Ma5Mr1836 (where the intermediate-scale instability can grow) has a much higher efficiency in converting the incoming flow kinetic energy into non-thermal energy in electrons in the downstream region. In fact, it is more efficient (by about 2 orders of magnitude) in accelerating electrons in comparison to the simulation with a higher $\\mathcal{M}_{\\rm A}$ (blue), suggesting that there is a much more efficient process at work that enables electrons to be efficiently accelerated in our low-$\\mathcal{M}_{\\rm A}$ model (red).\n\nMoreover, the Ma5Mr100 simulation (black) has the same $\\mathcal{M}_{\\rm A}$ and $\\mathcal{M}_{\\rm s}$ as the simulation shown in red but uses a lower mass ratio: $m_r=100$.\nAs shown in the top panel of Figure~\\ref{fig:spectra300}, this results in a much smaller electron acceleration efficiency in this simulation.\nAdditionally, the heating of electrons in the downstream region of the simulation with a reduced mass ratio (black) is much smaller compared to the simulations with a realistic mass ratio (red and blue)\\footnote{In Appendix~\\ref{app:temp}, we show that the normalized temperature, $k_{\\rm B} T_s\/m_s c^2$, of the thermal part of the plasma population of species $s$ is proportional to the value of $u\/c$ where the $u^4 f(u)$ is maximized, see Equation~\\eqref{Eq:TR}.}.\nThat is, the top panel of Figure~\\ref{fig:spectra300} demonstrates that using a lower mass ratio results not only in an artificially suppressed electron acceleration efficiency but also leads to an erroneous electron heating in the downstream region.\n\nAs we will now argue, the differences in the acceleration efficiency of electrons as well as for particle heating at shocks is a direct consequence of the differences in the nature of the unstable small-scale modes.\nThe bottom panel of Figure~\\ref{fig:spectra300} shows that the small-scale power of the perpendicular magnetic field in the downstream region is much larger in the simulation model Ma5Mr1836 in comparison to the simulations Ma5Mr100 (black) and Ma21Mr1836 (blue).\nIn the latter two cases, the small-scale unstable modes are not comoving with the incoming flow, and thus can be easily quenched, which leads to a much smaller amount of small-scale perpendicular magnetic power in the downstream regions.\n\nTo demonstrate the growth of these small scale wavemodes, in Figure~\\ref{fig:KLS} we plot the time evolution of the average perpendicular magnetic field for small (dashed) and large (solid) scale wave modes in various simulations. To this end, we take the power of the perpendicular magnetic field as shown in the bottom panel of Figure~\\ref{fig:spectra300} and compute the average for larger scales, i.e., for $kc\/\\omega_i<2$ and for small scales, i.e., for $2<kc\/\\omega_i<20$.\nThe top panel of Figure~\\ref{fig:GrowthRate} predicts that in the linear regime, both small and large scale magnetic perturbations grow with similar rates in all simulations at the shock transition region. \nThe right-hand panel of Figure~\\ref{fig:KLS} demonstrates that this prediction is in an excellent agreement with the growth of magnetic perturbations in various simulations at the shock transition region. \n\n\nOn the other hand, the left-hand panel of Figure~\\ref{fig:KLS} demonstrates the crucial importance of the difference in the  nature of the grown small scale wavemodes.\nAs shown in the bottom panel of Figure~\\ref{fig:GrowthRate}, only for the Ma5Mr1836 simulation (red), the small-scale wavemodes are comoving with the incoming flow. This means that these modes are not easily quenched in the shock transition region, which is demonstrated by the ability of maintaining a high level of small-scale fluctuations as a result of continued growth of the instability, which explains the much larger efficiency of electron acceleration in this simulations, see Figure~\\ref{fig:spectra300}. \nMoreover, much of this small-scale electro-magnetic power is transferred to the downstream region as can be inferred from the left-hand panel of Figure~\\ref{fig:KLS}.\nIn the case of the simulations without the intermediate-scale instability (Ma5Mr100 and Ma21Mr1836), much of the small-scale power is quenched and thus a much smaller fraction of this is transferred to the downstream region of the shock.\n\n\n\n\nIn the next sections, we show that the use of a lower mass ratio also leads to erroneous heating and acceleration efficiency  in the shock front (transition) region, and that holds true throughout the evolution of the simulations (as seen in Figures~\\ref{fig:Temp} and \\ref{fig:Eff}).\n\n\n\\section{Heating of the shocked plasmas}\n\\label{sec:thermal}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig06.pdf}\n\\caption{Top: shown is the time evolution of downstream (left) and shock front (right) temperatures of ions (dashed) and electrons (solid) in various simulations. Dotted red (black) line in the top left-hand panel indicate the predicted MHD ion temperature for $m_r = 1836 ~ (100)$ as given in Equation~\\eqref{eq:kTiV}. \nBottom: evolution of the electron-to-ion temperature ratio of the plasma in the downstream (left) and shock front (right) regions in various simulations.\n\\label{fig:Temp}\n}\n\\end{figure*}\n\n\nAll simulations presented here only have a parallel background magnetic field and are characterized by the same sonic Mach number $\\mathcal{M}_s \\sim 365$.\nAfter shock formation, electrons and ions in the downstream and shock front regions are heated. Modeling the saturation temperatures for electron and ions  in such collisionless shocks is very important for understanding the observations of SNRs~\\citep{Vink+2015}, clusters of galaxies~\\citep{Russell+2012}, and the warm hot intergalactic medium which contains a significant fraction of the baryon in the universe~\\citep{Bykov+2008}.\n\n\\subsection{Models for saturation temperatures}\n\nThe MHD model for strong parallel shocks predicts that all the upstream drifting kinetic energy is converted into thermal energy in the post-shock rest frame.\nThe expected temperature can be computed using the Rankine-Hugoniot jump conditions for a strong, parallel MHD shock in the regime of high sonic Mach numbers, $\\mathcal{M}_s\\gg1$ \\citep[e.g.,][]{boyd}, so that  the temperature of the shocked regimes (downstream and shock front regimes) is given by\n\\begin{align}\nk_{\\rm B} T =  \\frac{3}{16} \\mu m_i \\varv_\\mathrm{sh}^2\n  = \\frac{1}{3} \\mu m_i \\varv_u^2 = \\frac{1}{600} m_r m_e c^2,\n  \\label{eq:kTi}\n\\end{align}\nwhere $\\varv_\\mathrm{sh}=4\\varv_u \/3$ is the upstream velocity in the shock frame, $\\varv_u=0.1c$ is our adopted value for the upstream velocity in the simulation frame, we adopted the adiabatic index $\\Gamma_\\mathrm{ad}=5\/3$ appropriate for our case, and $\\mu = 1\/2$ is the mean molecular weight of an electron-proton fluid.\nThe MHD assumption implies that the ion and electron temperatures equilibrate on negligible timescales so that $T_e=T_i$ in this model.\nThat is, we expect the temperature in the shocked regimes to depend on the adopted mass ratio as follows:\n\\begin{equation}\nk_{\\rm B}  T_{i} =  k_{\\rm B}  T_{e} =  \\left\\{ \n  \\begin{aligned}\n    &3.06~~~ m_e c^2\n    &&\\mbox{  for  } m_r = 1836,\\\\\n    &0.166~ m_e c^2\n    &&\\mbox{  for  } m_r = 100.\\\\\n  \\end{aligned}\n  \\right.\n    \\label{eq:kTiV}\n\\end{equation}\nThese values are shown with dotted red ($m_r=1836$) and black ($m_r=100$) lines in the top left-hand panel of Figure~\\ref{fig:Temp}.\n\nBecause these shocks are collisionless, the MHD assumption of strong collisional relaxation is not fulfilled and the electrons may not equilibrate with the ions.\nTherefore, another theoretical model for saturation temperatures can be obtained by assuming that thermalization of the initial flow kinetic energy occurs separately for ions and electron \\citep{Vink+2015}.\nThis implies that the electron-to-ion temperature $T_e\/T_i = m_e\/m_i$, and for our simulations this predicts $k_{\\rm B} T_e  = 0.0016 ~ m_e c^2$ (where we used Equation~\\ref{eq:kTi}). \n\n\n\\subsection{Saturation temperatures in simulations}\n\nIn Figure~\\ref{fig:Temp} we show the evolution of the temperature in the downstream (left) and shock front (right) regions in various simulations.\nThe top panel of Figure~\\ref{fig:Temp} shows that a significant fraction of the incoming flow kinetic energy is converted to thermal heating. However, unlike the prediction of MHD shocks, the electron and ion temperatures differ.\nIn all simulations, the ion temperatures in the downstream and shock front regions saturate slightly below the MHD prediction because a fraction of the incoming flow kinetic energy is channeled into magnetic energy (as seen in Section~\\ref{sec:Bamp}) and non-thermal energy of cosmic ray ions and electrons (as we show in Section~\\ref{sec:nothermal}).\n\nFor ions, we find $k_{\\rm B}T_{i}\\sim2.5~m_e c^2$ (for $m_r = 1836$, which is only 22.4\\% lower than the MHD prediction) and $k_{\\rm B}T_{i}\\sim0.1~m_e c^2$ (for $m_r = 100$, which is 66\\% lower than the MHD prediction).\nOn the other hand, for electrons, saturation temperatures have a larger mismatch with the MHD predicted temperatures.\n \nWhile the mismatch of the post-shock temperature in the case of a realistic mass ratio can be approximately accounted for by efficient ion acceleration and energy retained in electromagnetic fluctuations, the offset in the case of a mass ratio of $m_r = 100$ is puzzling and cannot be explained by energy in non-thermal components, which only add up to 5\\%. We conjecture that the missing plasma instabilities as a result of the artificially small mass ratio precludes exciting sufficient electromagnetic modes that provide the necessary scattering centers for dissipating all the kinetic energy of ions.\n\nFigure~\\ref{fig:Temp} also shows that assuming an independent evolution for ions and electrons is not correct since the initial flow kinetic energy (mainly from the ions) excites magnetic field wavemodes which both heat and accelerate electrons to an energy density that, in total,  is much larger than that of the initial flow kinetic energy for the electrons.\nThe top panel of Figure~\\ref{fig:Temp} shows that the saturation temperature for ions (electrons) is lower (larger) than predicted from such models, and the bottom panels show that the prediction for the electron-to-ions temperature ratio of such models is inconsistent with all simulations.\n\nIn agreement with the prediction from MHD shocks, simulations with a realistic ion-to-electron mass ratio have saturation temperatures of ions and electrons that are independent of the {\\alf}ic Mach number $\\mathcal{M}_{\\rm A}$. This is demonstrated by the excellent agreement between blue and red (dashed and solid) curves in Figure~\\ref{fig:Temp} albeit with $T_e < T_i$.\nThe saturation temperature of electrons is the same in the downstream and shock front regions, which is also approximately true for ions. This means that all heating occurs in the shock front region and {\\it no further} heating occurs in the downstream region for ions or electrons.\nThe lower panels of Figure~\\ref{fig:Temp} show that, at saturation, $T_e\/T_i = 0.4$ in the downstream regime for the two simulations that use a realistic mass ratio and $T_e\/T_i = 1.4$ for the simulation with $m_r=100$.\n\nIn comparison to simulations with a realistic mass ratio, the simulation that uses a low mass ratio of $m_r=100$ results in {\\it incorrect} saturation temperatures for ions (by factor of $\\sim 25$) and electrons (by factor of $\\sim 7.4$) at both, the downstream and shock front regions as shown in the top panel of Figure~\\ref{fig:Temp}. \nThis low saturation temperature for electrons leads to strong suppression in electron acceleration efficiency as we show in Section~\\ref{sec:nothermal}.\nMoreover, the use of a lower mass ratio results in a wrong electron-to-ion temperature ratio at saturation, i.e., $T_e>T_i$ as can be seen from the bottom panel of Figure~\\ref{fig:Temp}.\nSaturation temperatures where $T_e < T_i$ are observed for example in Balmer-dominated shocks~\\citep{van-Adelsberg+2008}.\nThat is, $T_e\/T_i>1$ found in the simulation with low ion-to-electron mass ratio disagrees with observed saturation temperature ratios.\nSurprisingly, for the simulation with $m_r=100$, the ion and electron temperatures continue to increase from the shock front region to the downstream region. This indicate that further heating occurs in the downstream region of this simulation in disagreement with the results from the simulations with a realistic mass ratio.\n\n\n\\section{Formation of non-thermal particles}\n\\label{sec:nothermal}\n\n\\begin{figure*}\n\\includegraphics[width=18.5cm]{Plots\/Fig07.pdf}\n\\caption{Top: the time evolution of the acceleration efficiency (as defined in Equation~\\ref{Eq:Eff-sh}) in the downstream (left) and shock front (right) region for ions (dashed) and electrons (solid) in various simulations.\nBottom: the time evolution of the fraction of non-thermal energy of electrons relative to that of ions ($K_{\\rm ei}$) in the downstream (left) and shock front (right) regions in various simulations.\n\\label{fig:Eff}\n}\n\\end{figure*}\n\n\n\nAs shown above, the upstream kinetic energy is channeled into magnetic energy (Section~\\ref{sec:Bamp}), non-thermal energy (Section~\\ref{sec:nth1}) and thermal energy (Section~\\ref{sec:thermal}) in the downstream and shock front regions of the system. \nHere we quantify the amount of energy carried by non-thermal ions and electrons in these regions in Figure~\\ref{fig:Eff}.\nIn Appendix~\\ref{app:temp}, we show that the fraction of particles with non-thermal energy can be quantified in various ways, and here we focus on how much energy density is carried by these particles when compared to the upstream kinetic energy density, i.e., using Equation~\\eqref{Eq:Eff-sh}.\nWe define non-thermal particles to be particles with $u>5u_m$~\\citep{Caprioli2014a,Xu2020}, where $u=\\gamma \\varv$ is the spatial-part of the 4-velocity, $\\gamma$ is the Lorenz factor, and $u_m$ is defined as the value of $u$ where $u^4 f(u)$ is maximized.\nAs an example, particles to the right of the blue vertical lines in Figure~\\ref{fig:Thermal} are considered to form the non-thermal part of the momentum distribution. \nThis quantification is done for different regions as a function of time and was used to compute the time evolution of the efficiency shown in the top panel of Figure~\\ref{fig:Eff}.\nThe top panel shows the time evolution of the efficiency $\\epsilon_{\\rm sh}$, i.e., the percentage of upstream kinetic energy density that is channeled into non-thermal electrons and ions in the downstream (right) and shock front regions (left).\nBy contrast, the bottom panel shows the time evolution of the ratio of non-thermal energy of electrons to that of ions, $K_{\\rm ei}$, in various simulations also in the downstream (left) and shock front (right) regions.\n\nThe non-thermal energy density of ions (dashed curves) is about 20-30\\% of the upstream kinetic energy density, and this result is roughly independent of the ion-to-electron mass ratio $m_r$ at the shock front region.\nThe simulation with a higher $\\mathcal{M}_{\\rm A}$ has a slightly higher efficiency in producing non-thermal ions at the shock front (transition) region.\nMoreover, in the downstream region, it contains a higher fraction of non-thermal ions indicating that this simulation is more efficient in scattering ions that are accelerated in the shock front (transition) region back to the downstream region.\n\nOn the other hand, the fraction that is channeled into non-thermal electrons is strongly dependent on both, $\\mathcal{M}_{\\rm A}$ and $m_r$.\nThe top panel of Figure~\\ref{fig:Eff} shows that, at the shock front, the simulation with $\\mathcal{M}_{\\rm A}=5.2$ and a realistic mass ratio efficiently accelerates electrons, leading to 0.05--0.1\\% of initial flow kinetic energy converted into non-thermal electrons, while in the simulation with a higher $\\mathcal{M}_{\\rm A}$ and a realistic mass ratio, a smaller fraction (0.01--0.05\\%) of the upstream kinetic energy is channeled into non-thermal electrons.\nThe simulation with a reduced mass ratio leads to about 2-3 orders of magnitude reduction in electron acceleration efficiency.\n\nClearly, the simulation where the intermediate-scale instability is allowed to grow at the shock front region (red) is much more efficient in confining the accelerated electrons by scattering them back to the downstream regions, leading to a much larger non-thermal electron energy in the downstream region in comparison to the simulation with a higher $\\mathcal{M}_{\\rm A}$ as seen in the left-hand panel of Figure~\\ref{fig:Eff}.\nMoreover, the simulation with a low mass ratio (black) leads to significant reduction in the energy channeled into non-thermal electrons in the downstream region.\n\nAs shown in the lower left-hand panel of Figure~\\ref{fig:Eff}, in the downstream region, the simulation that allows for growth of comoving ion-cyclotron waves (red) has a 2-3 orders of magnitude larger non-thermal electron-to-ion energy ratio, $K_{\\rm ei}$, in comparison to that in the simulations where the condition for the intermediate-scale instability is not satisfied.\n\n\n\\section{Discussion}\n\\label{sec:dis}\n\nIn this section, we discuss the impact of a finite thermal temperature on the growth of the intermediate-scale instability, followed by a comparison of our proposed electron pre-acceleration mechanism in comparison to the mechanism proposed by \\citet{2015phrvl.114h5003p}. Finally, we discuss how our results can be used to possibly infer a lower limit on the amplitude of the magnetic field at shocks of supernova remnants.\n\n \n\n\\subsection{Impact of finite plasma beta}\n\n\nThe dispersion relation used in Section~\\ref{sec::theory} to predict the unstable modes at the shock transition region assumes a cold background of electrons and ions.\nThus, it is of great importance to consider the impact of the finite plasma beta $\\beta_s$ on the growth of the intermediate-scale instability, where  $\\beta_s \\equiv 2 \\varv_{{\\rm th},s}^2\/\\varv_{{\\rm A},s}^2$, the square of the thermal speed is $\\varv_{{\\rm th},s}^2 \\equiv k_B T_s\/m_s$ and $s=e,i$ for electrons and ions respectively.\n\nThe dispersion relation studied in \\citet{sharp2} and presented in Section~\\ref{sec::theory} are computed assuming $\\beta_i =\\beta_e=0$ while the simulation presented by \\citet{sharp2} has $\\beta_e = \\beta_i = 2$. Interestingly, the growth rates of the analytical derivation in the cold-background limit and the simulations with the finite $\\beta$ values show an excellent agreement, thus indicating that there is no impact of  $\\beta_e$ or $\\beta_i$ on the maximum growth rate of the intermediate-scale instability.\n\n\\citet{sharp2} argue that the damping due to thermal ion-cyclotron damping (at $k \\varv_{\\rm th,i} = (1$--$2) \\Omega_i$) could at most impact one peak of the instability if parameters are fine-tuned and thus typically it cannot impact instability growth for typical astrophysical plasma conditions. Similarly, it is strainght-forward to show that this is also true for thermal electron-cyclotron damping because the separation between the two peaks of the intermediate-scale instability does not solely depend on the ion-to-electron mass ratio.\n\nFor the shock problem studied in the present paper, there are two very different regimes of the beta factor: a low-beta regime in the pre-shock region and a high-beta regime in the shock transition zone. In the pre-shock region, we have $\\beta_e = \\beta_i = 1.28\\times10^{-4}$ ($\\mathcal{M}_{\\rm A}=5.3$)  and $\\sim 2 \\times 10^{-3}$ $(\\mathcal{M}_{\\rm A} = 21.3$). The case of $\\beta_e \\gtrsim1$ would imply a much faster growth of the intermediate-scale instability in comparison to the growth rate of the gyro-scale instability because of the dependence of the growth rate on $\\varv_\\perp$ as given in equation (6) of \\citet{sharp2}. Thus, we expect that in this case the instability will have an even larger impact on the mechanism of electron acceleration but postpone a detailed study of this to future works.\n\nAt the shock transition zone of the simulations with $m_r=1836$, the electron are hot and the temperature is such that $k_{\\rm B} T_e \\sim m_e c^2$. Therefore, we obtain $\\beta_e = 1.74$ for the $\\mathcal{M}_{\\rm A} = 5.3$ simulation and $\\beta_e = 27.88$ for the $\\mathcal{M}_{\\rm A} = 21.3$ simulation, i.e., in both cases, the electron beta is larger than one. However, only in the simulation with $\\mathcal{M}_{\\rm A} = 5.3$, the relative drift is such that the instability condition is fulfilled and thus allows for the growth of the intermediate-scale instability. This allows resonant interactions between electrons and unstable modes and results in much larger acceleration efficiency as shown in the simulation.\n\n\n\\subsection{On the electron acceleration mechanism}\n\nThe proposed mechanism for pre-accelerating electrons works by exciting intermediate-scale waves at the shock transition region.\nThese sub-ion skin-depth (ion-cyclotron) waves are comoving with the incoming upstream plasma (see bottom panel of Figure~\\ref{fig:GrowthRate}) and hence are coherently transported to the downstream region (see Figure~\\ref{fig:KLS}).\nIn the downstream and shock transition regions, the hot electrons scatter off of these unstable waves  and the waves reflected at the contact discontinuity (see top-left panel of Figure~\\ref{fig:XT-evol}). This leads to the high acceleration efficiency shown in the red simulation (with $\\mathcal{M}_{\\rm A} = 5.3$) as seen in Figures~\\ref{fig:spectra300} and \\ref{fig:Eff}, possibly in a first-order Fermi type process.\n\nThe simulation with $\\mathcal{M}_A=21.3$ (depicted in blue) shows excitation of sub-ion skin-depth (Whistler) waves at the shock front region. However, these modes are not transported to the downstream region (see Figure~\\ref{fig:KLS}), which thus results in a much lower electron acceleration efficiency.\nThat is, this simulation shows that for a parallel shock geometry, electron pre-acceleration due to scattering with Whistler waves has a much lower efficiency in comparison to that in the red simulation (see Figure~\\ref{fig:Eff}).\n\n\nWe emphasize that our proposed mechanism for electron pre-acceleration does not depend on ion acceleration.\nThis is clearly manifested by the large fraction of pre-accelerated electrons before we observe any significant ion acceleration in the  simulation with $\\mathcal{M}_{\\rm A} = 5.3$, as shown in the top-panel (red curves) of Figure~\\ref{fig:spectra300}.\nOn the other hand, the mechanism proposed by~\\citet{2015phrvl.114h5003p} for electron pre-acceleration relies on the amplification of Bell modes in the shock pre-cursor driven by the propagation of accelerated ions and thus the combination of SDA and scattering with Bell modes resulted in electron pre-acceleration and injection in DSA cycles.\n\n\\subsection{Applications to supernova remnants}\n\\label{sec:SNR}\nWhile we have clearly demonstrated the importance of the intermediate-scale instability for thermalizing and accelerating electrons in our PIC simulations, we will now turn to discuss the potential relevance of this instability in observations of astrophysical shocks. Perhaps the cleanest astrophysical object is the supernova remnant SN~1006, which enables testing our ideas of the prevailing plasma instabilities that are responsible for electron scattering and acceleration at the quasi-parallel shock conditions we encounter at the polar cap regions of that remnant shock. \\citet{Winner+2020} perform three-dimensional MHD simulations of the Sedov Taylor explosion and evolve the electron distribution function while accounting for magnetic obliquity-dependent electron acceleration. To this end, their method finds and characterizes the MHD shocks, injects a pre-defined electron power-law spectrum into the post-shock region \\citep{Winner+2019}, and explores the effects of varying the magnetic amplification factor on the surface brightness maps as well as the multi-frequency spectrum of SN~1006.\n\nMatching the radial emission profiles and the gamma-ray spectrum  requires a volume-filling, turbulently amplified magnetic field of $B\\approx35~\\mu$G in the downstream region of the parallel shock and that the Bell-amplified magnetic field \\citep{Bell2004} is starting to get damped in the further post-shock region \\citep[see figure~2 of][]{Winner+2020}. The exact value of the Bell amplification factor $f_\\mathrm{B}$ barely changes the multi-frequency spectrum so that we obtain a post-shock Alfv\\'en velocity of $\\varv_\\mathrm{A}=B f_\\mathrm{B}\/\\sqrt{\\mu_0\\mu m_p n}\\approx200~f_\\mathrm{B}~\\mathrm{km~s}^{-1}$, where $\\mu=1.25$ is the mean molecular weight of the warm interstellar medium, $m_p$ is the proton rest mass, and $n=0.12~\\mathrm{cm}^{-3}$. While the Bell amplified field is maximized in the immediate post-shock regime \\citep{Caprioli2014b}, the turbulently amplified magnetic field keeps rising in the post-shock regime \\citep{Ji+2016}, so that it is appropriate to set $f_\\mathrm{B}=1$ while noting that the turbulently amplifying field on its route to saturation is replacing the Bell-amplified field as it is damping further downstream.\n\nAdopting the lab frame shock velocity of SN~1006 at the current epoch, $\\varv_s=3000~\\mathrm{km~s}^{-1}$, we obtain a post-shock Alfv\\'enic Mach number of\n\\begin{align}\n  \\mathcal{M}_\\mathrm{A}=\\frac{\\varv_s}{\\varv_\\mathrm{A}}=15.2<\\frac{\\sqrt{m_r}}{2},\n\\end{align}\nwhich obeys the condition for exciting the intermediate-scale instability and thus should enable efficient electron acceleration at the polar cap regions of the parallel shocks of SN~1006 \\citep[see figure~4 of][]{Winner+2020}.\n\nIn fact, efficient electron acceleration at parallel shocks enables us to put a lower limit on the post-shock magnetic field, \n\\begin{align}\n  \\label{eq:Bmin}\n  B&>2 \\varv_s \\sqrt{\\frac{\\mu_0\\rho}{m_r A}}=B_\\mathrm{min}\\\\\n  &= 22.7~\\left(\\frac{\\varv_s}{3000~\\mathrm{km~s}^{-1}}\\right)\n  \\left(\\frac{n}{A\\, 0.1~\\mathrm{cm}^{-3}}\\right)^{1\/2}\\,\\mu\\mathrm{G},\n\\end{align}\nwhere $m_r$ is the proton-to-electron mass ratio and $A$ is the atomic mass number of the element responsible for driving the intermediate~scale instability. Hence, if the plasma is composed of abundant ${}^{56}\\mathrm{Fe}$ ions, the minimum post-shock magnetic field is lowered to $3~\\mu$G for the same shock parameters.\nFor heavy ions to dominate the growth of the intermediate instability, we require them to be very abundant because the growth rate, $\\Gamma$, of the intermediate-scale instability depends on the number density of ions via $\\Gamma \\propto n_{\\rm Fe}^{1\/3}$.\n\n\n\\section{Conclusions}\n\n\\label{sec:concl}\n\n\nIn collisionless shocks, electrons are heated to energies much larger than the kinetic energy of upstream  electrons impinging on the shock. However, in non-relativistic shocks, their Larmor radii fall short of those of ions so that another acceleration mechanism is needed to boost their energies to the point where they can actively participate in the process of DSA.\nPreviously suggested mechanisms, which were based on driving whistler waves unstable by hot downstream or cold upstream electrons, require high values of the {\\alf}ic Mach number and were shown to not work in PIC simulations \\citep{Niemiec+2012}. \nIn this paper we consider a new mechanism for electron pre-acceleration in quasi-parallel, non-relativistic electron-ion shocks that is based on driving ion-cyclotron waves unstable by means of ions that are drifting with their upstream velocity through the shock transition zone. The corresponding intermediate-scale instability \\citep{sharp2} only works for low values of the {\\alf}ic Mach number at the shock front, $\\mathcal{M}_{\\rm A}^f < \\sqrt{m_i\/m_e} \\approx 21$, which is the condition for the instability to operate.\n\nWe present results from three 1D3V PIC simulations for parallel electron-ion shocks with sonic Mach number $\\mathcal{M}_{\\rm s} \\sim 365$:\nthe first (red) simulation uses a realistic ion-to-electron mass ratio of $m_r=1836$ and provides favorable conditions for exciting the intermediate scale instability.\nThe second (black) simulation employs identical physical initial conditions but has an artificially lower value for the ion-to-electron mass ratio, $m_r=100$, which violates the instability condition.\nIn the third (blue) simulation, the condition is also violated by using a higher value of $\\mathcal{M}_{\\rm A}$ and $m_r=1836$.\nHighlight results include:\n\\begin{itemize}\n\n\\item Only the simulation that grows the intermediate scale instability scatters hot electrons in the downstream off of driven ion-cyclotron waves. We demonstrate that this efficiently pre-accelerates electrons to energies that enable them to participate in DSA cycles and yields a non-thermal power-law distribution of electrons (see Figure~\\ref{fig:spectra300}).\n\n\\item This effective electron acceleration comes about because the excited ion-cyclotron waves at the shock front are comoving with the upstream plasma and hence, survive advection into the downstream. In consequence, the amplitude of perpendicular magnetic field fluctuations, which are resonantly scattering hot electrons, is substantially increased in comparison to the other simulations that preclude instability growth (see Figures~\\ref{fig:spectra300} and \\ref{fig:KLS}).\n\n\\item The simulation with the higher value of $\\mathcal{M}_{\\rm A}$ shows a reduction, by more than 2 orders of magnitude, in the efficiency of electron acceleration (see Figure~\\ref{fig:Eff}). However, the electrons in the downstream are heated to the same temperature as the red simulation (see Figure~\\ref{fig:Temp}).\n  \n\\item The simulation with the lower mass ratio ($m_r=100$) results in much lower heating of electrons in the downstream region and thus an even lower electron acceleration efficiency (see Figures~\\ref{fig:Eff} and \\ref{fig:Temp}). We conclude that accurate PIC simulations of collisionless electron-ion shocks require realistic mass ratios.\n  \n\\end{itemize}\n\nThese findings put the magnetic amplification processes at shocks into the limelight because of the strict instability criterion that favors low values of $\\mathcal{M}_{\\rm A}$ and as such, is able to relate efficient electron acceleration with a lower limit on the downstream magnetic field strength (see Equation~\\ref{eq:Bmin}). Most importantly, this paper provides an important cornerstone in our understanding of diffusive shock acceleration of electrons, and potentially enables us to construct an analytical subgrid model for including electron acceleration into modern MHD models.\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Anatoly Spitkovsky for discussions on various aspects of the simulations. We also acknowledge comments by Damiano Caprioli and the anonymous referee.\nM.S., R.L., T.T., and C.P. acknowledge support by the European Research Council under ERC-CoG grant CRAGSMAN-646955.\nThe authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre (www.lrz.de). This research was in part supported by the Munich Institute for Astro- and Particle Physics (MIAPP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u00b4s Excellence Strategy \u2013 EXC-2094 \u2013 390783311.\n\n\n\\begin{appendix}\n\n\n\n\n\\section{Distribution of relativistic particles}\n\\label{app:temp}\n\nHere we review the shape of the momentum distribution of charged particles in thermal equilibrium, and the impact of the existence of a non-thermal tail.\nWe discuss how it can be characterized in such cases, and different ways to quantify the fraction of density and energy in its non-thermal tail.\n\n\\subsection{Average thermal energy}\n\nIn thermal equilibrium, the rest-frame (R) isotropic momentum distribution of relativistic particles (with elementary mass $m_s$) is given by the Maxwell-J{\\\"u}ttner distribution~\\citep{Wright1975}\n\\begin{eqnarray}\n\\label{Eq:distribution}\nf_{\\rm th}(\\vec{u}) = \\frac{ n_R e^{-\\gamma\/\\tilde{T}_R}  }{4 \\pi \\tilde{T}_R K_2(1\/\\tilde{T}_R)} ,\n\\end{eqnarray}\nwhere $K_2$ is the modified Bessel function of the second kind, the Lorentz-factor is $\\gamma = \\sqrt{1+ \\vec{u} \\cdot \\vec{u}}$, and $\\tilde{T}_R$ is the thermodynamical equilibrium temperature in the rest frame\\footnote{\nIt is worth mentioning here that because the phase-space volume is invariant under Lorentz transformation, this distribution takes the same form in any other frame, e.g., in the laboratory frame \\citep{Wright1975}.} normalized by $m_s c^2\/k_{\\rm B}$, all velocities are normalized with the speed of light $c$, and $n_R$ is the rest-frame number density of the particles.\nTherefore, the average (thermal + rest-mass) energy per particle, normalized by $m_s c^2$, is\n\n\\begin{eqnarray}\n\\langle \\bar{E} \\rangle &=&  \\int d^3u \\gamma \\frac{f_{\\rm th}(\\vec{u})}{n_R} = 4 \\pi  \\int_0^{\\infty}  du  \\frac{u^2 \\gamma e^{-\\gamma\/\\tilde{T}_R}  }{4 \\pi \\tilde{T}_R K_2(1\/\\tilde{T}_R)}\n\\nonumber \\\\\n&=&\n\\frac{1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\int_0^{\\infty}  du u^2 \\gamma e^{-\\gamma\/\\tilde{T}_R} \n\\nonumber \\\\\n&=&\n\\frac{-1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\frac{d}{d(1\/\\tilde{T}_R)} \\int_0^{\\infty}  du u^2  e^{-\\gamma\/\\tilde{T}_R}\n\\nonumber \\\\\n&=&\n\\frac{-1}{ \\tilde{T}_R K_2(1\/\\tilde{T}_R)} \\frac{d ( \\tilde{T}_R K_2(1\/\\tilde{T}_R) )}{d(1\/\\tilde{T}_R)} \n=3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)}.\n~~~~~~~ \n\\end{eqnarray}\nThat is, the average thermal energy per particle is\n\\begin{eqnarray}\n\\label{Eq:Eth}\n\\langle E_{\\rm th} \\rangle = \\left( 3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)} - 1 \\right) m_s c^2.\n\\end{eqnarray}\n\n\n\\subsection{Equilibrium rest-frame temperature}\nFor a distribution of particle momenta, we can find $\\tilde{T}_R$ (assuming a thermal equilibrium of most low energy particles) as follows:\n\\begin{eqnarray}\n\\label{Eq:du4fu}\n\\frac{d }{du} \\left[  u^4 f_{\\rm th}  \\right]  =  \\frac{ n_R  u^3 \\left(4 \\sqrt{u^2+1} \\tilde{T}_R-u^2\\right)  }{4 \\pi \\tilde{T}_R^2 K_2(1\/\\tilde{T}_R) \\sqrt{u^2+1}}  e^{-\\frac{\\sqrt{u^2+1}}{\\tilde{T}_R}} .\n\\end{eqnarray}\nThat is, using the value of $u$ for which $u^4 f(u)$ is maximum ($u_m$), we can determines $\\tilde{T}_R$ using\n\\begin{eqnarray}\n\\label{Eq:TR}\n\\tilde{T}_R = \\frac{  u_m^2 }{ 4 \\sqrt{1+u_m^2}}.\n\\end{eqnarray} \n\n\n\\subsection{Momentum distribution with non-thermal particles}\n\n\n\nHere, we consider the full momentum distribution $f_{\\rm full}$ that contains non-thermal and high-energy particles. \nThe main assumption adopted here is that {\\it{all low energy particles are in thermal equilibrium}}.\nThat is, all non-thermal particle are such that $u \\gg u_m$, which means the value of $u$ at which the derivative in Equation~\\eqref{Eq:du4fu} is zero is always outside the range for which the non-thermal parts of the distribution is non-zero.\nTherefore, the value of $u_m$ and hence $\\tilde{T}_R$, computed from Equation~\\eqref{Eq:TR}, is still the same.\nAn example of such a case is shown in Figure~\\ref{fig:Thermal}.\n\nAssuming that the non-thermal part of the distribution follows a power law with a typical index of -$4$, the full distribution can be approximated as follows,\n\\begin{eqnarray}\n\\label{Eq:Ffull}\nf_{\\rm full}(u) = \\frac{n_R}{1+\\alpha}\n\\left[\n\\frac{e^{-\\gamma\/\\tilde{T}_R}}{4 \\pi k_2 (1\/\\tilde{T}_R)}\n+\n\\alpha\n\\frac{u_1 u_2 u^{-4}}{4 \\pi (u_1 - u_2) }\\theta_1 \\theta_2\n\\right],\n\\end{eqnarray}\nwhere $\\alpha$ is the fractional number density of non-thermal particles which is implicitly assumed to be small, i.e., $\\alpha \\ll 1$, $\\theta_1 = \\theta(u-u_1)$, $\\theta_2 = \\theta(u_2 -u)$, and $u_1$ and $u_2$ ($u_2>u_1$) is the range in which the distribution takes a power-law form.\nThe distribution in Equation~\\eqref{Eq:Ffull} is discontinuous at $u=u_1$. However, in reality the thermal and $u^{-4}$ parts of the distribution are connected with a suprathermal distribution (with a logarithmic slope steeper than $-4$) that starts at $u \\sim 4 u_m$ to $ 5 u_m$ \\citep{Caprioli2014a}.\nTherefore, the suprathermal distribution does not change the relation between $u_m$ and $\\tilde{T}_R$ (Equation~\\ref{Eq:TR}).\n\nThe inclusion of the non-thermal part leads to a lower fraction of particles in thermal equilibrium and hence lowers proportionately the value of $E_{\\rm th}$ at a given $\\tilde{T}_R$. Namely\n\\begin{eqnarray}\n\\label{Eq:Eth2}\n\\langle E_{\\rm th} \\rangle_{\\rm full}  \\approx \\frac{1}{1+\\alpha} \\left( 3 \\tilde{T}_R +\\frac{K_1 \\left(1\/\\tilde{T}_R \\right)}{K_2\\left(1\/\\tilde{T}_R \\right)} - 1 \\right) m_s c^2.\n\\end{eqnarray}\n\nThe value of $\\alpha$ can be determined by comparing the value of $f(u_m)$ and $f_{\\rm full}(u_m)$, where the value of \n$f(u_m)$ is computed from Equation~\\eqref{Eq:distribution} at $u_m$ where the expression $u^4 f_{\\rm full}(u)$ is maximized, and\n$f_{\\rm full}(u_m)$ is computed from the normalized histogram of the particles' normalized momenta ($u$) in the simulation.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{Plots\/Fig08.pdf}\n\\caption{\nMomentum distribution for ions (red dashed) and electrons (red solid) at $t~ \\Omega_i=325$ for the Ma5Mr1836 simulation, where we first transform momenta of particles to the plasma rest frame before we compute the momentum spectra.\nSolid black lines show the analytical Maxwell-J{\\\"u}ttner distribution scaled appropriately, $u^4 f(u)$ (as given by Equation~\\ref{Eq:distribution}). The normalized temperature (given by Equation~\\ref{Eq:TR}) is determined from the location of the peaks ($u_m^i$ for ions and $u_m^e$ for electrons).\nParticle energies with $u>5u_m$ (to the right of blue lines) are considered to be non-thermal.\n\\label{fig:Thermal}\n}\n\\end{figure}\n\n\n\\subsection{Acceleration efficiency}\n\n\n\nWe note here that $\\alpha = n_{\\rm non-th}\/n_{\\rm th}$, where $n_{\\rm th}$ is the number density of particles in thermal equilibrium, and $n_{\\rm non-th}$ is the number density of non-thermal particles. Thus, \nthe rest-frame total number density is $n_R = n_{\\rm th} + n_{\\rm non-th}$.\nThat is, we can define the efficiency of acceleration as the fraction of non-thermal particles ($\\epsilon_n$) measured in per cent via\n\\begin{eqnarray}\n\\label{Eq:EffN}\n\\epsilon_n \\equiv \\frac{n_{\\rm non-th}}{n_R} \\times 100 = \\frac{\\alpha}{1+\\alpha} \\times 100  .\n\\end{eqnarray}\n\nWe can also define the acceleration efficiency by the fractional energy carried by non-thermal particles ($\\epsilon_E$) in per cent as\n\\begin{eqnarray}\n\\label{Eq:EffE}\n\\epsilon_E & \\equiv & \n\\frac{ E_{\\rm tot} - \\langle E_{\\rm th} \\rangle_{\\rm full} }{ E_{\\rm tot} } \\times 100  \n=\n\\frac{E_{\\rm non-th}}{E_{\\rm tot}} \\times 100 ,\n\\end{eqnarray}\nwhere $ E_{\\rm tot} = \\langle \\gamma -1 \\rangle m_s c^2$ is the average of $(\\gamma-1)$ of all thermal and non-thermal particles.\n\nAn equally important definition of the acceleration efficiency is how much of the upstream plasma kinetic energy is channeled into non-thermal energy. In this case, the efficiency of acceleration in per cent is defined as\n\\begin{eqnarray}\n\\label{Eq:Eff-sh}\n\\epsilon_{\\rm sh}\n& \\equiv &\n\\frac{ E_{\\rm non-th}   }{ \n0.5  ~ (m_e + m_i) \\varv_{\\rm sh}^2 } \\times 100 ,\n\\end{eqnarray}\nwhere $\\varv_{\\rm sh}$ is the upstream plasma drifting speed in the shock rest-frame.\nWe compute the non-thermal energy density, $E_{\\rm non-th}$, as the average (per particle) energy with $u>5u_m$ \\citep[][]{Xu2020}. The vertical blue lines in Figure~\\ref{fig:Thermal} show such cuts.\nThat is, particles with velocities $u>5u_m$ are assumed to form the non-thermal part of the distribution function.\n\n\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{Introduction}\n\nThe classification problem for group actions on operator algebras has a long history dating back to the hallmark results of Connes \\cite{Connes73, Connes74, Connes75, Connes76, Connes77} about injective factors that employed ideas of a dynamical nature in a crucial way.\nThe subsequent pioneering work by Jones \\cite{Jones80}, Ocneanu \\cite{Ocneanu85} and others \\cite{SutherlandTakesaki89, KawahigashiSutherlandTakesaki92, KatayamaSutherlandTakesaki98, Masuda13} completely unraveled the structure of countable amenable group actions on injective factors and paved the way for related further developments in subfactor theory \\cite{Jones83, Popa94, Popa10, Masuda99}.\nWhen the acting group is nonamenable, then the structure of its actions on the hyperfinite II$_1$-factor is already less managable, as a theorem of Jones \\cite{Jones83oc} and other subsequent stronger \\emph{no-go theorems} such as \\cite{Popa06, BrothierVaes15} later demonstrated.\n\nConnes' seminal paper \\cite{Connes76} was the first of many influential works in operator algebras to make use of a kind of touchstone object (in his case the hyperfinite II$_1$-factor $\\mathcal{R}$) and to begin the classification approach by showing that every object to be classified absorbs this touchstone object.\nMore specifically, Connes' approach begins with a structural result asserting that every injective II$_1$-factor is \\emph{McDuff} --- i.e., tensorially absorbs $\\mathcal R$; see \\cite{McDuff70} --- which is then used further to show that each such factor is isomorphic to $\\mathcal R$.\nIn Ocneanu's pioneering work to classify outer actions of an arbitrary countable amenable group $G$ on $\\mathcal{R}$, he likewise proves at an early point in the theory that each such action (even without assuming outerness) absorbs the trivial $G$-action on $\\mathcal R$ tensorially, which he then exploits for his more sophisticated classification theorem.\nAlthough one would generally need injectivity of a factor to arrive at a satisfactory classification theory about (outer) $G$-actions on it, this early part of Ocneanu's theory works in remarkable generality.\nThe precise statement and a (comparably) self-contained proof using the methods of this article, which we included for the reader's convenience, is Theorem~\\ref{theorem:model-absorption}.\n\nIf one is concerned with C$^*$-algebraic considerations related to the \\emph{equivariant Jiang--Su stability problem} (see \\cite[Conjecture A]{Szabo21si}), the current methods always find a way to exploit Oceanu's aforementioned theorem in one form or another, usually involving to some degree Matui--Sato's property (SI) technique \\cite{MatuiSato12acta, MatuiSato12, MatuiSato14, Sato19}.\nLooking at the state-of-the-art at present \\cite{GardellaHirshbergVaccaro, Wouters21}, the key difficulties arise from pushing these methods to the case where a group action $G\\curvearrowright A$ on a C$^*$-algebra induces a complicated $G$-action on the traces of $A$.\nIn particular, it is generally insufficient for such considerations to only understand $G$-actions on $\\mathcal{R}$, but one rather needs to have control over $G$-actions on more general tracial von Neumann algebras.\nThis C$^*$-algebraically motivated line of investigation led us to ask the following question that is intrinsic to von Neumann algebras:\n\n\\begin{question}\nLet $G$ be a countable amenable group and $M$ a separably acting finite von Neumann algebra with $M\\cong M\\bar{\\otimes}\\mathcal R$.\nIs it true that every action $\\alpha: G\\curvearrowright M$ is cocycle conjugate to $\\alpha\\otimes\\operatorname{id}_{\\mathcal R}: G\\curvearrowright M\\bar{\\otimes}\\mathcal R$?\n\\end{question}\n\nAlthough Ocneanu's original results confirm this in full generality when $M$ is a factor,\\footnote{While Ocneanu's work \\cite{Ocneanu85} only contains an explicit proof for so-called centrally free actions $\\alpha$, his comment following \\cite[Theorem 1.2]{Ocneanu85} suggests an alternative approach to avoid this assumption. In several papers, the more general version without central freeness is also attributed to Ocneanu.} it turned out to be not so straightforward to resolve this question, despite common folklore wisdom in the subject suggesting that the factor case ought to imply the general case.\nSome classification results in the literature \\cite{JonesTakesaki84, SutherlandTakesaki85} imply that the above has a positive answer when $M$ is injective, but relying on this has two drawbacks.\nFirstly, the question we are trying to answer is by design much weaker than a hard classification result, so it would be desirable to have a proof not relying on such a powerful theorem, in particular when an assumption such as injectivity may not even be needed.\nSecondly, there is a subtle gap in the proof of \\cite[Lemma 4.2]{SutherlandTakesaki85}.\nWe are indebted to Stefaan Vaes for pointing this out to us in the context of the above question and for outlining a sketch of proof on how to correct this, which became a sort of blueprint for the main result of the fourth section.\n\nIn contemporary research by C$^*$-algebraists, the aforementioned results by Sutherland--Takesaki are still used to provide a partial answer to the above question, for example in \\cite{GardellaHirshbergVaccaro}. In light of the previous discussion, the present article aims to give a self-contained and --- dare we say also relatively elementary --- approach to answer this question instead.\nIn fact we can treat it in greater generality than posed above, without restrictions on the type of $M$ and in the setting of amenable actions of arbitrary discrete groups.\nThe following can be viewed as our main result; see Theorem~\\ref{thm:general-amenable-McDuff}.\n\n\\begin{theoremi} \\label{theorem-A}\nLet $G$ be a countable discrete group and $M$ a von Neumann algebra with separable predual such that $M \\cong M\\bar{\\otimes} \\mathcal{R}$. \nThen every amenable action $\\alpha\\colon G \\curvearrowright M$ is cocycle conjugate to $\\alpha \\otimes \\mathrm{id}_\\mathcal{R}\\colon G\\curvearrowright M\\bar{\\otimes} \\mathcal{R}$.\n\\end{theoremi}\n\nAlong the way, our methodology employs dynamical variants of McDuff-type properties analogous to the theory of strongly self-absorbing C$^*$-dynamics \\cite{Szabo18ssa}, which can and is treated in the more general setting of continuous actions of locally compact groups; see Definitions~\\ref{def:strong-absorption} and \\ref{def:ssa-action}.\n\n\\begin{defii}\nLet $G$ be a second-countable locally compact group.\nAn action $\\delta: G\\curvearrowright\\mathcal R$ is called \\emph{strongly self-absorbing}, if there exists an isomorphism $\\Phi: \\mathcal R\\to\\mathcal R\\bar{\\otimes}\\mathcal R$, a $(\\delta\\otimes\\delta)$-cocycle $\\mathbb{U}: G\\to\\mathcal{U}(\\mathcal R\\bar{\\otimes}\\mathcal R)$ and a sequence of unitaries $v_n\\in\\mathcal{U}(\\mathcal R\\bar{\\otimes}\\mathcal R)$ such that\n\\[\nv_n(x\\otimes 1_{\\mathcal R})v_n^* \\to \\Phi(x) \\quad\\text{and}\\quad v_n(\\delta\\otimes\\delta)_g(v_n)^* \\to \\mathbb{U}_g\n\\]\nin the strong operator topology for all $x\\in\\mathcal R$ and $g\\in G$, the latter uniformly over compact subsets in $G$.\n\\end{defii}\n\nFor such actions we prove the following dynamical generalization of McDuff's famous theorem \\cite{McDuff70}; see Theorem~\\ref{thm:general-McDuff}.\n\n\\begin{theoremi} \\label{theorem-C}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual and let $\\delta: G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen $\\alpha$ is cocycle conjugate to $\\alpha\\otimes\\delta$ if and only if there exists a unital equivariant $*$-homomorphism $(\\mathcal R,\\delta) \\to (M_{\\omega,\\alpha},\\alpha_\\omega)$, where the latter denotes the induced $G$-action on the asymptotic centralizer algebra of $M$.\n\\end{theoremi}\n\nOur initial methodology inspired by the theory of C$^*$-dynamics is only well-suited to build all the aforementioned (and other related) theory in the setting of (actions on) semi-finite von Neumann algebras.\nAfter the first preliminary section, the second and third section are dedicated to proving Theorem~\\ref{theorem-C} in the special case of semi-finite von Neumann algebras.\nThe fourth section then builds on some of this theory, combined with the original ideas by Sutherland--Takesaki \\cite{SutherlandTakesaki85} related to disintegrating a $G$-action to an action of its transformation groupoid induced on the center. This culminates in our main technical result of that section --- a kind of measurable local-to-global principle for absorbing a given strongly self-absorbing action, Theorem~\\ref{thm:main_technical} --- which is then used to prove a stronger version of Theorem~\\ref{theorem-A} for actions on semi-finite von Neumann algebras. \n \nThe general main results are then obtained in the fifth section with the help of Tomita--Takesaki theory.\nIt is in this step that it becomes obvious why we want to treat Theorem~\\ref{theorem-C} beyond the case of discrete groups.\nNamely, if $\\alpha\\colon G\\curvearrowright M$ is an action as in Theorem~\\ref{theorem-A} on a von Neumann algebra that is not semi-finite, we may consider the extended action $\\tilde{\\alpha}\\colon G\\curvearrowright\\tilde{M}$ on the (semi-finite) continuous core.\nHowever, in order to conclude that $\\alpha$ absorbs $\\delta$ with the help of Tomita--Takesaki theory, it is not sufficient to argue that $\\tilde{\\alpha}$ absorbs $\\delta$, but one actually needs to verify this absorption for certain enlargements of these actions to continuous actions of $G\\times\\mathbb{R}$, which in any case requires Theorem~\\ref{theorem-C} for non-discrete groups.\nFortunately this can all be arranged to work and we end the last section with the proofs of our main results.\n\n\n\\section{Preliminaries}\n\nThroughout the paper, $\\omega$ denotes a fixed free ultrafilter on $\\mathbb{N}$ and $G$ denotes a second-countable, locally compact group.\nLet $M$ be a von Neumann algebra with predual $M_*$. For $x \\in M$ and $\\phi \\in M_*$ we define elements $x\\phi, \\phi x$ and $[x,\\phi] \\in M_*$ by $(x\\phi)(y) = \\phi(yx)$, $(\\phi x)(y) = \\phi(xy)$ for all $y \\in M$ and $[x,\\phi] = x\\phi - \\phi x$. Moreover, for $x \\in M$ and $\\phi \\in M_*$ we set $\\|x\\|_{\\phi} = \\phi(x^*x)^{1\/2}$ and $\\|x\\|_{\\phi}^\\# = \\phi(x^*x + xx^*)^{1\/2}$. When $\\phi$ is a faithful normal state, the norms $\\|\\cdot\\|_\\phi$ and $\\|\\cdot\\|_\\phi^\\#$ induce the strong and strong-$*$ topology on bounded sets, respectively.\nMore generally, when $\\phi$ is a normal weight on $M$, we define $\\|x\\|_{\\phi}:= \\phi(x^*x)$ for all $x$ contained in the left-ideal\n\\[\\{x  \\in M \\mid \\phi(x^*x) < \\infty\\}.\\]\nRecall that $M$ is called $\\sigma$-finite if it admits a faithful normal state.\n\\subsection{Ultrapowers of von Neumann algebras}\n\nWe start with a reminder on the Ocneanu ultrapower of a von Neumann algebra and related concepts.\nThis originates in \\cite[Section 2]{Connes74} and \\cite[Section 5]{Ocneanu85}, but the reader is also referred to \\cite{AndoHaagerup14} for a thorough exposition on ultrapower constructions.\n\n\\begin{definition}\nLet $M$ be a $\\sigma$-finite von Neumann algebra. We define the subset $\\mathcal{I}_\\omega(M) \\subset \\ell^\\infty(M)$ by\n\\begin{align*}\n\\mathcal{I}_\\omega(M) &= \\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid x_n \\rightarrow 0 \\text{ $*$-strongly as } n \\rightarrow \\omega\\}\\\\\n&= \\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid \\lim_{n \\rightarrow \\omega}\\|x_n\\|_{\\phi}^\\# =0 \\text{ for some faithful normal state } \\phi \\text{ on } M\\}.\n\\end{align*}\nDenote \\[\\mathcal{N}_\\omega(M)=\\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid (x_n)_{n \\in \\mathbb{N}} \\mathcal{I}_\\omega(M) \\subset \\mathcal{I}_\\omega(M), \\text{ and } \\mathcal{I}_\\omega(M)(x_n)_{n \\in \\mathbb{N}} \\subset \\mathcal{I}_\\omega(M)\\},\\] \n\\[\\mathcal{C}_\\omega(M) =\\{(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M) \\mid \\lim_{n \\rightarrow \\omega}\\|[x_n,\\phi]\\| = 0 \\text{ for all } \\phi \\in M_*\\}.\\]\nThen \\[{\\mathcal{I}_\\omega(M) \\subset \\mathcal{C}_\\omega(M) \\subset \\mathcal{N}_\\omega(M)}.\\] \nThe \\emph{Ocneanu} ultrapower $M^\\omega$ of $M$ is defined as\n\\[M^\\omega := \\mathcal{N}_\\omega(M)\/\\mathcal{I}_\\omega(M),\\]\nand the \\emph{asymptotic centralizer} $M_\\omega$ of $M$ is defined as\n\\[M_\\omega := \\mathcal{C}_\\omega(M)\/\\mathcal{I}_\\omega(M).\\]\nThese are both von Neumann algebras. \nAny faithful normal state $\\phi$ on $M$ induces a faithful normal state $\\phi^\\omega$ on $M^\\omega$ via the formula\n\\[\\phi^\\omega((x_n)_{n \\in \\mathbb{N}}) = \\lim_{n \\rightarrow \\omega} \\phi(x_n).\\]\nThe restriction of $\\phi^\\omega$ to $M_\\omega$ is a tracial state.\n\\end{definition}\n\n\\begin{remark}\nSince the constant sequences are easily seen to be contained in $\\mathcal{N}_\\omega(M)$, one considers $M$ as a subalgebra of $M^\\omega$.\nIf $\\lim_{n \\rightarrow \\omega}\\|[x_n,\\phi]\\| = 0$ for all $\\phi \\in M_*$, then $\\lim_{n \\rightarrow \\omega}\\|[x_n,y]\\|_\\phi^\\# = 0$ for all $y \\in M$ by  \\cite[Proposition 2.8]{Connes74}.\n  In this way we get a natural inclusion $M_\\omega \\subset M^\\omega \\cap M'$.\n   That same proposition also shows that in order to check whether a sequence $(x_n)_n$ in $\\ell^\\infty(M)$ satisfies $\\lim_{n \\rightarrow \\omega}\\|[x_n,\\psi]\\| = 0$ for all $\\psi \\in M_*$, it suffices to check if this is true for just a single faithful normal state $\\phi$ and to check if $\\lim_{n \\rightarrow \\omega}\\|[x_n,y]\\|^\\#_\\phi =0$ for all $y \\in M$.\n    This shows that $M_\\omega = M^\\omega \\cap M'$ whenever $M$ admits a faithful normal tracial state. The same is then true for all semi-finite von Neumann algebras with separable predual (for example by \\cite[Lemma~2.8]{MasudaTomatsu16}).\n\\end{remark}\n\n\\begin{definition}\nA continuous action $\\alpha\\colon G \\curvearrowright M$ of a second-countable locally compact group on a von Neumann algebra is a homomorphism $G \\to \\mathrm{Aut}(M)$, $g \\mapsto \\alpha_g$ such that\n\\[\n\\lim_{g \\rightarrow 1_G}\\|\\varphi \\circ \\alpha_g - \\varphi\\|=0 \\text{ for all } \\varphi \\in M_*.\n\\]\nBy \\cite[Proposition~X.1.2]{Takesaki03}, this is equivalent to the map being continuous for the point-weak-$*$ (or equivalently, point-weak, point-strong,$\\hdots$) topology.\nIn most contexts we omit the word ``continuous'' as it will be implicitly understood that we consider some actions to be continuous.\nIn contrast, we will explicitly talk of an algebraic $G$-action when we are considering an action of $G$ viewed as a discrete group.\n\\end{definition}\n\nGiven an action $\\alpha\\colon G \\curvearrowright M$, the induced algebraic actions $\\alpha^\\omega\\colon G \\rightarrow \\mathrm{Aut}(M^\\omega)$ and $\\alpha_\\omega \\colon G \\rightarrow \\mathrm{Aut}(M_\\omega)$ are usually not continuous.\nThe remainder of this subsection is devoted (for lack of a good literature reference) to flesh out the construction of their `largest' von Neumann subalgebras where the action is sufficiently well-behaved for our needs, called the \\emph{$(\\alpha, \\omega)$-equicontinuous parts} (see Definition \\ref{def:equicontinuous_parts}).\nThese constructions are based on \\cite[Section 3]{MasudaTomatsu16}, where the special case $G=\\mathbb{R}$ is considered. \n\n\\begin{definition}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$.\nFix a faithful normal state $\\phi$ on $M$. An element $(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M)$ is called \\emph{$(\\alpha, \\omega)$-equicontinuous} if for every $\\varepsilon>0$, there exists a set $W \\in \\omega$ and an open neighborhood $1_G\\in U \\subset G$ such that\n\\[\n\\sup_{n\\in W} \\sup_{g\\in U} \\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\varepsilon .\n\\]\nWe denote the set of $(\\alpha, \\omega)$-equicontinuous sequences by $\\mathcal{E}^\\omega_\\alpha$.\n\\end{definition}\n\n\\begin{remark}\nThe definition above does not depend on the faithful normal state chosen.\nWhenever $\\phi$ and $\\psi$ are two faithful normal states on $M$, one has for every $\\varepsilon>0$ some $\\delta>0$ such that for all $x \\in (M)_1$, $\\|x\\|_\\phi^\\# < \\delta$ implies $\\|x\\|_\\psi^\\# < \\varepsilon$. \n\\end{remark}\n\n\\begin{lemma}\\label{lemma:epsilon_delta_sequences}\nLet $M$ be a von Neumann algebra with faithful normal state $\\phi$. For all $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{N}_\\omega(M)$ the following holds: \nFor any $\\varepsilon >0$ and compact set $\\Psi \\subset M_*^+$ there exists a $\\delta >0$ and $W \\in \\omega$ such that if $y \\in (M)_1$ and $\\|y\\|_\\phi^\\#< \\delta$, then $\\sup_{\\psi\\in \\Psi} \\|x_n y\\|_{\\psi}^\\#  <\\varepsilon$ and $\\sup_{\\psi\\in \\Psi} \\|y x_n\\|_{\\psi}^\\#  <\\varepsilon$ for all $n \\in W$. \n\\end{lemma}\n\\begin{proof} We prove this by contradiction. \nSuppose that there exists $\\varepsilon >0$ and a compact set $\\Psi \\subset M_*^+$ such that for any $k \\in \\mathbb{N}$ there exists a $y_k \\in (M)_1$ with $\\|y_k\\|_\\phi^\\# < 1\/k$ but the following set belongs to $\\omega$: \n\\[\nA_k := \\Big\\{ n \\in\\mathbb{N} \\ \\Big| \\ \\sup_{\\psi \\in \\Psi}\\|x_ny_k\\|_\\psi^\\#\\geq \\varepsilon \\text{ or } \\sup_{\\psi \\in \\Psi}\\|y_k x_n\\|_\\psi^\\# \\geq \\varepsilon \\Big\\}.\n\\]\nDefine $W_0 := \\mathbb{N}$ and $W_k := A_1 \\cap \\hdots \\cap A_k \\cap [k, \\infty)$ for $k \\geq 1$. \nThese all belong to $\\omega$. \nFor each $n \\in \\mathbb{N}$ define $k(n)$ as the unique number $k \\geq 0$ with $n \\in W_k \\setminus W_{k+1}$. Put $z_n := y_{k(n)}$ if $k(n) \\geq 1$, else put $z_n:=1_M$. \nNote that for all $n \\in W_m$ we get that $\\|z_n\\|_\\phi^\\# = \\|y_{k(n)}\\|_\\phi^\\# <\\frac{1}{k(n)} \\leq \\frac{1}{n}$. \nTherefore, it holds that $(z_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)$. \nSince $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{N}_\\omega(M)$, it follows that also $(x_nz_n)_{n\\in \\mathbb{N}}$ and $(z_nx_n)_{n\\in \\mathbb{N}}$ belong to $\\mathcal{I}_\\omega(M)$. \nHence we get that for all $\\psi \\in \\Psi$ \n\\[\\lim_{n \\rightarrow \\omega} \\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right) = 0.\\]\nSince $\\Psi$ is compact, we also have \n\\[\\lim_{n \\rightarrow \\omega} \\sup_{\\psi \\in \\Psi}\\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right) = 0.\\]\nThis gives a contradiction, since our choice of $z_n$ implies that for all $n \\in W_1$ \n\\[\\sup_{\\psi \\in \\Psi}\\left(\\|x_nz_n\\|_\\psi^\\# + \\|z_nx_n\\|_\\psi^\\#\\right)\\geq \\varepsilon.\\]\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with action $\\alpha:G \\curvearrowright M$.\nFor any two sequences $(x_n)_{n \\in \\mathbb{N}}, (y_n)_{n \\in \\mathbb{N}} \\in  \\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ it follows that $(x_ny_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$. \n\\end{lemma}\n\\begin{proof}\nWithout loss of generality we may assume $\\sup_{n \\in \\mathbb{N}} \\|x_n\\| \\leq \\frac{1}{2}$ and $\\sup_{n \\in \\mathbb{N}} \\|y_n\\| \\leq \\frac{1}{2}$.\nFix a faithful normal state $\\phi$ on $M$. Let $K \\subset G$ be a compact neighbourhood of the neutral element.\nTake $\\varepsilon >0$ arbitrarily.\nBy Lemma \\ref{lemma:epsilon_delta_sequences} there exists $\\delta>0$ and $W_1 \\in \\omega$ such that for every $z \\in (M)_1$ with $\\|z\\|_\\phi^\\# < \\delta$ one has \n\\[\n\\sup_{g \\in K}\\|x_n z\\|_{\\phi \\circ \\alpha_g}^\\#< \\frac{\\varepsilon}{2} \\text{ and } \\|zy_n\\|_\\phi^\\# < \\frac{\\varepsilon}{2} \\text{ for all } n \\in W_1.\n\\] \nSince $(x_n)_{n\\in \\mathbb{N}}$ and $(y_n)_{n\\in \\mathbb{N}}$ both belong to $\\mathcal{E}_\\alpha^\\omega$, we can find an open $U \\subset K$ containing the neutral element, and a $W_2 \\in \\omega$ such that\n\\[\n\\sup_{n \\in W_2}\\sup_{ g \\in U}\\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\delta \\text{, and}\n\\]\n\\[\\sup_{n \\in W_2}\\sup_{ g \\in U}\n\\|\\alpha_{g^{-1}}(y_n) - y_n\\|_\\phi^\\# < \\delta.\n\\]\nThen for $g \\in U$ and $n \\in W_1 \\cap W_2$ we have\n\\begin{align*}\n\\|\\alpha_g(x_n) \\alpha_g(y_n) - x_ny_n\\|_\\phi^\\# &\\leq \\|\\alpha_g(x_n)(\\alpha_g(y_n) - y_n)\\|_\\phi^\\# + \\|(\\alpha_g(x_n) - x_n)y_n\\|_\\phi^\\#\\\\\n&= \\|x_n(\\alpha_{g^{-1}}(y_n) - y_n)\\|_{\\phi \\circ \\alpha_g}^\\# + \\|(\\alpha_g(x_n) - x_n)y_n\\|_\\phi^\\#\\\\\n& < \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{2} = \\varepsilon.\n\\end{align*}\nThis ends the proof.\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha: G \\curvearrowright M$.\nThen:\n\\begin{enumerate}[label=\\textup{(\\arabic*)},leftmargin=*]\n\\item \\label{lem:equicont-algebra:1}\nSuppose $(x_n)_{n\\in \\mathbb{N}}, (y_n)_{n\\in \\mathbb{N}} \\in \\ell^\\infty(M)$ satisfy $(x_n-y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)$.\nThen $(x_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}^\\omega_\\alpha$ if and only if $(y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{E}^\\omega_\\alpha$.\n\\item \\label{lem:equicont-algebra:2}\n$\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ is an $\\alpha$-invariant C$^*$-subalgebra of $\\ell^\\infty(M)$. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} Fix a faithful normal state $\\phi$ on $M$.\nWe first prove \\ref{lem:equicont-algebra:1}.\nLet $\\varepsilon>0$.\nWe can choose $W \\in \\omega$ and an open neighborhood $1_G \\in U \\subset G$ such that \n\\[\n\\sup_{n\\in W} \\sup_{g\\in U} \\|\\alpha_g(x_n) - x_n\\|_\\phi^\\# < \\frac{\\varepsilon}{2}.\n\\] \nWithout loss of generality we may assume that $K=\\overline{U}$ is compact.\nConsider $s_n := \\sup_{g \\in K} \\|x_n - y_n\\|_{\\phi \\circ \\alpha_g}^\\#$. Since $K$ is compact and ${(x_n-y_n)_{n\\in \\mathbb{N}} \\in \\mathcal{I}_\\omega(M)}$, we have $\\lim_{n \\rightarrow \\omega} s_n = 0$.\nHence, after possibly replacing $W$ by a smaller set in the ultrafilter, we can assume that $s_n < \\varepsilon\/4$ for all $n \\in W$.\nWe may conclude for all $g \\in U$ and $n \\in W$ that\n\\begin{align*}\n\\|\\alpha_g(y_n) - y_n\\|_\\phi^\\# &\\leq \\|\\alpha_g(y_n) - \\alpha_g(x_n)\\|_\\phi^\\# + \\|\\alpha_g(x_n)- x_n\\|_\\phi^\\# + \\|x_n - y_n\\|_\\phi^\\#\\\\\n&\\leq 2s_n + \\frac{\\varepsilon}{2} < \\varepsilon.\n\\end{align*}\nSince $\\varepsilon>0$ was arbitrary, $(y_n)_{n\\in \\mathbb{N}}$ belongs to $\\mathcal{E}_\\alpha^\\omega$.\n\nLet us prove \\ref{lem:equicont-algebra:2}.\nClearly $\\mathcal{E}_\\alpha^\\omega$ is a $*$-closed, norm-closed linear subspace of $\\ell^\\infty(M)$. \nThe previous lemma shows that $\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M)$ is closed under multiplication. \nTo see that $\\mathcal{E}_\\alpha^\\omega$ is $\\alpha$-invariant, take $(x_n)_{n \\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$ and $h \\in G$. Take $\\varepsilon >0$. \nWe can find an open neighborhood $1_g \\in U \\subset G$ and $W \\in \\omega$ such that one has\n\\[\\sup_{n \\in W}\\sup_{g \\in U}\\|\\alpha_g(x_n) - x_n\\|_{\\phi \\circ \\alpha_h}^\\# < \\varepsilon.\\] \nThen for all $g \\in hUh^{-1}$ and $n \\in W$ we observe\n\\[\n\\|\\alpha_g(\\alpha_h(x_n)) - \\alpha_h(x_n)\\|_\\phi^\\# = \\|\\alpha_{h^{-1}gh}(x_n) - x_n\\|_{\\phi \\circ \\alpha_h}^\\# < \\varepsilon.\n\\]\nThis shows that $(\\alpha_h(x_n))_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$.\n\\end{proof}\n\n\\begin{definition}\\label{def:equicontinuous_parts}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$. \nWe define ${M_\\alpha^\\omega := (\\mathcal{E}_\\alpha^\\omega \\cap \\mathcal{N}_\\omega(M))\/\\mathcal{I}_\\omega}$ and ${M_{\\omega, \\alpha}:= M_\\alpha^\\omega \\cap M_\\omega}$. \nWe call them the \\emph{$(\\alpha,\\omega)$-equicontinuous parts} of $M^\\omega$ and $M_\\omega$, respectively. \n\\end{definition}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$.\nThen $M_\\alpha^\\omega$ and $M_{\\omega, \\alpha}$ are von Neumann algebras.  \n\\end{lemma}\n\\begin{proof}\nWe show that $M_\\alpha^\\omega$ is a von Neumann algebra by showing that its unit ball is closed with respect to the strong operator topology in $M^\\omega$. \nThen it automatically follows that $M_{\\omega, \\alpha} = M_\\alpha^\\omega \\cap M_\\omega$ is also a von Neumann algebra. \nTake a sequence $(X_k)_k \\in (M_\\alpha^\\omega)_1$ that strongly converges to $X \\in (M^\\omega)_1$. Fix a faithful normal state $\\phi$ on $M$ and a compact neighbourhood of the neutral element $K \\subset G$. \nThen the function $K \\rightarrow (M^\\omega)_*$ given by $g \\mapsto \\phi^\\omega \\circ \\alpha_g^\\omega$ is continuous (because $\\phi^\\omega \\circ \\alpha_g^\\omega = (\\phi \\circ \\alpha_g)^\\omega)$. \nHence, the set $\\{\\phi^\\omega \\circ \\alpha_g^\\omega\\}_{g \\in K}$ is compact and thus $\\lim_{n \\rightarrow \\infty} \\sup_{g \\in K} \\|X_n - X\\|^\\#_{\\phi^\\omega \\circ \\alpha_g^\\omega} = 0$.\nFix $\\varepsilon >0$. \nPick representing sequences $(x_k(n))_{n\\in \\mathbb{N}}$ and $(x(n))_{n\\in \\mathbb{N}}$ for the elements $X_k$ and $X$, respectively, such that $\\|x_k(n)\\| \\leq 1$, $\\|x(n)\\| \\leq 1$, for all $k,n \\in \\mathbb{N}$.\nThen we can find $k_0 \\in \\mathbb{N}$  and $W_1 \\in \\omega$ such that \n\\[\n\\sup_{n\\in W_1} \\sup_{g\\in K} \\|x_{k_0}(n) - x(n)\\|_{\\phi\\circ \\alpha_g}^\\# < \\frac{\\varepsilon}{3}.\n\\]\nSince $(x_{k_0}(n))_{n\\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$, we can find an open neighborhood $1_G \\in U \\subset K$ and $W_2 \\in \\omega$ such that\n\\[\n\\sup_{n\\in W_2} \\sup_{g\\in U} \\|\\alpha_g(x_{k_0}(n)) - x_{k_0}(n)\\|_\\phi^\\# < \\frac{\\varepsilon}{3}.\n\\]\nThen for all $g \\in U$ and $n \\in W_1 \\cap W_2$ it holds that\n\\begin{align*}\n\\|\\alpha_g(x(n)) - x(n)\\|_\\phi^\\# & \\leq \\|x(n) - x_{k_0}(n)\\|_{\\phi \\circ \\alpha_g}^\\# + \\|\\alpha_g(x_{k_0}(n)) - x_{k_0}(n)\\|_\\phi^\\# + \\|x_{k_0}(n) - x(n)\\|_\\phi^\\#\\\\\n&< \\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3}+\\frac{\\varepsilon}{3} = \\varepsilon. \n\\end{align*} \nThis shows that $(x(n))_{n \\in \\mathbb{N}} \\in \\mathcal{E}_\\alpha^\\omega$, or in other words $X \\in M_\\alpha^\\omega$.\n\\end{proof}\n\n\\begin{lemma}\nLet $M$ be a $\\sigma$-finite von Neumann algebra with an action $\\alpha\\colon G \\curvearrowright M$ of a second-countable locally compact group. \nThen $\\alpha^\\omega$ restricted to $M_\\alpha^\\omega$ and $\\alpha_\\omega$ restricted to $M_{\\omega,\\alpha}$ are continuous $G$-actions.\n\\end{lemma}\n\\begin{proof}\nFix a faithful normal state $\\phi$ on $M$. Since $\\phi^\\omega$ is faithful, $\\{a\\phi^\\omega \\mid a \\in M_\\alpha^\\omega\\}$ is dense in $(M_\\alpha^\\omega)_*$. For $a \\in M_\\alpha^\\omega$ and $g \\in G$ one has\n\\begin{align*}\n\\|(a\\phi^\\omega)  \\circ \\alpha^\\omega_g - a \\phi^\\omega \\|_{(M_\\alpha^\\omega)_*} & \\leq \\|\\alpha_{g^{-1}}^\\omega(a)(\\phi^\\omega \\circ \\alpha_g^\\omega - \\phi^\\omega)\\|_{(M_\\alpha^\\omega)_*} \\|+ \\|(\\alpha_{g^{-1}}^\\omega(a) - a)\\phi^\\omega\\|_{(M_\\alpha^\\omega)_*}\\\\\n& \\leq \\|a\\| \\, \\|\\phi \\circ \\alpha_g - \\phi\\|_{M_*} + \\|\\alpha_{g^{-1}}^\\omega(a) - a\\|_{\\phi^\\omega}.\n\\end{align*}\nWhen $g \\rightarrow 1_G$, this expression converges to zero because $\\alpha$ is a continuous $G$-action and $a \\in M_\\alpha^\\omega$. \nThis shows that $\\alpha^\\omega$ restricts to a genuine continuous $G$-action on $M_\\alpha^\\omega$, so the same is true for the restriction of $\\alpha_\\omega$ to $M_{\\omega, \\alpha}$. \n\\end{proof}\n\n\\begin{lemma} \\label{lemma:lifting_invariance_compact_sets}\nLet $M$ be a von Neumann algebra with a faithful normal state $\\phi$ and an action $\\alpha\\colon G \\curvearrowright M$. \nLet $z \\in M_\\alpha^\\omega$, $\\varepsilon >0$, $K \\subset G$ a compact set and suppose that $\\| \\alpha_g^\\omega(z)-z\\|_{\\phi^\\omega}^\\# \\leq \\varepsilon$ for all $g \\in K$.\nIf $(z_n)_{n\\in \\mathbb{N}}$ is any bounded sequence representing $z$, then \n\\[\n\\lim_{n \\rightarrow \\omega} \\max_{g \\in K} \\| \\alpha_g(z_n)-z_n\\|_\\phi^\\# \\leq \\varepsilon.\n\\]\n\\end{lemma}\n\\begin{proof}\nLet $\\delta>0$.\nThen for each $g \\in K$ there exists an open neighborhood $g \\in U \\subset G$ and $W_g \\in \\omega$ such that\n\\[\\sup_{n \\in W_g}\\sup_{h \\in U}\\|\\alpha_h(z_n) - \\alpha_g(z_n)\\|_\\phi^\\# < \\delta.\\]\nSince this obviously yields an open cover of $K$ and $K$ is compact, we can find finitely many elements $g_1, \\hdots, g_N \\in K$ and an open covering $K \\subset \\cup_{i=1}^N U_j$ with $g_j \\in U_j$ and some $W_1 \\in \\omega$ such that for $j=1, \\hdots, N$ we have\n\\[\\sup_{n \\in W_1}\\sup_{g \\in U_j}\\|\\alpha_{g}(z_n) - \\alpha_{g_j}(z_n)\\|_\\phi^\\# < \\delta.\\]\nSince $\\max_{g \\in K}\\| \\alpha_g^\\omega(z)-z\\|_{\\phi^\\omega}^\\# \\leq \\varepsilon$, there exists $W_2 \\in \\omega$ such that for all $n \\in W_2$ and $j=1, \\hdots, N$ \n\\[\\|\\alpha_{g_j}(z_n) - z_n\\|_\\phi^\\# \\leq \\varepsilon+\\delta.\\]\nHence, for an arbitrary $g \\in K$, there is some $j \\in \\{1, \\hdots, N\\}$ such that $g \\in U_j$ and\n\\[\\| \\alpha_g(z_n)-z_n\\|_\\phi^\\# \\leq \\|\\alpha_g(z_n) - \\alpha_{g_j}(z_n)\\|_\\phi^\\# + \\|\\alpha_{g_j}(z_n) - z_n\\|_\\phi^\\# \\leq 2\\delta+ \\varepsilon\\]\nfor all $n \\in W_1 \\cap W_2.$\nSince $\\delta$ was arbitrary, this proves the claim. \n\\end{proof}\n\n\\subsection{Cocycle morphisms}\n\n\\begin{definition}[cf.\\ {\\cite[Definition 1.10]{Szabo21cc}}]\nLet $\\alpha\\colon G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions of a second-countable locally compact group on von Neumann algebras. \nA \\emph{cocycle morphism} from $(M,\\alpha)$ to $(N,\\beta)$ is a pair $(\\phi,\\mathbbm{u})$, where $\\phi\\colon M \\rightarrow N$ is a unital normal $*$-homomorphism and $\\mathbbm{u}:G\\rightarrow \\mathcal{U}(N)$ is a continuous map (in the strong operator topology) such that for all $g,h \\in G$ we have\n\\[\n\\mathrm{Ad}(\\mathbbm{u}_g) \\circ \\beta_g \\circ \\phi = \\phi \\circ \\alpha_g \\quad\\text{and}\\quad \\mathbbm{u}_g \\beta_g(\\mathbbm{u}_h) = \\mathbbm{u}_{gh}.\n\\]\nIn the special case where $\\mathbbm{u}$ is the trivial map, we identify $\\phi$ with $(\\phi,1)$ and call $\\phi$ equivariant.\n\\end{definition}\n\n\\begin{remark} \\label{rem:cocycle-category}\nAs the arguments in \\cite[Subsection 1.3]{Szabo21cc} show, the above endows the class of continuous $G$-actions on von Neumann algebras with a categorical structure, whereby the Hom-sets are given by cocycle morphisms.\nThe composition is given via\n\\[\n(\\psi,\\mathbbm{v}) \\circ (\\phi,\\mathbbm{u}) := (\\psi \\circ \\phi, \\psi(\\mathbbm{u}) \\mathbbm{v})\n\\]\nfor any pair of cocycle morphisms\n\\[\n(M, \\alpha) \\overset{(\\phi,\\mathbbm{u})}{\\longrightarrow} (N, \\beta) \\quad \\text{ and } \\quad (N, \\beta)  \\overset{(\\psi,\\mathbbm{v})}{\\longrightarrow} (L, \\gamma).\n\\]\nWe see furthermore that a cocycle morphism $(\\phi, \\mathbbm{u})\\colon(M, \\alpha) \\rightarrow (N, \\beta)$ is invertible if and only if $\\phi$ is a $*$-isomorphism of von Neumann algebras, in which case we have $(\\phi, \\mathbbm{u})^{-1} = (\\phi^{-1}, \\phi^{-1}(\\mathbbm{u})^*)$.\nIf this holds, we call $(\\phi, \\mathbbm{u})$ a \\emph{cocycle conjugacy}.\nWe call two actions $\\alpha$ and $\\beta$ \\emph{cocycle conjugate}, denoted as $\\alpha \\simeq_{\\mathrm{cc}} \\beta$, if there exists a cocycle conjugacy between them.\n\\end{remark}\n\n\\begin{example} \\label{ex:inner-cc}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action.\nThen every unitary $v \\in \\mathcal{U}(M)$ gives rise to a cocycle conjugacy \n\\[\n\\big( \\mathrm{Ad}(v), (v \\alpha_g(v)^*)_{g \\in G} \\big) \\colon (M, \\alpha) \\rightarrow (M, \\alpha).\\]\nWe will also write this simply as $\\mathrm{Ad}(v)$ when it is clear from context that we are talking about cocycle morphisms. \nWhen $\\beta\\colon G \\curvearrowright N$ is another action and ${(\\phi, \\mathbbm{u})\\colon (M, \\alpha) \\rightarrow (N, \\beta)}$ is a cocycle conjugacy, then\n\\[ (\\phi,\\mathbbm{u}) \\circ \\mathrm{Ad}(v) = \\mathrm{Ad}(\\phi(v)) \\circ (\\phi, \\mathbbm{u}).\\] \n\\end{example}\n\n\n\\begin{definition}\nLet $\\alpha:G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions on finite von Neumann algebras $M$ and $N$.\nLet $\\tau_N$ be a faithful normal tracial state on $N$.\nLet $(\\phi,\\mathbbm{u})$ and $(\\psi,\\mathbbm{v})$ be two cocycle morphisms from $(M,\\alpha)$ to $(N,\\beta)$. We say that $(\\phi,\\mathbbm{u})$ and $(\\psi,\\mathbbm{v})$ are \\emph{approximately unitarily equivalent} if there exists a net of unitaries $w_\\lambda \\in \\mathcal{U}(N)$ such that $\\|w_\\lambda\\phi(x)w_\\lambda^*-\\psi(x)\\|_{\\tau_N} \\to 0$ for all $x\\in M$ and $\\max_{g\\in K} \\| w_\\lambda \\mathbbm{u}_g \\beta_g(w_\\lambda)^* - \\mathbbm{v}_g\\|_{\\tau_N} \\rightarrow 0$ for every compact set $K\\subseteq G$.\nWe denote the relation of approximately unitary equivalence by $\\approx_{\\mathrm{u}}$.\n\\end{definition}\n\n\\section{One-sided intertwining}\nIn this section we prove a version of \\cite[Lemma~2.1]{Szabo18ssa} for group actions on semi-finite von Neumann algebras. First we prove the following intermediate lemma:\n\n\\begin{lemma}\\label{lemma:point_strong_dense_subset}\nLet $M, N$ be von Neumann algebras, and let $\\tau_N$ be  a faithful, normal, semi-finite trace on $N$. \nConsider a sequence of $*$-homomorphisms $(\\theta_n\\colon M \\rightarrow N)_{n \\in \\mathbb{N}}$ and a $*$-isomorphism $\\theta:M \\rightarrow N$ such that $\\tau_N \\circ \\theta = \\tau_N \\circ \\theta_n$ for all $n \\in \\mathbb{N}$. \nLet $X \\subset (M)_1$ be a dense subset in the strong operator topology that contains a sequence of projections $(p_n)_{n \\in \\mathbb{N}}$ converging strongly to $1_M$ with $\\tau_N (\\theta(p_n)) < \\infty$. \nIf $\\theta_n (x) \\rightarrow \\theta(x)$ strongly as $n \\rightarrow \\infty$ for every $x \\in X$, then $\\theta_n \\rightarrow \\theta$ in the point-strong topology as $n \\rightarrow \\infty$.\n\\end{lemma}\n\\begin{proof}\nTake $y \\in (M)_1$. Since the sequence $(\\theta(p_n))_{n \\in \\mathbb{N}}$ converges strongly to $1_N$, it suffices to show that for all $k \\in \\mathbb{N}$\n\\[ (\\theta(y)-\\theta_n(y)) \\theta(p_k) \\rightarrow 0 \\text{ strongly as } n \\rightarrow \\infty.\\]\nFix $k \\in \\mathbb{N}$ and $a \\in N$ such that $\\tau_N(a^*a) < \\infty$. Given $\\varepsilon>0$, there exists $x \\in X$ such that \n\\[\\|\\theta(x-y)\\theta(p_k)\\|_{\\tau_N} < \\frac{\\varepsilon}{4\\|a\\|}.\\]\n Then there exists $n_0 \\in \\mathbb{N}$ such that for all $n \\geq n_0$ \n \\[\\|(\\theta(x p_k) - \\theta_n ( x p_k))a\\|_{\\tau_N} < \\frac{\\varepsilon}{4} \\text{ and } \\|(\\theta(p_k)-\\theta_n(p_k))a\\|_{\\tau_N} < \\frac{\\varepsilon}{4}.\\] \nFor all $n \\geq n_0$ we then get that\n\\begin{align*}\n\\|(\\theta(y) - \\theta_n(y))\\theta(p_k)a\\|_{\\tau_N} &\\leq \\|\\theta(x-y)\\theta(p_k)a\\|_{\\tau_N} + \\|\\theta_n(x-y)\\theta_n(p_k)a\\|_{\\tau_N}\\\\\n&\\quad + \\|(\\theta(xp_k) - \\theta_n(xp_k))a\\|_{\\tau_N} + \\|\\theta_n(y)(\\theta(p_k) - \\theta_n(p_k))a\\|_{\\tau_N}\\\\\n&< 2\\|a\\| \\|\\theta(x-y)\\theta(p_k)\\|_{\\tau_N} +\\varepsilon\/4 + \\|(\\theta(p_k)-\\theta_n(p_k))a\\|_{\\tau_N} \\\\\n&< \\varepsilon.\n\\end{align*}\nAs $k$ and $a$ were arbitrary, this proves the claim.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:one-sided_intertwining} \nLet $M$ and $N$ be two von Neumann algebras with separable predual and faithful normal semi-finite traces $\\tau_{M}$ and $\\tau_{N}$, respectively. \nLet $\\alpha\\colon G \\curvearrowright M$ and $\\beta\\colon G \\curvearrowright N$ be two actions.\nLet $\\rho\\colon (M, \\alpha) \\rightarrow (N, \\beta)$ be a unital equivariant normal $*$-homomorphism with $\\tau_N \\circ \\rho = \\tau_M$.\nSuppose there exists a faithful normal state $\\phi$ on $N$ and a sequence of unitaries $(w_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{U}(N)$ satisfying\n\\begin{enumerate}[leftmargin=*,label=$\\bullet$]\n\\item $\\mathrm{Ad}(w_n) \\circ \\rho \\to \\rho$ in the point-strong topology; \n\\item For all $y \\in (N)_1$ there exists a sequence $(x_n)_{n \\in \\mathbb{N}} \\subset (M)_1$ such that $y -  w_n\\rho(x_n)w_n^* \\rightarrow 0$ in the strong operator topology;\n\\item $\\max_{g \\in K} \\|\\beta_g(w_n^*) - w_n^*\\|_\\phi \\rightarrow 0$ for every compact subset $K \\subseteq G$.\n\\end{enumerate}\nThen $\\rho(\\mathcal{Z}(M))=\\mathcal{Z}(N)$ and there exists a cocycle conjugacy $(\\theta,\\mathbbm{v})$ between $\\alpha$ and $\\beta$ with $\\theta|_{\\mathcal{Z}(M)}=\\rho|_{\\mathcal{Z}(M)}$.\nIn case $\\tau_N$ is finite, the existence of such a sequence of unitaries for $\\phi = \\tau_N$ is equivalent to the condition that $\\rho$ is approximately unitarily equivalent to a cocycle conjugacy. \n\\end{lemma}\n\\begin{proof}\nWe note right away that the first two conditions above can always be tested on self-adjoint elements, hence one can equivalently state them with the strong-$*$ topology. \nDenote\n\\[\\mathfrak{m} := \\{x \\in M \\mid \\tau_M(x^*x) < \\infty\\} \\subset M.\\]\nWe let $L^2(M,\\tau_M)$ denote the GNS-Hilbert space of $M$ with respect to $\\tau_M$.\nSimilarily, we use the notation $L^2(N, \\tau_N)$.\n\nChoose a countable subset $X = \\{x_n\\}_{n \\in \\mathbb{N}}$ in $(M)_1$ such that $X \\cap \\mathfrak{m}$ is $\\|\\cdot\\|_{\\tau_M}$-dense in $(\\mathfrak{m})_1$. \nTake a strongly dense sequence $\\{y_n\\}_{n \\in \\mathbb{N}}$ in $(N)_1$.\nChoose an increasing sequence of compact subsets $K_n \\subseteq G$ such that the union is all of $G$. \n\nWe are going to create a map $\\theta: M\\to N$ via an inductive procedure. \nFor the first step, we choose $x_{1,1} \\in (M)_1$ and $z_1 \\in \\mathcal{U}(N)$ such that\n\\begin{itemize}\n\\item $ \\|z_1 \\rho(x_1) z_1^* - \\rho(x_1)\\|^\\#_\\phi \\leq 1\/2$;\n\\item $\\|y_1 - z_1\\rho(x_{1,1})z_1^*\\|_\\phi \\leq 1\/2$;\n\\item $\\max_{g \\in K_1} \\|\\beta_g(z_1^*) - z_1^*\\|_\\phi \\leq 1\/2$.\n\\end{itemize}\nNow assume that after the $n$-th step of the induction we have found $z_1, \\hdots, z_n \\in \\mathcal{U}(N)$ and ${\\{x_{l,j}\\}_{j \\leq l \\leq n} \\subset (M)_1}$ such that\n\\begin{enumerate}\n\\item $\\|z_n \\rho(x_j) z_n^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $j= 1, \\hdots, n$; \\label{eq:commutation_dense}\n\\item $\\|z_n \\rho(x_{l,j}) z_n^* - \\rho(x_{l,j})\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $l=1, \\hdots, n-1$ and $j = 1, \\hdots, l$; \\label{eq:commutation_close_elements}\n\\item $ \\|  z_{n-1}^* \\hdots z_1^*  y_j z_1 \\hdots z_{n-1} - z_n\\rho(x_{n,j})z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ for $j=1, \\hdots, n$; \\label{eq:close_elements}\n\\item \n$\\max_{g \\in K_n} \\|\\beta_g(z_n^*) - z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{n-1})} \\leq 2^{-n}$ and\n\n$\\max_{g \\in K_n} \\|\\beta_g(z_n^*) - z_n^*\\|_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_{n-1}))} \\leq 2^{-n}$.\n \\label{eq:invariance}\n\n\\label{eq:approximate_fixedness}\n\\end{enumerate}\nThen by our assumptions we can find $z_{n+1} \\in \\mathcal{U}(N)$ and $\\{x_{n+1,j}\\}_{j \\leq n+1}\\subset (M)_1$ such that\n\\begin{itemize}\n\\item $\\|z_{n+1} \\rho(x_j)z_{n+1}^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $j= 1, \\hdots, n+1$;\n\\item $\\|z_{n+1} \\rho(x_{l,j})z_{n+1}^* - \\rho(x_{l,j})\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $l=1, \\hdots, n$ and $j = 1, \\hdots, l$;\n\\item $ \\|z_{n}^* \\hdots z_1^*y_jz_1 \\hdots z_{n} - z_{n+1}\\rho(x_{n+1,j})z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ for $j=1, \\hdots, n+1$;\n\\item $\\max_{g \\in K_{n+1}} \\|\\beta_g(z_{n+1}^*)- z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_n)} \\leq 2^{-(n+1)}$ and\n\n$\\max_{g \\in K_{n+1}} \\|\\beta_g(z_{n+1}^*)- z_{n+1}^*\\|_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_n))} \\leq 2^{-(n+1)}$. \n\\end{itemize}\nWe carry on inductively and obtain a sequence of unitaries $(z_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{U}(N)$ and a family $\\{ x_{n,j} \\}_{n\\in\\mathbb{N}, j\\leq n}\\subset (M)_1$.\nFor each $n \\in \\mathbb{N}$, we define $u_n=z_1 \\hdots z_n$ and the normal $*$-homomorphism ${\\theta_n\\colon M \\rightarrow N}$ by $\\theta_n = \\mathrm{Ad}(u_n) \\circ \\rho$.\n\nFor $n > m$ and $j=1, \\hdots, m+1$ we get\n\\begin{align*}\n\\|\\theta_n(x_j) - \\theta_m(x_j)\\|^\\#_{\\phi} &\\leq \n\\sum_{k=m}^{n-1} \\|\\theta_{k+1}(x_j) - \\theta_k(x_j)\\|^\\#_\\phi\\\\\n&= \\sum_{k=m}^{n-1} \\|z_{k+1}\\rho(x_j)z_{k+1}^* - \\rho(x_j)\\|^\\#_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_k)}\\\\\n\\overset{\\ref{eq:commutation_dense}}&{\\leq}  \\sum_{k=m}^{n-1} 2^{-k-1}.\n\\end{align*}\nWe see that for all $j\\in \\mathbb{N}$ the sequence $(\\theta_n(x_j))_{n \\in \\mathbb{N}}$ is norm-bounded and Cauchy with respect to $\\|\\cdot\\|_\\phi^\\#$.\n This means that it converges to some element in $N$ in the strong-$*$-operator topology.\n  A similar calculation using \\ref{eq:commutation_close_elements} shows that\nfor $n > m \\geq l \\geq j$  \n\\begin{equation}\\label{eq:convergence_close_elements}\\|\\theta_n(x_{l,j}) - \\theta_m(x_{l,j})\\|^\\#_{\\phi} < \\sum_{k=m}^{n-1} 2^{-k-1}, \\end{equation}\nso the sequence $(\\theta_n(x_{l,j}))_{n \\in \\mathbb{N}}$ also converges in the strong-$*$-operator topology for all $j \\leq l$.\n Since $\\theta_n$ is a $*$-homomorphism for all $n \\in \\mathbb{N}$, we conclude that, restricted to the C$^*$-algebra $A\\subset M$ generated by $\\{x_n\\}_{n \\in \\mathbb{N}} \\cup \\{x_{l,j}\\}_{j \\leq l}$, the sequence $(\\theta_n)_{n \\in \\mathbb{N}}$ converges point-$*$-strongly to a $*$-homomorphism $\\theta'\\colon A \\rightarrow N$.\nSince $A$ contains a $\\|\\cdot\\|_{\\tau_M}$-dense subset of $\\mathfrak{m}$, and clearly $\\tau_N\\circ\\theta'=\\tau_M|_A$, there is a unique isometry $T\\colon L^2(M, \\tau_M) \\rightarrow L^2(N, \\tau_N)$ induced from the formula $T[a]=[\\theta'(a)]$ for all $a\\in A\\cap\\mathfrak{m}$.\n Then the normal $*$-homomorphism\n\\[\\theta \\colon M \\rightarrow N \\colon x \\mapsto T x T^*\\]\n extends $\\theta'$ and $\\left(\\theta_n \\big \\lvert_\\mathfrak{m}\\right)_{n \\in \\mathbb{N}}$ converges point-strongly to $\\theta\\lvert_\\mathfrak{m}$.\n\nWe claim that $\\theta$ is an isomorphism.\nClearly $\\tau_N\\circ\\theta=\\tau_M$ and so $\\theta$ is injective.\nBy applying \\ref{eq:close_elements} we find for all $m \\geq j$ that\n\\begin{equation*}\n\\|\\theta_m(x_{m,j}) - y_j\\|_\\phi = \\|z_m\\rho(x_{m,j})z_m^* -  z_{m-1}^* \\hdots z_1^*  y_j z_1 \\hdots z_{m-1}\\|_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_{m-1})} < 2^{-m}.\n\\end{equation*}\nCombining this with \\eqref{eq:convergence_close_elements} for $l=m$ and $n \\rightarrow \\infty$ we find that\n\\[\\|\\theta(x_{m,j}) - y_j\\|_\\phi \\leq \\|\\theta'(x_{m,j}) - \\theta_m(x_{m,j})\\|_\\phi + \\|\\theta_m(x_{m,j}) - y_j\\|_\\phi \\leq 2^{-m} + 2^{-m} = 2^{-m+1}.\\]\nSince the $y_j$ are strongly dense in the unit ball of $N$ and $\\theta$ is normal, this implies surjectivity of $\\theta$.\nBy Lemma~\\ref{lemma:point_strong_dense_subset} it then follows that $\\theta_n\\to \\theta$ point-strongly as $n\\to\\infty$.\nSince $\\theta_n$ is a unitary perturbation of $\\rho$ for each $n$, this implies $\\rho|_{\\mathcal{Z}(M)}=\\theta_n|_{\\mathcal{Z}(M)}\\to\\theta|_{\\mathcal{Z}(M)}$ and in particular $\\rho(\\mathcal{Z}(M))=\\theta(\\mathcal{Z}(M))=\\mathcal{Z}(N)$.\n\nFor $n > m$ and $g \\in K_{m+1}$ we have\n\\begin{align*}\n&\\|z_1 \\hdots z_n \\beta_g(z_n^* \\hdots z_1^*) - z_1 \\hdots z_m \\beta_g(z_m^*\\hdots z_1^*)\\|_{\\phi}^\\#\\\\\n&\\leq \\sum_{k=m}^{n-1} \\|z_1 \\hdots z_k(z_{k+1}\\beta_g(z_{k+1}^*) - 1) \\beta_g(z_k^*\\hdots z_1^*)\\|_{\\phi}^\\#\\\\\n&=\\sum_{k=m}^{n-1} \\big( \\|\\beta_g(z_{k+1}^*) - z_{k+1}^*\\|^2_{\\phi \\circ \\mathrm{Ad}(\\beta_g(z_1 \\hdots z_k))} + \\|\\beta_g(z_{k+1}^*) - z_{k+1}^*\\|^2_{\\phi \\circ \\mathrm{Ad}(z_1 \\hdots z_k)} \\big)^{1\/2}\\\\\n\\overset{\\ref{eq:invariance}}&{\\leq} \\sqrt{2} \\sum_{k=m}^{n-1} 2^{-(k+ 1)}.\n\\end{align*}\nFrom this calculation we see that for every $g \\in G$ the sequences $(z_1 \\hdots z_n \\beta_g(z_n^*\\hdots z_1^*))_{n \\in \\mathbb{N}}$ are Cauchy with respect to $\\|\\cdot\\|^\\#_\\phi$, with uniformity on compact sets. \nIt follows that for every $g \\in G$, the strong-$*$ limit $\\mathbbm{v}_g = \\lim_{n \\rightarrow \\infty} u_n \\beta_g(u^*_n)$ exists in $\\mathcal{U}(N)$ and that this convergence is uniform (w.r.t.\\ $\\|\\cdot\\|^\\#_\\phi$) on compact sets. \nSince $\\beta$ is point-strong continuous, this implies the continuity of the assignment $g \\mapsto \\mathbbm{v}_g$. \n\nMoreover, for each $g \\in G$ and $x \\in M$ we have the equalities of limits with respect to the strong operator topology:\n\\begin{align*}\n(\\theta \\circ \\alpha_g) (x) &= \\lim_{n \\rightarrow \\infty} (\\mathrm{Ad}(u_n) \\circ \\rho \\circ \\alpha_g)(x)\\\\\n&= \\lim_{n \\rightarrow \\infty} (\\mathrm{Ad}(u_n) \\circ \\beta_g \\circ \\rho) (x)\\\\\n&= \\lim_{n \\rightarrow \\infty} u_n \\beta_g(u_n^*)\\beta_g(u_n \\rho(x) u_n^*) \\beta_g(u_n) u_n^*\\\\\n&= (\\mathrm{Ad}(\\mathbbm{v}_g) \\circ \\beta_g \\circ \\theta) (x).\n\\end{align*}\nIt follows that $(\\theta,\\mathbbm{v})$ is a cocycle conjugacy. \n\nFor the last part of the statement, assume that $\\tau_N$ is finite.\nThen our previous calculations show that in the above situation, $\\rho$ is approximately unitarily equivalent to $\\theta$.\nConversely, suppose $\\rho$ is approximately unitarily equivalent to a cocycle conjugacy $(\\theta,\\mathbbm{v})$.\nIn particular, there exists a sequence $(u_n)_{n \\in \\mathbb{N}} \\in \\mathcal{U}(N)$ such that $\\|u_n\\rho(x)u_n^* -  \\theta(x)\\|_{\\tau_N} \\rightarrow 0$ for all $x \\in M$ and $\\|u_n \\beta_g(u_n^*) - \\mathbbm{v}_g\\|_{\\tau_N} \\rightarrow 0$ uniformly over compact subsets of $G$.\nChoose a sequence $\\{y_n\\}_{n \\in \\mathbb{N}} \\subset (N)_1$ that is strongly dense in $(N)_1$. For all $k,n \\in \\mathbb{N}$ define $x_{n,k} = \\theta^{-1}(u_ny_ku_n^*)$. Then choose an increasing sequence $(m(n))_{n \\in \\mathbb{N}} \\subset \\mathbb{N}$ such that\n \\begin{equation*}\n \\lim_{n \\rightarrow \\infty} \\| \\theta(x_{n,k}) - u_{m(n)} \\rho (x_{n,k}) u_{m(n)}^*\\|_{\\tau_N} = 0 \\quad \\text{for } k \\in \\mathbb{N}.\\end{equation*}\nDefine $w_n := u_n^*u_{m(n)}$. One can check that these satisfy the assumptions in the lemma. \n\\end{proof}\n\n\\section{Strongly self-absorbing actions}\n\n\\begin{definition}[cf.\\ {\\cite[Definition 5.1]{Szabo21cc}}] \\label{def:strong-absorption}\nLet $\\alpha:G \\curvearrowright M$ and $\\delta\\colon G \\curvearrowright N$ be two actions of a second-countable locally compact group on finite von Neumann algebras $M$ and $N$ with separable predual.\nWe say that $\\alpha$ \\emph{strongly absorbs} $\\delta$ if the equivariant embedding\n\\[\\mathrm{id}_M \\otimes 1_N\\colon (M, \\alpha) \\rightarrow (M \\bar{\\otimes} N, \\alpha \\otimes \\delta)\\]\nis approximately unitarily equivalent to a cocycle conjugacy.\n\\end{definition}\n\n\\begin{definition} \\label{def:ssa-action}\nLet $\\delta\\colon G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nWe say that $\\delta$ is \\emph{strongly self-absorbing}, if $\\delta$ strongly absorbs $\\delta$.\n\\end{definition}\n\n\\begin{definition}\nLet $\\alpha\\colon G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nWe say $\\alpha$ has \\emph{approximately inner half-flip} if the two equivariant embeddings\n\\[\n\\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}, 1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}\\colon (\\mathcal{R}, \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} \\mathcal{R}, \\alpha \\otimes \\alpha)\n\\]\nare approximately unitarily equivalent (as cocycle morphisms).\n\\end{definition}\n\n\\begin{remark}\nIt is well-known that any type II$_1$ von Neumann algebra $N$ with approximately inner half-flip in the above sense (with $G=\\{1\\}$) must be isomorphic to $\\mathcal R$. Indeed, it is clear that $N$ must have trivial center. Then $N \\cong \\mathcal{R}$ follows from \\cite[Theorem 5.1]{Connes76} under the stronger condition that the flip automorphism on $N\\bar{\\otimes} N$ is approximately inner, but the weaker condition is seen to be enough via Connes' theorem and the obvious modification of the proof of \\cite[Proposition 2.8]{EffrosRosenberg78} that shows the semi-discreteness of $N$.\n\\end{remark}\n\n\\begin{example}\\label{example:trivial_action_R}\nFor any second-countable locally compact group $G$, the trivial action $\\mathrm{id}_\\mathcal{R}\\colon G \\curvearrowright \\mathcal{R}$ has approximately inner half-flip as a consequence of the flip automorphism on a tensor product of matrix algebras $M_n(\\mathbb{C}) \\otimes M_n(\\mathbb{C})$ being inner. \nIt is also seen to be a strongly self-absorbing action.\n\\end{example}\n\n\\begin{theorem} \\label{theorem:sufficient_criterium_strong_absorption}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action on a semi-finite von Neumann algebra with separable predual.\nSuppose that $\\delta\\colon G \\curvearrowright \\mathcal{R}$ is an action with approximately inner half-flip such that there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (M_{\\omega,\\alpha}, \\alpha_\\omega)$.\nThen there exists a cocycle conjugacy $(\\theta,\\mathbbm{v}): (M,\\alpha) \\to (M\\bar{\\otimes}\\mathcal{R},\\alpha \\otimes \\delta)$ with $\\theta|_{\\mathcal{Z}(M)}=\\operatorname{id}_{\\mathcal{Z}(M)}\\otimes 1_{\\mathcal R}$.\nIf $M$ is finite, then $\\alpha$ strongly absorbs $\\delta$.\n\\end{theorem}\n\\begin{proof}\nFix a faithful normal state $\\phi$ on $M$.\nLet $\\pi\\colon (\\mathcal{R},\\delta) \\rightarrow (M_{\\omega,\\alpha},\\alpha_{\\omega})$ be a unital equivariant $*$-homomorphism.\nWe obtain an induced a map on the algebraic tensor product\n\\[\n\\mathcal{R} \\odot M \\rightarrow M^\\omega_\\alpha \\text{ via } x \\otimes m \\mapsto \\pi(x) m.\n\\]\nSince for each $m \\in M_+$ the map $x \\mapsto \\phi^\\omega(\\pi(x)m)$ defines a positive tracial functional on $\\mathcal{R}$,\n we see that it must be equal to some multiple of the unique tracial state $\\tau$ on $\\mathcal{R}$ and hence,\n  we get for each $x \\in \\mathcal{R}$ and $m \\in M$ that \n\\[\\phi^\\omega(\\pi(x) m) = \\tau(x) \\phi(m) = (\\tau \\otimes \\phi)(x \\otimes m).\\]\nSo we see that the map sends the faithful normal state $\\tau \\otimes \\phi$ to $\\phi^\\omega$ and hence,\n it extends to a unital normal $*$-homomorphism $\\mathcal{R} \\bar{\\otimes} M \\rightarrow M^\\omega_\\alpha$,\n which moreover is $(\\delta \\otimes \\alpha)$-to-$\\alpha^\\omega$ equivariant.   \nIn this way we get a unital equivariant normal $*$-homomorphism\n\\[(\\mathcal{R} \\bar{\\otimes} \\mathcal{R} \\bar{\\otimes}M, \\delta \\otimes \\delta \\otimes \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} M^\\omega_\\alpha, \\delta \\otimes \\alpha^\\omega),\\]\ngiven by $x_1 \\otimes x_2 \\otimes m \\mapsto x_1 \\otimes (\\phi(x_2) m)$. \nComposing with the canonical inclusion map $\\iota\\colon \\mathcal{R} \\bar{\\otimes} M^\\omega_\\alpha \\mapsto (\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta\\otimes\\alpha}$ we get a unital and equivariant normal $*$-homomorphism\n\\[\\Phi\\colon (\\mathcal{R} \\bar{\\otimes} \\mathcal{R} \\bar{\\otimes} M, \\delta \\otimes \\delta \\otimes \\alpha) \\rightarrow ((\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta \\otimes \\alpha}, (\\delta \\otimes \\alpha)^\\omega)\\]\nsuch that\n\\[\n\\Phi(x \\otimes 1_\\mathcal{R} \\otimes m) = x \\otimes m \\text{ for all } x \\in \\mathcal{R}, m \\in M,\n\\]\nand\n\\[\n\\Phi(1_\\mathcal{R} \\otimes \\mathcal{R} \\otimes M) \\subset \\iota(1_\\mathcal{R} \\otimes M^\\omega_\\alpha).\n\\]\nSince $\\delta$ has approximately inner half-flip,\n we can choose a sequence of unitaries $(v_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{R} \\bar{\\otimes} \\mathcal{R}$ such that \n $\\max_{g \\in K}\\|v_n - (\\delta \\otimes \\delta)_g(v_n)\\|_{\\tau \\otimes \\tau} \\rightarrow 0$ for all compact subsets $K \\subseteq G$ and \n ${\\|x \\otimes 1_\\mathcal{R} - v_n (1_\\mathcal{R} \\otimes x)v_n^*\\|_{\\tau \\otimes \\tau}  \\rightarrow 0}$ for all $x \\in \\mathcal{R}$. \nDefine $u_n := \\Phi(v_n \\otimes 1_M) \\subset (\\mathcal{R} \\bar{\\otimes} M)^\\omega_{\\delta \\otimes \\alpha}$. This sequence of unitaries satisfies\n\\begin{itemize} \n\\item $[u_n, 1_\\mathcal{R} \\otimes m] = \\Phi([v_n \\otimes 1_M, 1_{\\mathcal{R} \\bar{\\otimes} \\mathcal{R}} \\otimes m]) = 0$ for all $m \\in M$;\n\\item  $\\Phi(1_\\mathcal{R} \\otimes x \\otimes m) \\in \\iota(1_\\mathcal{R} \\otimes M^\\omega_\\alpha)$ and\n\\begin{align*}\n\\lim_{n \\rightarrow \\infty} u_n \\Phi(1_\\mathcal{R} \\otimes x \\otimes m) u_n^* &=\\lim_{n \\rightarrow \\infty} \\Phi((v_n \\otimes 1_M)(1_\\mathcal{R} \\otimes x \\otimes m)(v_n^* \\otimes 1_M))\\\\\n&=\\Phi(x \\otimes 1_\\mathcal{R} \\otimes m) \\\\&= x \\otimes m\n\\end{align*}\nwhere the limit is taken with respect to the strong operator topology;\n\\item $\\displaystyle \\max_{g \\in K} \\|u_n^* - (\\delta \\otimes \\alpha)^\\omega_g(u_n^*)\\|_{(\\tau \\otimes \\phi)^\\omega} = \\max_{g \\in K} \\|(v_n^* - (\\delta \\otimes \\delta)_g(v_n^*)) \\otimes 1_M\\|_{\\tau \\otimes \\tau \\otimes \\phi} \\to 0$ for all compact $K \\subseteq G$.\n\\end{itemize}\n\n Each $u_n$ can be lifted to a sequence of unitaries $(z_n^{(k)})_{k\\in \\mathbb{N}}$ in $\\mathcal{E}_{\\delta \\otimes \\alpha}^\\omega \\cap \\mathcal{N}_\\omega(\\mathcal{R} \\bar{\\otimes} M)$. Applying a diagonal sequence argument to the $(z_n^{(k)})_{k\\in \\mathbb{N}}$ and using Lemma~\\ref{lemma:lifting_invariance_compact_sets}, we can obtain a sequence of unitaries $(w_n)_{n \\in \\mathbb{N}}$ in $\\mathcal{R} \\bar{\\otimes} M$ such that\n\\begin{itemize}\n\\item $\\mathrm{Ad}(w_n)(1_\\mathcal{R} \\otimes m) - 1_\\mathcal{R} \\otimes m \\rightarrow 0$ strongly for all $m \\in M$.\n\\item $\\inf_{m \\in (M)_1}\\|x - w_n(1_\\mathcal{R} \\otimes m)w_n^*\\|_{\\tau \\otimes \\phi} \\rightarrow 0$ for $x \\in (\\mathcal{R} \\bar{\\otimes}M)_1$.\n\\item $\\max_{g \\in K} \\|w_n^* - (\\delta \\otimes \\alpha)_g(w_n^*)\\|_{\\tau \\otimes \\phi} \\rightarrow 0$ for every compact subset $K \\subseteq G$.\n\\end{itemize}\nWe conclude that the map $1_\\mathcal{R} \\otimes \\mathrm{id}_M\\colon (M, \\alpha) \\rightarrow (\\mathcal{R} \\bar{\\otimes} M, \\delta \\otimes \\alpha)$ satisfies all the necessary conditions to apply Lemma \\ref{lemma:one-sided_intertwining}. This completes the proof.\n\\end{proof}\n\n\\begin{theorem} \\label{theorem:equivalence_ssa}\nLet $\\delta:G \\curvearrowright \\mathcal{R}$ be an action on the hyperfinite II$_1$-factor.\nThen $\\delta$ is strongly self-absorbing if and only if it has approximately inner half-flip and there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$. \n\\end{theorem}\n\\begin{proof}\nThe `if' direction follows immediately from the previous proposition.\nTo prove the other direction, we assume that $\\delta$ is strongly self-absorbing and reproduce an argument analogous to \\cite[Proposition 1.4]{TomsWinter07} and \\cite[Proposition 5.5]{Szabo21cc}.\nDenote the unique tracial state on $\\mathcal{R}$ by $\\tau$. Let $(\\phi,\\mathbbm{u})\\colon (\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R} \\bar{\\otimes} \\mathcal{R}, \\delta \\otimes \\delta)$ be a cocycle conjugacy and $u_n \\in \\mathcal{U}(\\mathcal{R} \\bar{\\otimes} \\mathcal{R})$ a sequence of unitaries such that\n\\begin{equation}\\label{eq:approx_coboundary} \\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\| u_n(\\delta \\otimes \\delta)_g(u_n^*) - \\mathbbm{u}_g\\|_{\\tau \\otimes \\tau} = 0 \\text{ for every compact } K \\subseteq G, \\text{ and }\\end{equation} \n\\[\n\\lim_{n \\rightarrow \\infty} \\|\\phi(x) - u_n(x \\otimes 1) u_n^*\\|_{\\tau \\otimes \\tau} = 0 \\text{ for all } x \\in \\mathcal{R}.\n\\]\nNote that \n\\[\\mathrm{Ad}(u_n^*) \\circ \\phi \\circ \\delta_g = \\mathrm{Ad}(u_n^*\\mathbbm{u}_g (\\delta \\otimes \\delta)_g(u_n)) \\circ (\\delta \\otimes \\delta)_g \\circ \\mathrm{Ad}(u_n^*) \\circ \\phi.\\]\nAs a consequence of \\eqref{eq:approx_coboundary}, then for every compact $K \\subseteq G$ one has \n\\begin{equation}\\label{eq:approximate_invariance_perturbation}\\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\sup_{x \\in (\\mathcal{R})_1} \\|(\\mathrm{Ad}(u_n^*) \\circ \\phi \\circ \\delta_g)(x) - ((\\delta \\otimes \\delta)_g \\circ \\mathrm{Ad}(u_n^*) \\circ \\phi)(x)\\|_{\\tau \\otimes \\tau} = 0.\\end{equation}\nIn particular, applying this to $x = \\phi^{-1}(u_n)$ and using that $(\\tau \\otimes \\tau) = \\tau \\circ \\phi^{-1}$ yields\n\\[\\lim_{n \\rightarrow \\infty} \\max_{g \\in K}\\|(\\mathrm{Ad}(\\phi^{-1}(u_n^*)) \\circ \\delta_g) (\\phi^{-1}(u_n)) -(\\phi^{-1}\\circ (\\delta \\otimes \\delta)_g)(u_n)\\|_\\tau = 0.\\]\nCombining this with \\eqref{eq:approx_coboundary} again, one gets\n\\begin{equation}\\label{eq:approx_coboundary_inverse}\n \\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\| \\phi^{-1}(u_n^*)\\, \\delta_g(\\phi^{-1}(u_n)) - \\phi^{-1}(\\mathbbm{u}^*_g)\\|_{\\tau \\otimes \\tau} = 0 \\text{ for every compact } K \\subseteq G.\n\\end{equation}\nFirst, we prove that $\\delta$ has approximately inner half-flip. \nDefine the cocyle morphism $(\\psi, \\mathbbm{v}) := (\\phi, \\mathbbm{u})^{-1}  \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R})$. Note that\n\\begin{align*}\n1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R} \n&= (\\phi, \\mathbbm{u}) \\circ (\\psi, \\mathbbm{v})\\\\\n&\\approx_{\\mathrm{u}} (\\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}) \\circ (\\psi, \\mathbbm{v})\\\\\n&= (\\psi \\otimes 1_\\mathcal{R}, \\mathbbm{v} \\otimes 1).\n\\end{align*} \nApplying the equivariant flip automorphism to both sides of this equivalence, we get that\n\\begin{equation}\\label{eq:approx_equivalence_first_factor_embedding} \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R} \\approx_\\mathrm{u} (1_\\mathcal{R} \\otimes \\psi, 1 \\otimes \\mathbbm{v}) .\\end{equation}\nWe also get\n\\begin{align*}\n(\\psi \\otimes 1_\\mathcal{R}, \\mathbbm{v} \\otimes 1) &= (\\phi^{-1} \\otimes \\mathrm{id}_\\mathcal{R}, \\phi^{-1}(\\mathbbm{u})^* \\otimes 1) \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R})\\\\\n\\overset{\\eqref{eq:approx_equivalence_first_factor_embedding}}&{\\approx_{\\mathrm{u}}} (\\phi^{-1} \\otimes \\mathrm{id}_\\mathcal{R}, \\phi^{-1}(\\mathbbm{u})^* \\otimes 1) \\circ (1_\\mathcal{R} \\otimes 1_\\mathcal{R} \\otimes \\psi, 1\\otimes 1 \\otimes \\mathbbm{v})\\\\\n&=(1_\\mathcal{R} \\otimes \\psi, \\phi^{-1}(\\mathbbm{u})^* \\otimes \\mathbbm{v})\\\\\n\\overset{\\eqref{eq:approx_coboundary_inverse}}&{\\approx_{\\mathrm{u}}} (1_\\mathcal{R} \\otimes \\psi, 1 \\otimes \\mathbbm{v}).\n\\end{align*}\nBy transitivity we get that ${1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}\\approx_\\mathrm{u} \\mathrm{id}_\\mathcal{R} \\otimes 1_\\mathcal{R}}$.\n\nNext we prove the existence of a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$. Define the sequence of trace-preserving $*$-homomorphisms \n\\[\\chi_n = \\phi^{-1} \\circ \\mathrm{Ad}(u_n) \\circ (1_\\mathcal{R} \\otimes \\mathrm{id}_\\mathcal{R}).\\] \nWe conclude from \\eqref{eq:approximate_invariance_perturbation} that for all $x \\in \\mathcal{R}$\n\\[\\lim_{n \\rightarrow \\infty} \\max_{g \\in K} \\|\\delta_g(\\chi_n(x)) - \\chi_n(\\delta_g(x))\\|_{\\tau} =0.\\]\nFrom this and the fact that all $\\chi_n$ are trace-preserving it also follows that $(\\chi_n(x))_{n \\in \\mathbb{N}}$ belongs to $\\mathcal{E}^\\omega_\\alpha$.\nMoreover, for any $x,y \\in \\mathcal{R}$\n\\begin{align*}\n\\lim_{n \\rightarrow \\infty} \\|[x, \\chi_n(y)] \\|_\\tau\n&= \\lim_{n \\rightarrow \\infty}\\|[\\phi(x), u_n(1 \\otimes y) u_n^*]\\|_{\\tau \\otimes \\tau}\\\\\n&= \\lim_{n \\rightarrow \\infty}\\|u_n[x \\otimes 1, 1 \\otimes y] u_n^*\\|_{\\tau \\otimes \\tau}\\\\\n&=0. \n\\end{align*}\nSo the $\\chi_n$ induce a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$.\n\\end{proof}\n\nThe following can be seen as a direct generalization of the famous McDuff theorem \\cite{McDuff70} to actions on semi-finite von Neumann algebras.\n\n\\begin{corollary} \\label{cor:equivalence_equivariant_McDuff}\nLet $\\alpha:G \\curvearrowright M$ be an action on a semi-finite von Neumann algebra with separable predual and let $\\delta:G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen the following are equivalent:\n\\begin{enumerate}[leftmargin=*,label=\\textup{(\\arabic*)}]\n\\item There exists a cocycle conjugacy $(\\theta,\\mathbbm{v})\\colon (M,\\alpha) \\to (M\\bar{\\otimes}\\mathcal{R},\\alpha \\otimes \\delta)$ with $\\theta|_{\\mathcal{Z}(M)}=\\operatorname{id}_{\\mathcal{Z}(M)}\\otimes 1_{\\mathcal R}$; \\label{prop:McDuff:1}\n\\item $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$; \\label{prop:McDuff:2}\n\\item There exists a unital equivariant $*$-homomorphism $(\\mathcal{R},\\delta) \\rightarrow (M_{\\omega,\\alpha}, \\alpha_\\omega)$. \\label{prop:McDuff:3}\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\nThe implication \\ref{prop:McDuff:1}$\\Rightarrow$\\ref{prop:McDuff:2} is tautological.\nSince strong self-absorption implies approximately inner half-flip by Proposition~\\ref{theorem:equivalence_ssa}, the implication \\ref{prop:McDuff:3}$\\Rightarrow$\\ref{prop:McDuff:1} follows from Proposition~\\ref{theorem:sufficient_criterium_strong_absorption}.\n\nIn order to prove \\ref{prop:McDuff:2}$\\Rightarrow$\\ref{prop:McDuff:3}, it is enough to show that there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow ((M \\bar{\\otimes} \\mathcal{R})_{\\omega,\\alpha \\otimes \\delta}, (\\alpha \\otimes \\delta)_\\omega)$.\nWe know there exists a unital equivariant $*$-homomorphism $(\\mathcal{R}, \\delta) \\rightarrow (\\mathcal{R}_{\\omega,\\delta}, \\delta_\\omega)$ by Proposition \\ref{theorem:equivalence_ssa}.\nSince the latter is unitally and equivariantly contained in $((M \\bar{\\otimes} \\mathcal{R})_{\\omega,\\alpha \\otimes \\delta}, (\\alpha \\otimes \\delta)_\\omega)$, this finishes the proof.\n\\end{proof}\n\nThe following lemma is a straightforward application of the noncommutative Rokhlin Theorem of Masuda \\cite[Theorem 4.8]{Masuda13}\\footnote{This Rokhlin Theorem is actually a variant of Ocneanu's noncommutative Rokhlin Theorem \\cite[Theorem 6.1]{Ocneanu85}, and the proof of Masuda's version is essentially the same as Ocneanu's proof. While it is possible to deduce what we need from Ocneanu's Theorem, here we cite Masuda's version for convenience of the reader, as it is directly applicable and there is no need to deal with $\\varepsilon$-paving families of $G$.}\n\n\\begin{lemma}\\label{lem:approx-central-embeddings}\nLet $\\alpha\\colon G \\curvearrowright M$ be an action of a countable discrete group on a McDuff factor (i.e., $M \\cong M \\bar{\\otimes} \\mathcal{R}$) with separable predual.\nLet $N\\subseteq G$ be the normal subgroup consisting of all elements $g\\in G$ such that $\\alpha_{\\omega,g} \\in\\operatorname{Aut}(M_\\omega)$ is trivial.\nSuppose that the quotient group $G_0=G\/N$ is amenable with quotient map $\\pi: G\\to G_0$.\nLet $\\delta: G_0\\curvearrowright\\mathcal R$ be an action with induced $G$-action $\\delta_\\pi=\\delta\\circ\\pi$. \nThen there exists an equivariant unital $*$-homomorphism $(\\mathcal R,\\delta_\\pi)\\to (M_\\omega,\\alpha_\\omega)$.\n\\end{lemma}\n\\begin{proof}\nConsider the induced faithful action $\\gamma: G_0\\curvearrowright M_\\omega$ via $\\gamma_{gN}=\\alpha_{\\omega,g}$.\nThen clearly the claim is equivalent to finding a $G_0$-equivariant unital $*$-homomorphism $(\\mathcal R,\\delta)\\to (M_\\omega,\\gamma)$.\nLet us introduce some notation.\nLet $(x_n)_{n \\in \\mathbb{N}} \\in \\ell^\\infty(M)$ be a sequence representing an element $X \\in M_\\omega$. Then we set $\\tau_\\omega(X) = \\lim_{n \\rightarrow \\omega} x_n$,  where the limit is taken in the $\\sigma$-weak topology.\nSince $M$ is a factor and $\\tau_\\omega(X)$ is central, this limit belongs to $\\mathbb{C}$. For any $\\phi \\in M_*$ we have\n\\[\\phi^\\omega(X) = \\lim_{n \\rightarrow \\omega}\\phi(x_n) = \\phi(\\tau_\\omega(X)) = \\tau_\\omega(X).\\]\nIn particular, $\\tau_\\omega$ defines a normal faithful tracial state on $M_\\omega$ and we denote $\\|X\\|_1 = \\tau_\\omega(|X|)$.\n\nSince $M$ is McDuff we can find a unital $*$-homomorphism $\\Phi: \\mathcal{R} \\rightarrow M_\\omega$. Fix $\\varepsilon >0$ and a symmetric finite subset $F \\subset\\joinrel\\subset G_0$ containing the neutral element.\nBy \\cite[Lemmas 5.6 and 5.7]{Ocneanu85} we are allowed to apply \\cite[Theorem 4.8]{Masuda13} to the action $\\gamma\\colon G_0\\curvearrowright M_\\omega$.\nSo if $S \\subset\\joinrel\\subset G_0$ is a finite $(F,\\varepsilon)$-invariant subset, then there exists a partition of unity of projections $\\{E_s\\}_{s \\in S} \\subset M_\\omega$ such that\n\\begin{align}\n\\sum_{s \\in g^{-1}S \\cap S} \\|\\gamma_{g}(E_s) - E_{gs}\\|_1 &< 4\\varepsilon^{1\/2} \\text{ for all } g \\in F; \\label{eq:r-lemma1}\\\\\n\\sum_{s \\in S\\setminus g^{-1}S} \\|E_s\\|_1 &< 3\\varepsilon^{1\/2} \\text{ for all } g \\in F;\\label{eq:r-lemma2}\\\\\n[E_s, \\gamma_h(X)] &= 0 \\text{ for all } s \\in S,\\ h\\in G_0,\\ X \\in \\Phi(\\mathcal{R}).\\label{eq:commutation_image}\n\\end{align}\nDefine \n\\[\n\\Psi: \\mathcal{R} \\rightarrow M_\\omega \\text{ via } \\Psi(x) = \\sum_{s \\in S} \\gamma_{s}(\\Phi(\\delta_{s}^{-1}(x))) E_s.\n\\]\nThis is a unital trace-preserving $*$-homomorphism because the projections $E_s$ form a partition of unity and condition \\eqref{eq:commutation_image}.\nFor $g \\in F$ and $x \\in \\mathcal{R}$ we use conditions \\eqref{eq:r-lemma1} and \\eqref{eq:r-lemma2} to observe\n\\begin{align*}\n\\|\\gamma_{g}(\\Psi(x)) - \\Psi(\\delta_g(x))\\|_1 &=  \\Big\\| \\sum_{s \\in S} \\gamma_{gs}(\\Phi(\\delta_s^{-1}(x))) \\gamma_{g}(E_s) - \\sum_{s \\in g^{-1}S} \\gamma_{gs}(\\Phi(\\delta_{s}^{-1}(x))) E_{gs} \\Big\\|_1\n\\\\\n&\\leq \\sum_{s \\in S \\cap g^{-1}S}  \\left\\| \\gamma_{gs}(\\Phi(\\delta_s^{-1}(x))) ( \\gamma_{g}(E_s) - E_{gs}) \\right\\|_1\\\\\n& \\qquad + \\sum_{s \\in S\\setminus g^{-1}S} \\|\\gamma_s(\\Phi(\\delta_s^{-1}(x))) E_s\\|_1 + \\sum_{s \\in S \\setminus gS} \\|\\gamma_{s}(\\Phi(\\delta_{g^{-1}s}^{-1}(x)))E_s\\|_1 \\\\\n&< 10\\varepsilon^{1\/2}\\|x\\|.\n\\end{align*}\nSince we can do this for arbitrary $\\varepsilon >0$ and $F \\subset\\joinrel\\subset G$, the claim follows via a standard reindexing trick.\n\\end{proof}\n\nThe following result recovers a famous result due to Ocneanu \\cite[Theorem 1.2 and following remark]{Ocneanu85} as well as his uniqueness theorem of outer actions of amenable groups on $\\mathcal R$.\nWe include this proof for the reader's benefit as it is comparably elementary with the methods established so far.\n\n\\begin{theorem} \\label{theorem:model-absorption}\nLet $G$ and $G_1$ be countable discrete groups with $G_1$ amenable.\nLet $\\delta: G_1\\curvearrowright\\mathcal R$ be an outer action and $\\alpha\\colon G \\curvearrowright M$ an action on a semi-finite McDuff factor (i.e.\\ $M \\cong M \\bar{\\otimes} \\mathcal{R}$) with separable predual.\nThen:\n\\begin{enumerate}[label=\\textup{(\\roman*)},leftmargin=*]\n\\item $\\delta$ is strongly self-absorbing and cocycle conjugate to any other outer action $G_1\\curvearrowright\\mathcal R$.\\label{theorem:model-absorption:1}\n\\item Suppose $H\\subseteq G$ is a normal subgroup containing all elements $g\\in G$ such that $\\alpha_{\\omega,g}$ is trivial.\nSuppose $G_1=G\/H$ with quotient map $\\pi: G\\to G_1$.\nThen $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta_\\pi$. \\label{theorem:model-absorption:2}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\\ref{theorem:model-absorption:1}:\nLet $\\tau$ be the unique tracial state on $\\mathcal R$, which we may use to define the 1-norm $\\|\\cdot\\|_1=\\tau(|\\cdot|)$ on $\\mathcal R$.\nSet $\\delta^{(2)}=\\delta\\otimes\\delta: G_1\\curvearrowright \\mathcal R\\bar{\\otimes}\\mathcal R=:\\mathcal R^{(2)}$, which is also an outer action.\nSince the flip automorphism $\\sigma$ on $\\mathcal R^{(2)}$ is known to be approximately inner, we may pick a unitary $U\\in\\mathcal{U}(\\mathcal R^{(2)\\omega})$ with $UxU^*=\\sigma(x)$ for all $x\\in\\mathcal R^{(2)}$.\n\nBy \\cite[Theorem 3.2]{Connes77}, the induced action $\\delta^{(2)\\omega}: G_1\\curvearrowright\\mathcal R^{(2)}_\\omega$ is faithful.\nWe may hence argue exactly as in the proof of Lemma~\\ref{lem:approx-central-embeddings} and apply Masuda's noncommutative Rokhlin lemma.\nSo let $F\\subset\\joinrel\\subset G_1$ be a symmetric finite set and $\\varepsilon>0$.\nIf $S \\subset\\joinrel\\subset G_1$ is a finite $(F,\\varepsilon)$-invariant subset, then there exists a partition of unity of projections $\\{E_s\\}_{s \\in S} \\subset \\mathcal R^{(2)}_\\omega$ such that\n\\begin{align}\n\\sum_{s \\in g^{-1}S \\cap S} \\|\\delta^{(2)\\omega}_{g}(E_s) - E_{gs}\\|_1 &< 4\\varepsilon^{1\/2} \\text{ for all } g \\in F; \\label{eq:r-lemma1-}\\\\\n\\sum_{s \\in S\\setminus g^{-1}S} \\|E_s\\|_1 &< 3\\varepsilon^{1\/2} \\text{ for all } g \\in F;\\label{eq:r-lemma2-}\\\\\n[E_s, x] &= 0 \\text{ for all } s \\in S,\\ x\\in\\{ \\delta^{(2)\\omega}_h(U)\\}_{h\\in G_1}.\\label{eq:commutation_image-}\n\\end{align}\nDefine $W = \\sum_{s \\in S} \\delta^{(2)\\omega}_{s}(U) E_s$.\nThis is also a unitary in $\\mathcal R^{(2)\\omega}$ implementing the flip $\\sigma$ because the projections $E_s$ form a partition of unity and condition \\eqref{eq:commutation_image-}.\nFor $g \\in F$ we use conditions \\eqref{eq:r-lemma1-} and \\eqref{eq:r-lemma2-} to observe\n\\begin{align*}\n\\|\\delta^{(2)\\omega}_{g}(W) - W\\|_1 &=  \\Big\\| \\sum_{s \\in S} \\delta^{(2)\\omega}_{gs}(U) \\delta^{(2)\\omega}_{g}(E_s) - \\sum_{s \\in g^{-1}S} \\delta^{(2)\\omega}_{gs}(U) E_{gs} \\Big\\|_1\n\\\\\n&\\leq \\sum_{s \\in S \\cap g^{-1}S}  \\left\\| \\delta^{(2)\\omega}_{gs}(U) ( \\delta^{(2)\\omega}_{g}(E_s) - E_{gs}) \\right\\|_1\\\\ & \\qquad + \\sum_{s \\in S\\setminus g^{-1}S} \\| \\delta^{(2)\\omega}_{s}(U) E_s\\|_1 + \\sum_{s \\in S \\setminus gS} \\|\\delta^{(2)\\omega}_{s}(U) E_{s}\\|_1 \\\\\n&< 10\\varepsilon^{1\/2}.\n\\end{align*}\nSince we can do this for arbitrary $\\varepsilon >0$ and $F \\subset\\joinrel\\subset G_1$, we can use a reindexing trick to obtain a unitary $W\\in\\mathcal{U}((\\mathcal R^{(2)\\omega})^{\\delta^{(2)\\omega}})$ with $WxW^*=\\sigma(x)$ for all $x\\in\\mathcal R^{(2)}$.\nIn particular, $\\delta$ has approximately inner half-flip.\nIf we apply Lemma~\\ref{lem:approx-central-embeddings} for $G=G_1$, $N=\\{1\\}$ and $\\delta$ in place of $\\alpha$, it follows with Theorem~\\ref{theorem:equivalence_ssa} that $\\delta$ is strongly self-absorbing.\nIf $\\gamma\\colon G_1\\curvearrowright\\mathcal R$ is another outer action, then the same follows for $\\gamma$.\nBy applying Lemma~\\ref{lem:approx-central-embeddings} and Corollary~\\ref{cor:equivalence_equivariant_McDuff} twice, we obtain that $\\gamma$ and $\\delta$ absorb each other, hence they are cocycle conjugate.\n\n\\ref{theorem:model-absorption:2}:\nDefine $N$ to be the subgroup of all elements $g\\in G$ such that $\\alpha_{\\omega,g}$ is trivial, and set $G_0=G\/N$ with quotient map $\\pi^0: G\\to G_0$.\nBy assumption we have $N\\subseteq H$, hence $G_1$ can be viewed as a quotient of $G_0$ via a map $\\pi^{0\\to 1}: G_0\\to G_1$.\nThen $\\pi=\\pi^{0\\to 1}\\circ\\pi^0$ and the action $\\delta_{\\pi^{0\\to 1}}:=\\delta\\circ\\pi^{0\\to 1}$ is a $G_0$-action with $(\\delta_{\\pi^{0\\to 1}})_{\\pi^0}=\\delta_\\pi$.\nBy Lemma~\\ref{lem:approx-central-embeddings}, it follows that there exists an equivariant unital $*$-homomorphism $(\\mathcal R,\\delta_\\pi) \\to (M_\\omega,\\alpha_\\omega)$.\nSince $\\delta$ was strongly self-absorbing, so is $\\delta_\\pi$ as a $G$-action and the claim follows by Corollary~\\ref{cor:equivalence_equivariant_McDuff}.\n\\end{proof}\n\n\\section{Actions of discrete amenable groupoids}\n\nWe begin by recalling the definition of a discrete measured groupoid. This concept dates back to \\cite{Mackey63}. \n\n\\begin{definition}A discrete measured groupoid $\\mathcal{G}$ is a groupoid in the usual sense that carries the following additional structure:\n\\begin{enumerate}[label=$\\bullet$,leftmargin=*]\n\\item The groupoid $\\mathcal{G}$ is a standard Borel space and the units $\\mathcal{G}^{(0)} \\subset \\mathcal{G}$ form a Borel subset.\n\\item The source and target maps $s,t\\colon \\mathcal{G} \\rightarrow \\mathcal{G}^{(0)}$ are Borel and countable-to-one.\n\\item Define $\\mathcal{G}^{(2)}:= \\{(g,h) \\in \\mathcal{G} \\times \\mathcal{G} \\mid s(g) = t(h)\\}.$ The multiplication map $\\mathcal{G}^{(2)} \\rightarrow \\mathcal{G}\\colon (g,h) \\mapsto gh$ and the inverse map $\\mathcal{G}\\rightarrow \\mathcal{G}\\colon g \\mapsto g^{-1}$ are Borel.\n\\item $\\mathcal{G}^{(0)}$ is equipped with a measure $\\mu$ satisfying the following property.\nLet $\\mu_s$ and $\\mu_t$ denote the $\\sigma$-finite measures on $\\mathcal{G}$ obtained by integrating the counting measure over $s,t\\colon \\mathcal{G}\\rightarrow \\mathcal{G}^{(0)}$, respectively. Then $\\mu_s \\sim \\mu_t$. \n\\end{enumerate}\n\\end{definition}\n\\begin{example}\nAn important example of a discrete measured groupoid is the \\emph{transformation groupoid} associated to a non-singular action $G \\curvearrowright (X, \\mu)$ of a countable, discrete group $G$ on a standard measure space $(X, \\mu)$. In that case the unit space can be identified with $X$ and the measure $\\mu$ satisfies the necessary requirements. We denote this transformation groupoid by $G \\ltimes X$. \n\\end{example}\nWe assume the reader is familiar with the concept of amenability for discrete measured groupoids, see \\cite[Definition 3.2.8]{AnantharamanRenault00}. In particular, recall that a groupoid $\\mathcal{G}$ is amenable if and only if the associated equivalence relation\n\\[\n\\big\\{ \\big( s(g),t(g) \\big) \\mid g \\in \\mathcal{G} \\big\\}\n\\]\nand almost all associated isotropy groups\n\\[\\{g \\in \\mathcal{G} \\mid s(g) = t(g) = x\\} \\quad \\text{for } x \\in \\mathcal{G}^{(0)}\\]\nare amenable (see e.g.\\ \\cite[Corollary 5.3.33]{AnantharamanRenault00}). \nIn case of a non-singular action $G \\curvearrowright (X, \\mu)$, the associated transformation groupoid $G \\ltimes X$ is amenable if and only if the action is amenable in the sense of Zimmer (\\cite{Zimmer78, Zimmer77, Zimmer77b}).\n\n\\begin{remark}\nIn this paper we work with measurable fields of all kinds of separable structures, such as Polish spaces, Polish groups, von Neumann algebras with separable predual, and fields that can be derived from these.\nFor Polish groups the definition is explicitly given in \\cite{Sutherland85}, while the other notions can be defined in an analogous way.\nWe only consider the measurable setting and hence will often implicitly discard sets of measure zero whenever needed.\nThis means all measurable fields, groupoids and isomorphisms between measure spaces are defined up to sets of measure zero.\nBecause of this, all statements should be interpreted as holding only almost everywhere whenever appropriate.\nThis also means that we have no problem to apply the von Neumann measurable selection theorem (see e.g.\\ \\cite[Theorem 18.1]{Kechris95}) to obtain measurable sections after deletion of a suitable null set, and we will often omit the fine details related to such arguments.  \n\\end{remark}\n\n\\begin{definition}\nLet $\\mathcal{G}$ be a discrete measured groupoid with unit space $(X,\\mu)$. An \\emph{action} $\\alpha$ of $\\mathcal{G}$ on a measurable field $(B_x)_{x \\in X}$ of factors with separable predual is given by a measurable field of $*$-isomorphisms \\[\\mathcal{G} \\ni g \\mapsto \\alpha_g\\colon B_{s(g)} \\rightarrow B_{t(g)},\\] \nsatisfying $\\alpha_g \\circ \\alpha_h = \\alpha_{gh}$ for all $(g,h) \\in \\mathcal{G}^{(2)}$.\n\\end{definition}\n\n\\begin{definition}\\label{def:cc_groupoid_actions}\nLet $\\mathcal{G}$ be a discrete measured groupoid with unit space $(X,\\mu)$. Suppose that $\\alpha$ and $\\beta$ are actions of $\\mathcal{G}$ on the measurable fields of factors with separable predual $(B_x)_{x \\in X}$ and $(D_x)_{x \\in X}$, respectively. The actions are said to be \\emph{cocycle conjugate} if there exists a measurable field of $*$-isomorphisms $X \\ni x \\mapsto \\theta_x\\colon B_x \\rightarrow D_x$ and a measurable field of unitaries $\\mathcal{G} \\ni g \\mapsto w_g \\in \\mathcal{U}(D_{t(g)})$ satisfying  \n\\begin{align*}\n\\theta_{t(g)} \\circ \\alpha_g \\circ \\theta_{s(g)}^{-1} &= \\mathrm{Ad} w_g \\circ \\beta_g \\text{ for all } g \\in \\mathcal{G}\\\\\n  w_g \\beta_g(w_h) &= w_{gh} \\text{ for all } (g,h) \\in \\mathcal{G}^{(2)}.\n\\end{align*}\n\\end{definition}\n\n\\begin{example} \\label{ex:central_decomposition}\nLet $B$ be a von Neumann algebra acting on a separable Hilbert space $\\mathcal{H}$. Then we can centrally decompose $B$ as \n\\[ (B, \\mathcal{H}) = \\int_{X}^\\oplus (B_x, \\mathcal{H}_x)\\, d\\mu(x), \\]\nwhere $(X,\\mu)$ is a standard probability space such that $L^\\infty(X,\\mu) \\cong \\mathcal{Z}(B)$ (see e.g.\\ \\cite[Theorem IV.8.21]{Takesaki02}).\nIn this way we get a measurable field of factors $(B_x)_{x \\in X}$. When $B$ is of type I, II$_1$, II$_\\infty$ or III, every $B_x$ has the same type by \\cite[Corollary V.6.7]{Takesaki02}.\nWe claim that if $B \\cong B \\bar{\\otimes} \\mathcal{R}$, then every fibre $B_x$ is McDuff.\nPick a $*$-isomorphism $\\Phi\\colon B \\rightarrow B \\bar{\\otimes} \\mathcal{R}$.\nThen there exists (see for example \\cite[Theorem III.2.2.8]{Blackadar}) a unitary $U\\colon \\mathcal{H} \\otimes \\ell^2(\\mathbb{N}) \\rightarrow \\mathcal{H} \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N})$ such that the amplification of $\\Phi$ is spatial, i.e.\\ $\\Phi(b) \\otimes 1 = U(x \\otimes 1)U^*$.\nWe have the decompositions\n\\[(B \\otimes \\mathbb{C}, \\mathcal{H} \\otimes \\ell^2(\\mathbb{N})) =\\int_X^\\oplus (B_x \\otimes \\mathbb{C}, \\mathcal{H}_x \\otimes \\ell^2(\\mathbb{N})) \\, d \\mu(x),\\text{ and}\\]\n\\[(B \\bar{\\otimes} \\mathcal{R} \\otimes \\mathbb{C},\\mathcal{H} \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N}))  = \\int_X^\\oplus \\left(B_x \\bar{\\otimes}\\mathcal{R} \\otimes \\mathbb{C}, \\mathcal{H}_x \\otimes L^2(\\mathcal{R}, \\tau_\\mathcal{R}) \\otimes \\ell^2(\\mathbb{N})\\right) \\, d\\mu(x).\\]\nAs the amplification of $\\Phi$ necessarily maps the diagonal algebras (i.e.\\ the respective centers) to each other, we can use the fact that the disintegration is unique \\cite[Theorem 8.23]{Takesaki02}. In particular, this means every $B_x$ is isomorphic to some $B_y \\bar{\\otimes} \\mathcal{R}$ and hence, $B_x \\cong B_x \\bar{\\otimes} \\mathcal{R}$. \n\nNow suppose $\\alpha\\colon G \\curvearrowright B$ is an action of a countable discrete group. \nLet ${\\mathcal{G} = G \\ltimes X}$ denote the transformation groupoid associated to the action on $(X, \\mu)$ induced by $\\alpha$.\n Then $\\alpha$ can be disintegrated as an action $\\bar{\\alpha}$ of $\\mathcal{G}$ on the measurable field $(B_x)_{x \\in X}$ (see e.g.\\ \\cite[Corollary X.3.12]{Takesaki02}\\footnote{When the field of factors $(B_x)_{x \\in X}$ is constant (for example when $B$ is injective type II$_1$ and all $B_x$ are $\\mathcal{R}$), this construction dates back to \\cite{SutherlandTakesaki89}. \n There, the groupoid $\\mathcal{G}$ and action $\\bar{\\alpha}$ are also called the \\emph{ancillary groupoid} and \\emph{ancillary action} associated to $\\alpha$. }) such that given $b= \\int_x^\\oplus b_x\\,  d \\mu(x)$, we have\n\\[\\alpha_g(b)_{g \\cdot x} = \\bar{\\alpha}_{(g,x)}(b_{x}) \\text{ for } (g,x) \\in \\mathcal{G}.\\]\n\nAssume $\\beta\\colon G \\curvearrowright D$ is another action on a separably acting von Neumann algebra $(D, \\mathcal{K}) = \\int_X^\\oplus (D_x, \\mathcal{K}_x)\\, d\\mu(x)$, and assume that $\\beta$ induces the same action on $(X,\\mu)$ as $\\alpha$. \nLet $\\bar{\\beta}$ denote its decomposition as an action of $\\mathcal{G}$ on $(D_x)_{x \\in X}$. \nIf $\\bar{\\alpha}$ and $\\bar{\\beta}$ are cocycle conjugate in the sense of Definition~\\ref{def:cc_groupoid_actions}, then $\\alpha$ and $\\beta$ are cocycle conjugate as actions on von Neumann algebras.\nIndeed, let $X \\ni x \\mapsto \\theta_x\\colon A_x \\rightarrow B_x$ and $\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x})$ denote the measurable fields of $*$-isomorphisms and unitaries realizing a cocycle conjugacy between $\\bar{\\alpha}$ and $\\bar{\\beta}$.\nThis gives rise to a $*$-isomorphism $\\theta\\colon A \\rightarrow B$ given by $\\theta(a)_x = \\theta_x(a_x)$ for $a = \\int_X^\\oplus a_x\\, \\mathrm{d} \\mu(x) \\in A$, and for each $g \\in G$ we get a unitary $\\mathbbm{v}_g \\in \\mathcal{U}(B)$ by\n$(\\mathbbm{v}_g)_x = w_{(g, g^{-1}\\cdot x)}$. The pair $(\\theta, \\mathbbm{v})$ is a cocycle conjugacy.\n\nConversely, one can show that every cocycle conjugacy $(\\theta, \\mathbbm{v}):(B, \\alpha) \\rightarrow (D, \\beta)$ with $\\theta\\big\\lvert_{L^\\infty(X)} =  \\mathrm{id}\\big\\lvert_{L^\\infty(X)}$ gives rise to a cocycle conjugacy in the sense of Definition~\\ref{def:cc_groupoid_actions}.\n\\end{example}\n\nWe will subsequently need the following lemma (albeit only for discrete groups) about strongly self-absorbing actions.\n\n\\begin{lemma} \\label{lem:special-cc}\nLet $G_j$, $j=1,2$, be two second-countable locally compact groups with a continuous group isomorphism $\\phi: G_1\\to G_2$.\nLet $\\delta^{(j)}\\colon G_j\\curvearrowright\\mathcal R$, $j=1,2$, be two strongly self-absorbing actions and choose a cocycle conjugacy $(\\Phi,\\mathbb{U})\\colon (\\mathcal R,\\delta^{(2)})\\to (\\mathcal R\\bar{\\otimes}\\mathcal R,\\delta^{(2)}\\otimes\\delta^{(2)})$ that is approximately unitarily equivalent to $\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R}$.\n(Note that $\\phi$ allows us to identify $(\\Phi,\\mathbb{U}\\circ\\phi)$ with a cocycle conjugacy between the $G_1$-action $\\delta_\\phi:=\\delta^{(2)}\\circ\\phi$ and its tensor square.)\nLet $\\alpha^{(j)}\\colon  G_j\\curvearrowright M_j$, $j=1,2$, be two actions on separably acting von Neumann algebras.\nGiven a cocycle conjugacy\n\\[\n(\\theta,\\mathbbm{v})\\colon (M_1,\\alpha^{(1)})\\to \\big( M_2\\bar{\\otimes}\\mathcal{R}, (\\alpha^{(2)}\\otimes\\delta^{(2)})\\circ\\phi \\big),\n\\]\nand a conjugacy $\\Delta\\colon (\\mathcal R, \\delta^{(2)} \\circ \\phi)\\to(\\mathcal R,\\delta^{(1)})$,\nconsider the cocycle conjugacy of $G_1$-actions\n\\[\n(\\Psi,\\mathbb{V})= \\big( (\\theta, \\mathbbm{v})^{-1} \\otimes \\Delta \\big) \\circ \\big( \\mathrm{id}_{M_2} \\otimes (\\Phi, \\mathbbm{U}\\circ\\phi) \\big) \\circ (\\theta, \\mathbbm{v})\n\\]\nbetween $(M_1,\\alpha^{(1)})$ and $(M_1\\bar{\\otimes}\\mathcal R,\\alpha^{(1)}\\otimes \\delta^{(1)} )$.\nThen there exists a sequence of unitaries $y_n\\in\\mathcal{U}(M_1\\otimes\\mathcal R)$ such that\n\\[\n\\operatorname{Ad}(y_n)\\circ (\\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R})\\to \\Psi,\\quad \\operatorname{Ad}(y_n^*)\\circ\\Psi \\to \\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}\n\\]\npoint-strongly, and such that\n\\[\ny_n(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)^* \\to \\mathbb{V}_g,\\quad y_n^*\\mathbb{V}_g(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)\\to 1_{M_1\\otimes\\mathcal R}\n\\]\nin the strong operator topology for all $g\\in G$ and uniformly over compact sets.\n\\end{lemma}\n\\begin{proof}\nBy assumption, there is a sequence of unitaries $z_n\\in\\mathcal{U}(\\mathcal{R}\\bar{\\otimes}\\mathcal{R})$ such that \n\\begin{equation} \\label{eq:main-tech:ssa1}\n\\|\\Phi(x) - z_n(x\\otimes 1_{\\mathcal R})z_n^*\\|_2\\to 0\\quad\\text{for all } x\\in\\mathcal R.\n\\end{equation}\nand \n\\begin{equation} \\label{eq:main-tech:ssa2}\n\\max_{h\\in K} \\|\\mathbb{U}_h - z_n(\\delta^{(2)}\\otimes\\delta^{(2)})_h(z_n)^*\\|_2\\to 0 \\quad\\text{for every compact } K\\subseteq G_2.\n\\end{equation}\nBy definition, we have\n\\[\n\\Psi = (\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\Phi)\\circ\\theta\n\\]\nand\n\\[\n\\mathbb{V}_g = ({\\theta}^{-1}\\otimes\\Delta)\\Big( (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g) \\cdot (1_{M_2}\\otimes \\mathbb{U}_{\\phi(g)})\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big),\\quad g\\in G_1.\n\\]\nIf we consider the sequence of unitaries \n\\[\ny_n=(\\theta^{-1}\\otimes\\Delta)(1_{M_2}\\otimes z_n),\n\\]\nthen we can observe with \\eqref{eq:main-tech:ssa1} that\n\\[\n\\operatorname{Ad}(y_n^*)\\circ\\Psi \\to (\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R})\\circ\\theta = \\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}\n\\]\nas well as\n\\[\n\\operatorname{Ad}(y_n)\\circ(\\operatorname{id}_{M_1}\\otimes 1_{\\mathcal R}) = \\operatorname{Ad}(y_n)\\circ(\\theta^{-1}\\otimes\\Delta)\\circ (\\operatorname{id}_{M_2}\\otimes\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R})\\circ\\theta \\to \\Psi\n\\]\npoint-strongly.\nMoreover, given $g\\in G_1$, the fact that $(\\theta^{-1}, \\big( \\theta^{-1}(\\mathbbm{v}^*_g) \\big)_{g\\in G_1} )$ is the inverse of $(\\theta,\\mathbbm{v})$ leads to the equation $\\alpha^{(1)}_g\\circ\\theta^{-1}=\\theta^{-1}\\circ\\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\alpha^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}$.\nIf we combine this with \\eqref{eq:main-tech:ssa1} and \\eqref{eq:main-tech:ssa2}, we can see that\n\\[\n\\begin{array}{cl}\n\\multicolumn{2}{l}{ y_n^*\\mathbb{V}_g(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n) } \\\\\n=& y_n^*\\mathbb{V}_g\\cdot (\\theta^{-1}\\otimes\\Delta)\\big( \\operatorname{Ad}(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})(1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n)) \\big)  \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( (1_{M_2}\\otimes z_n)^*\\cdot (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g) \\cdot (1_{M_2}\\otimes \\mathbb{U}_{\\phi(g)}) \\cdots \\\\\n& \\phantom{ (\\theta^{-1}\\otimes\\operatorname{id}_{\\mathcal{R}})\\Big( }\\cdots (1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n))\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big) \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( \\underbrace{ \\big( \\operatorname{id}_{M_2}\\otimes (  \\operatorname{Ad}(z_n^*)\\circ\\Phi ) \\big) (\\mathbbm{v}_g) }_{\\to \\mathbbm{v_g}\\otimes 1_{\\mathcal R} } \n\\cdot \\big( 1_{M_2}\\otimes \\underbrace{ z_n^*\\mathbb{U}_{\\phi(g)}(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n) }_{\\to 1_{\\mathcal{R}\\otimes\\mathcal{R}} } \\big)\\cdot (\\mathbbm{v}_g^* \\otimes 1_{\\mathcal R}) \\Big) \\\\\n\\to & 1_{M_1\\otimes\\mathcal R} \n\\end{array}\n\\]\nin the strong operator topology, uniformly over compact subsets.\nAnalogously, we observe the convergence\n\\[\n\\begin{array}{cl}\n\\multicolumn{2}{l}{ y_n(\\alpha^{(1)}\\otimes\\delta^{(1)})_g(y_n)^* }\\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( (1_{M_2}\\otimes z_n) \\cdot \\operatorname{Ad}(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})(1_{M_2}\\otimes(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n^*) ) \\Big) \\\\\n=& (\\theta^{-1}\\otimes\\Delta)\\Big( \\underbrace{ \\operatorname{Ad}(1_{M_2}\\otimes z_n)(\\mathbbm{v}_g\\otimes 1_{\\mathcal R})}_{\\to (\\operatorname{id}_{M_2}\\otimes\\Phi)(\\mathbbm{v}_g)}\n \\cdot \\big( 1_{M_2}\\otimes \\underbrace{ z_n(\\delta^{(2)}\\otimes\\delta^{(2)})_{\\phi(g)}(z_n^*) }_{\\to \\mathbb{U}_{\\phi(g)} } \\big) \\cdot (\\mathbbm{v}_g^*\\otimes 1_{\\mathcal R}) \\Big) \\\\\n\\to & \\mathbb{V}_g\n\\end{array}\n\\]\nuniformly over compact sets.\nThis finishes the proof.\n\\end{proof}\n\nIn the proof of the main technical result of this section we will make use of the following variant, due to Popa--Shlyakhtenko--Vaes, of the cohomology lemmas in \\cite[Appendix]{JonesTakesaki84} and \\cite[Theorem 5.5]{Sutherland85}.\n\n\\begin{lemma}[{\\cite[Theorem 3.5]{PopaShlyakhtenkoVaes20}}] \\label{lemma:cohomology_lemma}\nLet $\\mathcal{S}$ be an amenable countable nonsingular equivalence relation on the standard probability space $(X,\\mu)$.\nLet $(G_x \\curvearrowright P_x)_{x \\in X}$ be a measurable field of continuous actions of Polish groups on Polish spaces, on which $\\mathcal{S}$ is acting by conjugacies: we have measurable fields of group isomorphisms ${\\mathcal{S} \\ni (x,y) \\mapsto \\gamma_{(x,y)}\\colon G_y \\mapsto G_x}$ and homeomorphisms $\\mathcal{S} \\ni (x,y) \\mapsto \\beta_{(x,y)}\\colon P_y \\mapsto P_x$ satisfying \n\\[\n\\gamma_{(x,y)} \\circ \\gamma_{(y,z)} = \\gamma_{(x,z)}, \\quad \\beta_{(x,y)} \\circ \\beta_{(y,z)} = \\beta_{(x,z)}, \\quad \\beta_{(x,y)} (g \\cdot \\pi) = \\gamma_{(x,y)}(g) \\cdot \\beta_{(x,y)}(\\pi)\n\\]\nfor all $(x,y), (y,z) \\in \\mathcal{S}$ and $g \\in G_y, \\pi \\in P_y$.\nLet $X \\ni x \\mapsto \\sigma'(x) \\in P_x$ be a measurable section. Assume that for all $(x,y) \\in \\mathcal{S}$, the element $\\sigma'(x)$ belongs to the closure of ${G_x \\cdot \\beta_{(x,y)}(\\sigma'(y))}$. \nThen there exists a measurable family $\\mathcal{S} \\ni (x,y) \\mapsto v(x,y) \\in G_x$ and a section $X \\ni x \\mapsto \\sigma(x) \\in P_x$ satisfying:\n\\begin{itemize}\n\\item $v$ is a 1-cocycle: $v(x,y) \\gamma_{(x,y)}(v(y,z)) = v(x,z)$ for all $(x,y), (y,z)\\in \\mathcal{S}$;\n\\item $v(x,y)\\cdot \\beta_{(x,y)}(\\sigma(y)) = \\sigma(x)$ for all $(x,y) \\in \\mathcal{S}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{remark} \\label{rem:Polish-spaces}\nBefore we state and prove our main technical result in detail, we would like to outline for what kind of input data the lemma above will be used.\nIn the situation considered below, the typical Polish space $P$ will be the space of cocycle conjugacies $(M,\\alpha)\\to (N,\\beta)$, where $\\alpha: H\\curvearrowright M$ and $\\beta: H\\curvearrowright N$ are actions of a countable discrete group $H$ on separably acting von Neumann algebras.\nHere we consider the topology defined by declaring that one has a convergence of nets $(\\theta^{(\\lambda)},\\mathbbm{v}^{(\\lambda)})\\to(\\theta,\\mathbbm{v})$ if and only if $\\mathbbm{v}^{(\\lambda)}_g\\to\\mathbbm{v}_g$ in the strong operator topology for all $g\\in H$, and $\\|\\varphi\\circ\\theta-\\varphi\\circ\\theta^{(\\lambda)}\\|\\to 0$ for all $\\varphi\\in N_*$.\nThis generalizes the well-known topology on the space of isomorphisms $M\\to N$ that one usually called the ``$u$-topology''; cf.\\ \\cite{Haagerup75, Winslow98}.\nThe typical Polish group acting on this Polish space would be the unitary group $\\mathcal{U}(N)$, which is equipped with the strong operator topology, where the action is defined via composition with the inner cocycle conjugacy as per Example~\\ref{ex:inner-cc} and Remark~\\ref{rem:cocycle-category}.\nIn other words, a unitary $w\\in\\mathcal{U}(N)$ moves the cocycle conjugacy $(\\theta,\\mathbbm{v})$ to $\\operatorname{Ad}(w)\\circ(\\theta,\\mathbbm{v}) = \\big( \\operatorname{Ad}(w)\\circ\\theta, (w\\mathbbm{v}\\beta_g(w)^*)_{g\\in H} \\big)$.\n\nIf we assume in addition that $M$ and $N$ are semi-finite, we may pick a faithful normal semi-finite tracial weight $\\tau$ on $N$.\nAssume that $(\\Psi,\\mathbb{V})\\in P$ is a cocycle conjugacy.\nThen it follows from \\cite[Proposition 3.7]{Haagerup75} that on the space of all isomorphisms $\\Psi': M\\to N$ with $\\tau\\circ\\Psi'=\\tau\\circ\\Psi'$, the $u$-topology coincides with the topology of point-strong convergence.\nAs a direct consequence, we may conclude the following. \nIf $(\\Phi,\\mathbb{U})\\in P$ is another cocycle conjugacy and there exists a net of unitaries $w_\\lambda\\in\\mathcal{U}(N)$ such that $w_\\lambda\\mathbb{V}_g\\beta_g(w_\\lambda)^*\\to \\mathbb{U}_g$ for all $g\\in H$ in the strong operator topology and $\\operatorname{Ad}(w_\\lambda)\\circ\\Psi\\to\\Phi$ point-strongly, then $(\\Phi,\\mathbb{U})\\in\\overline{ \\mathcal{U}(N)\\cdot (\\Psi,\\mathbb{V}) }$.\n\\end{remark}\n\nThe following can be seen as the main technical result of this section, which we previously referred to as a kind of measurable local-to-global principle.\n\n\\begin{theorem} \\label{thm:main_technical}\nLet $G \\curvearrowright (X,\\mu)$ be an amenable action (in the sense of Zimmer) of a countable, discrete group on a standard probability space.\nLet $\\alpha$ be an action of ${\\mathcal{G}:= G \\ltimes X}$ on a measurable field of semi-finite factors with separable predual $(B_x)_{x \\in X}$. \nDenote by ${X \\ni x \\mapsto H_x}$ the measurable field of isotropy groups. \nFor any action ${\\delta\\colon G \\curvearrowright \\mathcal{R}}$ on the hyperfinite II$_1$-factor, we define a tensor product action $\\alpha \\otimes \\delta$ of $\\mathcal{G}$ on the field of factors $(B_x \\bar{\\otimes} \\mathcal{R})_{x \\in X}$ by $(\\alpha \\otimes \\delta)_{(g,x)} = \\alpha_{(g,x)} \\otimes \\delta_g$.\nIf $\\delta$ is strongly self-absorbing, then the following are equivalent:\n \\begin{enumerate}[leftmargin=*,label=\\textup{(\\arabic*)}]\n \\item $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$; \\label{prop:main_technical:1}\n \\item For every $x \\in X$ we have $\\alpha{|_{H_x}} \\simeq_{\\mathrm{cc}} (\\alpha \\otimes \\delta){|_{H_x}}$ as actions of $H_x$ on $B_x$ and $B_x \\bar{\\otimes} \\mathcal{R}$. \\label{prop:main_technical:2}\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nWe note that by following the argument outlined in Example~\\ref{ex:central_decomposition} and by applying Corollary~\\ref{cor:equivalence_equivariant_McDuff}, we see that $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta$ implies that one can find a cocycle conjugacy between $\\alpha$ and $\\alpha\\otimes\\delta$ that induces the identity map on $X$.\nHence \\ref{prop:main_technical:1} implies \\ref{prop:main_technical:2}.\n\nIn order to prove the other implication, assume \\ref{prop:main_technical:2} holds. \nTo verify \\ref{prop:main_technical:1}, we will show the existence of a measurable field of $*$-isomorphisms ${X \\ni x \\mapsto \\theta_x \\colon B_x \\rightarrow B_x \\bar{\\otimes} \\mathcal{R}}$ and unitaries ${\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x} \\bar{\\otimes} \\mathcal{R})}$ such that\n\\begin{align}\n\\theta_{g \\cdot x} \\circ \\alpha_{(g,x)} \\circ \\theta_{x}^{-1} &= \\mathrm{Ad} \\big( w_{(g,x)} \\big) \\circ (\\alpha \\otimes \\delta)_{(g,x)} \\text{ for all } (g,x) \\in \\mathcal{G}\\label{eq:required_cc}\\\\\n  w_{(g,h \\cdot x)} (\\alpha \\otimes \\delta)_{(g,h \\cdot x)}(w_{(h,x)}) &= w_{(gh,x)} \\text{ for all } g,h \\in G, x \\in X.\\label{eq:required_cocycle}\n\\end{align}\nFor every $x \\in X$, denote by $P_x$ the Polish space of cocycle conjugacies from $(B_x, \\alpha|_{H_x})$ to $(B_x \\bar{\\otimes} \\mathcal{R}, (\\alpha \\otimes \\delta)|_{H_x})$ as per Remark~\\ref{rem:Polish-spaces}. \nIn this way, we get a measurable field of Polish spaces $X \\ni x \\mapsto P_x$. Note that by assumption the sets $P_x$ are all non-empty and hence, there exists some measurable section \n$ X \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) \\in P_x$.\nDefining $w_{(g,x)} := \\mathbbm{v}_{x}(g)$ for $g \\in H_x$, we get that --- although $w$ is not defined on all of $\\mathcal{G}$ yet --- the equations \\eqref{eq:required_cc}--\\eqref{eq:required_cocycle} are satisfied whenever they make sense.\nIn the rest of the proof we will show with the help of Lemma \\ref{lemma:cohomology_lemma} that there exists a (potentially different) section for which there exists a well-defined map $w$ on all of $\\mathcal{G}$ obeying conditions \\eqref{eq:required_cc}--\\eqref{eq:required_cocycle}.\n\nDenote by $\\mathcal{S}$ the countable non-singular orbit equivalence relation associated to $\\mathcal{G}$, i.e.,\n\\[\\mathcal{S}=\\{(g \\cdot x,x) \\mid (g,x) \\in \\mathcal{G}\\}.\\]\nAs $G \\curvearrowright (X,\\mu)$ is amenable, the relation $\\mathcal{S}$ is amenable and hence it follows by the Connes--Feldman--Weiss theorem \\cite{ConnesFeldmanWeiss81} that after neglecting a set of measure zero, there exists a partition of $X$ into $\\mathcal{S}$-invariant Borel subsets $X_0\\sqcup X_1$ such that the restriction of $\\mathcal{S}$ to $X_0$ has finite orbits and the restriction to $X_1$ is induced by a free non-singular action of $\\mathbb{Z}$. \nThis implies that the map $q \\colon \\mathcal{G} \\rightarrow \\mathcal{S}\\colon (g,x) \\mapsto (g \\cdot x,x)$ admits a measurable section, i.e., a measurable groupoid morphism $s \\colon \\mathcal{S} \\rightarrow \\mathcal{G}$ such that $q \\circ s = \\mathrm{id}_\\mathcal{S}$. \nTherefore, we can view $\\mathcal{G}$ as the semi-direct product of the field of groups $(H_x)_{x \\in X}$ and the measurable field of group isomorphisms $\\phi_{(x,y)}\\colon H_y \\rightarrow H_x$ given by $\\phi_{(x,y)}(g)= s(x,y) g s(x,y)^{-1}$.\nNote that $\\phi_{(x,y)} \\circ \\phi_{(y,z)} = \\phi_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$.\n\nThis means that in order to define a measurable field $\\mathcal{G} \\ni (g,x) \\mapsto w_{(g,x)} \\in \\mathcal{U}(B_{g \\cdot x} \\bar{\\otimes} \\mathcal{R})$ satisfying \\eqref{eq:required_cocycle}, it suffices to find measurable families of unitaries \n${v{(x,y)} \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})}$ for $(x,y) \\in \\mathcal{S}$ and \n$\\mathbbm{v}_x(g) \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ for $x \\in X, g \\in H_x$ such that\n\\begin{itemize}\n\\item $v$ is a cocyle for the action of $\\mathcal{S}$ on the field of factors $(B_x\\bar{\\otimes} \\mathcal{R})_{x \\in X}$ induced by $s$, i.e.,\n$v{(x,y)} (\\alpha \\otimes \\delta)_{s(x,y)}(v{(y,z)})=v(x,z)$ for all $(x,y), (y,z) \\in \\mathcal{S}$;\n\\item for each $x \\in X$, the family $(\\mathbbm{v}_x(g))_{g \\in H_x}$ defines a cocycle for the action $H_x \\curvearrowright B_x \\bar{\\otimes} \\mathcal{R}$;\n\\item $\\mathbbm{v}_x(g) = v{(x,y)} (\\alpha \\otimes \\delta)_{s(x,y)}\\left(\\mathbbm{v}_y\\left({\\phi_{(y,x)}(g)}\\right)\\right) (\\alpha \\otimes \\delta)_g\\left(v(x,y)^*\\right) $ for all $(x,y) \\in \\mathcal{S}$ and $g \\in H_x$. \n\\end{itemize}\nIf these conditions are met, then setting $w_{g s(x,y)} := \\mathbbm{v}_x(g)(\\alpha \\otimes \\delta)_g (v(x,y))$ for $(x,y) \\in \\mathcal{S}$ and $g \\in H_x$ yields condition \\eqref{eq:required_cocycle}.\nMoreover, in order for a measurable field of $*$-isomorphisms ${X \\ni x \\mapsto \\theta_x \\colon B_x \\rightarrow B_x \\bar{\\otimes} \\mathcal{R}}$ to satisfy \\eqref{eq:required_cc}, it then suffices to check that \n\\begin{itemize}\n\\item for each $x \\in X$, the pair $(\\theta_x, \\mathbbm{v}_x)$ is a cocycle conjugacy in $P_x$;\n\\item for $(x,y) \\in \\mathcal{S}$ we have $\\theta_x \\circ \\alpha_{s(x,y)} \\circ \\theta_y^{-1} = \\mathrm{Ad}(v(x,y))\\circ (\\alpha \\otimes \\delta)_{s(y,z)}$.\n\\end{itemize}\nWe introduce some notation to rephrase this in terms of the terminology of Lemma~\\ref{lemma:cohomology_lemma}. Consider the natural action $\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x$ given by composing a cocycle conjugacy with the inner one given by $\\mathrm{Ad}(u)$ for $u \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ as per Remark~\\ref{rem:Polish-spaces}.\nIn this way, we get a measurable field $(\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x)_{x \\in X}$ of continuous actions of Polish groups on Polish spaces.\nLet us convince ourselves that in the terminology of Lemma~\\ref{lemma:cohomology_lemma}, the equivalence relation $\\mathcal{S}$ acts by conjugacies on this field of actions.\nFirstly, we have a measurable field of group isomorphisms \n\\[\n{\\mathcal{S} \\ni (x,y) \\mapsto  \\gamma_{(x,y)} = (\\alpha \\otimes \\delta)_{s(x,y)}|_{\\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})} \\colon \\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})\\rightarrow \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})}\n\\]\nsuch that $\\gamma_{(x,y)} \\circ \\gamma_{(y,z)} = \\gamma_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$.\nThe latter formula holds as $\\alpha\\otimes\\delta$ was a $\\mathcal G$-action and $s: \\mathcal{S}\\to\\mathcal{G}$ is a section.\nSecondly, we have an action of $\\mathcal{S}$ on $(P_x)_{x \\in X}$ as follows.\nGiven $(x,y) \\in \\mathcal{S}$ and $(\\theta, \\mathbbm{v}) \\in P_y$, we define $\\beta_{(x,y)}(\\theta, \\mathbbm{v}) := (\\tilde{\\theta}, \\tilde{\\mathbbm{v}})$, where\n\\[\n\\tilde{\\theta} = (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\theta \\circ \\alpha_{s(y,x)} \\quad\\text{and}\\quad\n\\tilde{\\mathbbm{v}}(h) = (\\alpha \\otimes \\delta)_{s(x,y)}(\\mathbbm{v}({\\phi_{(y,x)}(h)})) \\text{ for } h \\in H_x.\n\\]\nThis construction yields a well-defined cocycle conjugacy in $P_x$, and we get a well-defined map $\\beta_{(x,y)}\\colon P_y \\rightarrow P_x$.\nTogether these maps combine to form a measurable field of homeomorphisms $\\mathcal{S} \\ni (x,y) \\mapsto \\beta_{(x,y)}\\colon P_y \\to P_x$ such that $\\beta_{(x,y)} \\circ \\beta_{(y,z)} = \\beta_{(x,z)}$ for all $(x,y), (y,z) \\in \\mathcal{S}$. \nThis formula holds once again because $\\alpha\\otimes\\delta$ and $\\alpha$ are $\\mathcal G$-actions and $s\\colon \\mathcal{S}\\to\\mathcal{G}$ is a section.\nMoreover, the maps $\\beta$ and $\\gamma$ are compatible with the measurable field of actions ${(\\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R}) \\curvearrowright P_x)_{x \\in X}}$ (as required for Lemma~\\ref{lemma:cohomology_lemma}), since for any $(x,y) \\in \\mathcal{S}, u \\in \\mathcal{U}(B_y \\bar{\\otimes} \\mathcal{R})$ and $(\\theta,\\mathbbm{v}) \\in P_y$ we may simply compare definitions and observe\n\\[\n\\beta_{(x,y)}(u \\cdot (\\theta,\\mathbbm{v})) = \\gamma_{(x,y)}(u) \\cdot \\beta_{(x,y)}(\\theta,\\mathbbm{v}).\n\\]\nHaving introduced all this data, our previous discussion can be rephrased. \nIn order to complete the proof, it suffices to find a measurable section $X \\ni x \\mapsto \\sigma(x) \\in P_x$ and a measurable family $\\mathcal{S} \\ni (x,y) \\mapsto v(x,y) \\in \\mathcal{U}(B_x \\bar{\\otimes} \\mathcal{R})$ such that\n\\begin{itemize}\n\\item $v(x,y) \\gamma_{(x,y)}(v(y,z)) = v(x,z)$ for all $(x,y), (y,z) \\in \\mathcal{S}$;\n\\item $v(x,y) \\cdot \\beta_{(x,y)}(\\sigma(y)) = \\sigma(x)$ for all $(x,y) \\in \\mathcal{S}$.\n\\end{itemize}\nBy Lemma~\\ref{lemma:cohomology_lemma}, such maps exist if we can merely show that\nthere exists a measurable section $X \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) \\in P_x$ such that for all $(x,y) \\in \\mathcal{S}$, the element $(\\theta_x, \\mathbbm{v}_x)$ belongs to the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R}) \\cdot \\beta_{(x,y)}(\\theta_y,\\mathbbm{v}_y)$.\nWe claim that this is indeed the case.\n\nConsider any measurable section $X \\ni x \\mapsto (\\theta'_x, \\mathbbm{v}'_x) \\in P_x$. \nAs $\\delta$ is strongly self-absorbing, we can fix a cocycle conjugacy $(\\Phi, \\mathbb{U})$ from $(\\mathcal{R}, \\delta)$ to $(\\mathcal{R}\\bar{\\otimes} \\mathcal{R}, \\delta \\otimes \\delta)$ that is approximately unitarily equivalent to $\\operatorname{id}_{\\mathcal R}\\otimes 1_{\\mathcal R}$.\nFor each $x \\in X$ we can define the map\n\\[\n\\Lambda_x\\colon P_x \\rightarrow P_x,\\quad (\\theta, \\mathbbm{v}) \\mapsto \\big( (\\theta, \\mathbbm{v})^{-1} \\otimes \\mathrm{id}_\\mathcal{R} \\big) \\circ \\big( \\mathrm{id}_{B_x} \\otimes (\\Phi, \\mathbb{U}) \\big) \\circ (\\theta, \\mathbbm{v})\n\\]\nThen we get a new measurable section\n\\[\nX \\ni x \\mapsto (\\theta_x, \\mathbbm{v}_x) := \\Lambda_x(\\theta'_x, \\mathbbm{v}'_x) \\in P_x.\n\\]\nWe claim that this section does the trick. Fix $(x,y) \\in \\mathcal{S}$. \nIf we denote $(\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x):= \\beta_{(x,y)}(\\theta_y, \\mathbbm{v}_y)$, we need to convince ourselves that the cocycle conjugacy $(\\theta_x, \\mathbbm{v}_x)$ is in the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R})\\cdot (\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x)$.\nFirst of all, we observe that the construction of $(\\theta_x, \\mathbbm{v}_x)=\\Lambda_x(\\theta_x', \\mathbbm{v}_x')$ can be seen as a special case of Lemma~\\ref{lem:special-cc} for $M_1=M_2=B_x$, $G=H_x$, $\\phi=\\operatorname{id}_{H_x}$ and $\\Delta=\\operatorname{id}_{\\mathcal R}$.\nHence we can find a sequence of unitaries $y_n\\in\\mathcal{U}(B_x\\otimes\\mathcal R)$ such that\n\\begin{equation} \\label{eq:main-tech:yn}\ny_n^*\\theta_x(b)y_n \\to b\\otimes 1_{\\mathcal R},\\quad y_n^*\\mathbbm{v}_x(h)(\\alpha\\otimes\\delta)_h(y_n) \\to 1_{B_x\\otimes\\mathcal R}\n\\end{equation}\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\n\nNext, by our previous notation, the group isomorphism $\\phi_{(x,y)}\\colon H_y\\to H_x$ is exactly the one so that the isomorphism of von Neumann algebras $\\alpha_{s(x,y)}\\colon B_y\\to B_x$ can be viewed as a (genuine) conjugacy between the $H_x$-actions $(\\alpha|_{H_y})_{\\phi_{(y,x)}}$ and $\\alpha|_{H_x}$.\nMoreover if $s(x,y)=(k,y)$ for $k\\in G$, then $(\\alpha\\otimes\\delta)_{s(x,y)}=\\alpha_{s(x,y)}\\otimes\\delta_k$ and $\\delta_k$ can be seen as a conjugacy between the $H_x$-actions $(\\delta|_{H_y})_{\\phi_{(y,x)}}$ and $\\delta|_{H_x}$.\n\nBy definition of $\\beta_{(x,y)}$, we have\n\\[\n\\begin{array}{ccl}\n\\tilde{\\theta}_x &=& (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\theta_y \\circ \\alpha_{s(y,x)} \\\\\n&=&  (\\alpha \\otimes \\delta)_{s(x,y)} \\circ \\big( {\\theta_y'}^{-1} \\otimes \\mathrm{id}_\\mathcal{R} \\big) \\circ \\big( \\mathrm{id}_{B_y} \\otimes \\Phi \\big) \\circ \\theta_y' \\circ \\alpha_{s(y,x)} \\\\\n&=& \\big( ({\\theta_y'}\\circ\\alpha_{s(y,x)})^{-1} \\otimes \\delta_k \\big) \\circ \\big( \\mathrm{id}_{B_y} \\otimes \\Phi \\big) \\circ (\\theta_y' \\circ \\alpha_{s(y,x)})\n\\end{array}\n\\]\nand for all $h\\in H_x$ with $g=\\phi_{(y,x)}(h)$, one has\n\\[\n\\begin{array}{ccl}\n\\tilde{\\mathbbm{v}}_x(h) \n&=& (\\alpha \\otimes \\delta)_{s(x,y)}\\Big( ({\\theta_y'}^{-1}\\otimes\\operatorname{id}_{\\mathcal R})\\Big( (\\operatorname{id}_{B_y}\\otimes\\Phi)(\\mathbbm{v}_y'(g)) \\cdot (1_{B_y}\\otimes \\mathbb{U}_g)\\cdot (\\mathbbm{v}_y'(g)^* \\otimes1_{\\mathcal R}) \\Big) \\Big) \\\\\n&=& \\big( ({\\theta_y'}\\circ\\alpha_{s(y,x)})^{-1}\\otimes\\delta_k \\big)\\Big( (\\operatorname{id}_{B_y}\\otimes\\Phi)(\\mathbbm{v}_y'(g)) \\cdot (1_{B_y}\\otimes \\mathbb{U}_g)\\cdot (\\mathbbm{v}_y'(g)^* \\otimes1_{\\mathcal R}) \\Big)\n\\end{array}\n\\]\nWe conclude that Lemma~\\ref{lem:special-cc} is applicable to the cocycle conjugacy $(\\tilde{\\theta}_x,\\tilde{\\mathbbm{v}}_x)$, where we insert $G_1=H_x$, $G_2=H_y$, $\\phi=\\phi_{(y,x)}$, $M_1=B_x$, $M_2=B_y$, $\\Delta=\\delta_k$ and the cocycle conjugacy $(\\theta'_y\\circ\\alpha_{s(y,x)}, \\mathbbm{v}'_y\\circ\\phi_{(y,x)})$ in place of $(\\theta,\\mathbbm{v})$.\nThis allows us to find a sequence of unitaries $w_n\\in\\mathcal{U}(B_x\\otimes\\mathcal R)$ satisfying\n\\begin{equation} \\label{eq:main-tech:wn}\nw_n(b\\otimes 1_{\\mathcal R})w_n^* \\to \\tilde{\\theta}_x(b),\\quad w_n(\\alpha\\otimes\\delta)_h(w_n)^* \\to \\tilde{\\mathbbm{v}}_x(h)\n\\end{equation}\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\nIf we consider both of the conditions \\eqref{eq:main-tech:yn} and \\eqref{eq:main-tech:wn} and keep in mind that $G$ is countable and $B_x$ is separable and semi-finite, with the help of Lemma \\ref{lemma:point_strong_dense_subset} it is possible to find an increasing sequence of natural numbers $m_n$ such that the resulting sequence of unitaries $z_n=w_ny_{m_n}^*$ satisfies\n\\[\nz_n\\theta_x(b)z_n^* \\to \\tilde{\\theta}_x(b),\\quad z_n\\mathbbm{v}_x(h)(\\alpha\\otimes\\delta)_h(z_n)^* \\to \\tilde{\\mathbbm{v}}_x(h)\n\\]\nin the strong operator topology for all $b\\in B_x$ and $h\\in H_x$.\nThen it follows from Remark~\\ref{rem:Polish-spaces} that $(\\theta_x, \\mathbbm{v}_x)$ indeed belongs to the closure of $\\mathcal{U}(B_x \\bar{\\otimes}\\mathcal{R})\\cdot (\\tilde{\\theta}_x, \\tilde{\\mathbbm{v}}_x)$.\nThis finishes the proof.\n\\end{proof}\n\n\\begin{definition}[see {\\cite[Definition~3.4]{Delaroche79}}]\nAn action $\\alpha \\colon G \\curvearrowright B$ of a countable discrete group on a von Neumann algebra is called amenable, if there exists an equivariant conditional expectation \n\\[\nP\\colon (\\ell^\\infty(G) \\bar{\\otimes} B, \\tau \\otimes \\alpha) \\rightarrow (B, \\alpha),\n\\]\nwhere $\\tau$ denotes the left translation action $G \\curvearrowright \\ell^\\infty(G)$.\n\\end{definition}\n\nBy \\cite{Delaroche82}, an action $\\alpha$ as above is amenable if and only if its restriction to $\\mathcal{Z}(B)$ is amenable, which is equivalent to the action on the measure-theoretic spectrum of the center being amenable in the sense of Zimmer.\nRecall that an automorphism $\\alpha\\in\\operatorname{Aut}(M)$ on a separably acting von Neumann algebra is properly centrally non-trivial, if for every non-zero projection $p\\in\\mathcal{Z}(M)$, the restriction of $\\alpha_\\omega$ on $pM_\\omega$ is non-trivial.\n\nThe following result contains Theorem~\\ref{theorem-A} for actions on semi-finite von Neumann algebras as a special case via $H=G$.\n\n\\begin{corollary} \\label{cor:McDuff-passes}\nLet $G$ be a countable discrete group and $B$ a semi-finite von Neumann algebra with separable predual such that $B \\cong B \\bar{\\otimes} \\mathcal{R}$. \nLet $\\alpha: G \\curvearrowright B$ be an amenable action.\nSuppose $H\\subseteq G$ is a normal subgroup such that for every $g\\in G\\setminus H$, the automorphism $\\alpha_g$ is properly centrally non-trivial.\nLet $G_1=G\/H$ with quotient map $\\pi: G\\to G_1$ and let $\\delta: G_1\\curvearrowright\\mathcal R$ be a strongly self-absorbing action.\nThen $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\delta_\\pi$.\n\\end{corollary}\n\\begin{proof}\nAdopt the notation from Example~\\ref{ex:central_decomposition} and Theorem~\\ref{thm:main_technical}.\nWe identify $\\alpha$ with an action of ${\\mathcal{G}:= G \\ltimes X}$ on a measurable field of semi-finite factors with separable predual $(B_x)_{x \\in X}$. \nDenote by ${X \\ni x \\mapsto H_x}$ the measurable field of isotropy groups.\nAmenability of the action $\\alpha$ implies that the action on $(X,\\mu)$ is amenable in the sense of Zimmer, \nwhich in turn implies amenability of the associated transformation groupoid. \nIn particular all isotropy groups $H_x$ are amenable. \n\nBy assumption on $H$ and \\cite[Theorem 9.14]{MasudaTomatsu16}, it follows for every $g\\in G\\setminus H$ and $\\mu$-almost all $x\\in X$ that either $g\\notin H_x$ or the automorphism $(\\alpha|_{H_x})_g$ on the McDuff factor $B_x$ is centrally non-trivial.\nIn other words, after discarding a null set from $X$, we may assume for all $x\\in X$ that for all $h\\in H_x\\setminus(H_x\\cap H)$, the automorphism $(\\alpha|_{H_x})_g$ on $B_x$ is centrally non-trivial.\nBy Theorem~\\ref{theorem:model-absorption}, we get that $(\\alpha|_{H_x})$ is cocycle conjugate to $(\\alpha\\otimes\\delta_\\pi)|_{H_x}$.\nThe claim then follows via Theorem~\\ref{thm:main_technical}.\n\\end{proof}\n\n\n\\section{Actions on arbitrary von Neumann algebras}\n\nIn this section we shall generalize some of the main results we obtained so far, namely Corollaries~\\ref{cor:equivalence_equivariant_McDuff} and \\ref{cor:McDuff-passes}, to the context of group actions on not necessarily semi-finite von Neumann algebras.\nThis uses standard results in Tomita--Takesaki theory, which allow us to reduce the generalize case to the semi-finite case considered in the previous sections.\nWe will henceforth assume that the reader is familiar with the basics of Tomita--Takesaki theory as well as the theory of crossed products (for a thorough treatment the reader should consult the book \\cite{Takesaki03}), although we are about to recall the specific points needed about the former for this section.\n\n\\begin{remark}[see {\\cite[Chapters VIII, XII]{Takesaki03}}] \\label{rem:tt-basics}\nLet $M$ be a separably acting von Neumann algebra.\nGiven a faithful normal state $\\varphi$ on $M$, we denote by $\\sigma^\\varphi: \\mathbb{R}\\curvearrowright M$ its associated \\emph{modular flow}.\nIf $\\psi$ is another faithful normal state on $M$, we denote by $(D\\psi: D\\varphi): \\mathbb{R}\\to\\mathcal{U}(M)$ the associated \\emph{Connes cocycle}, which is a $\\sigma^\\varphi$-cocycle satisfying $\\operatorname{Ad}(D\\psi: D\\varphi)_t\\circ\\sigma^\\varphi_t=\\sigma^\\psi_t$ for all $t\\in\\mathbb{R}$.\nThe crossed product von Neumann algebra $\\tilde{M}=M\\rtimes_{\\sigma^\\varphi}\\mathbb{R}$ is called the \\emph{continuous core} of $M$ and does not depend on the choice of $\\varphi$ up to canonical isomorphism.\nWith slight abuse of notation, we will consider $M$ as a von Neumann subalgebra in $\\tilde{M}$, and denote by $\\lambda^{\\sigma^\\varphi}: \\mathbb{R}\\to\\mathcal{U}(\\tilde{M})$ the unitary representation implementing the modular flow on $M$.\nThe continuous core $\\tilde{M}$ is always a semi-finite von Neumann algebra.\nGiven any automorphism $\\alpha\\in\\operatorname{Aut}(M)$, there is an induced \\emph{extended automorphism} $\\tilde{\\alpha}\\in\\operatorname{Aut}(\\tilde{M})$ uniquely determined by\n\\[\n\\tilde{\\alpha}|_M = \\alpha \\quad\\text{and}\\quad \\tilde{\\alpha}(\\lambda^{\\sigma^\\varphi}_t)=(D \\varphi\\circ\\alpha^{-1} : D\\varphi)_t\\cdot\\lambda^{\\sigma^\\varphi}_t,\\quad t\\in\\mathbb R.\n\\]\nThe assignment $\\alpha\\mapsto\\tilde{\\alpha}$ defines a continuous homomorphism of Polish groups.\nTherefore, given more generally a continuous action $\\alpha: G\\curvearrowright M$ of a second-countable locally compact group, we may induce the \\emph{extended action} $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}$.\nEvery extended automorphism on $\\tilde{M}$ has the property that it commutes with the dual flow $\\hat{\\sigma}^\\varphi: \\mathbb{R}\\curvearrowright\\tilde{M}$.\n\\end{remark}\n\n\n\\begin{prop} \\label{prop:cp-centralizer}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G\\curvearrowright M$ be an action on a separably acting von Neumann algebra.\nThen the normal inclusion $M\\subset M\\rtimes_\\alpha G$ induces (via componentwise application of representing sequences) a unital $*$-homomorphism\n$(M_{\\omega,\\alpha})^{\\alpha^\\omega} \\to (M\\rtimes_\\alpha G)_\\omega$.\n\\end{prop}\n\\begin{proof}\nAssume $M$ is represented faithfully on a Hilbert space $\\mathcal{H}$, and consider the canonical inclusion $\\pi\\colon M \\rtimes_\\alpha G \\rightarrow \\mathcal{B}(L^2(G)) \\bar{\\otimes} M \\subset \\mathcal{B}(L^2(G, \\mathcal{H}))$, which on $x \\in M \\subset M \\rtimes_\\alpha G$ is given by\n\\[ [\\pi(x) \\xi](g) = \\alpha_g^{-1}(x)\\xi(g), \\quad \\xi \\in L^2(G, \\mathcal{H}), g \\in G.\\]\nIf $(x_n)_{n \\in \\mathbb{N}}$ is any bounded sequence in $M$ representing an element $x \\in (M_{\\omega, \\alpha})^{\\alpha_\\omega}$, then the invariance of $x$ will guarantee that\n$\\pi(x_n) - (1 \\otimes x_n) \\rightarrow 0$ in the strong-$*$ operator topology as $n \\rightarrow \\omega$. Since $(1 \\otimes x_n)_{n \\in \\mathbb{N}}$ represents an element in $(\\mathcal{B}(L^2(G)) \\bar{\\otimes} M)_\\omega$, it follows that $(\\pi(x_n))_{n \\in \\mathbb{N}}$ represents an element in $(\\pi(M\\rtimes_\\alpha G))_\\omega$, so $x \\in (M\\rtimes_\\alpha G)_\\omega$. This finishes the proof.\n\\end{proof}\n\n\\begin{prop} \\label{prop:Takai-duality}\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual.\nLet $\\varphi$ be a faithful normal state on $M$.\nLet $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}=M\\rtimes_{\\sigma^\\varphi}\\mathbb R$ be the extended action on the continuous core as in Remark~\\ref{rem:tt-basics}.\nWith some abuse of notation, denote by $\\tilde{\\alpha}$ also the induced action $G\\curvearrowright\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$ by acting trivially on the canonical unitary representation implementing $\\hat{\\sigma}^\\varphi$.\nThen under the Takesaki--Takai duality isomorphism $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\cong \\mathcal B(L^2(\\mathbb R))\\bar{\\otimes} M$, the action $\\tilde{\\alpha}$ is cocycle conjugate to $\\operatorname{id}_{\\mathcal{B}(L^2(\\mathbb R))}\\otimes\\alpha$.\n\\end{prop}\n\\begin{proof}\nDenote $\\alpha'=\\operatorname{id}_{\\mathcal{B}(L^2(\\mathbb R))}\\otimes\\alpha$.\nWe apply Takesaki--Takai duality \\cite[Chapter X, Theorem 2.3(iii)]{Takesaki03} to understand the $G$-action $\\tilde{\\alpha}$.\nIf $M$ is represented faithfully on a Hilbert space $\\mathcal H$, then the natural isomorphism \n\\[\n\\Theta: \\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R = (M\\rtimes_{\\sigma^\\varphi}\\mathbb R)\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\to \\mathcal{B}(L^2(\\mathbb R)) \\bar{\\otimes} M\\subseteq \\mathcal{B}(L^2(\\mathbb R,\\mathcal H))\n\\]\nhas the following properties.\nLet $\\xi\\in L^2(\\mathbb R,\\mathcal H)$.\nFor $x\\in M$ and $s,t\\in\\mathbb R$ we have\n\\[\n[\\Theta(x)\\xi ](s) = \\sigma^\\varphi_{-s}(x)\\xi(s) \\quad\\text{and}\\quad [\\Theta(\\lambda^{\\sigma^\\varphi}_t)\\xi](s) = \\xi(s-t).\n\\]\nFurthermore, if the dual flow is given via the convention \n$\\hat{\\sigma}^\\varphi_t(\\lambda_s^{\\sigma^{\\varphi}}) = e^{its}\\lambda^{\\sigma^{\\varphi}}_s$, then we also have\n\\[\n[\\Theta(\\lambda^{\\hat{\\sigma}^\\varphi}_t)\\xi](s)=e^{its}\\xi(s).\n\\]\nThen we obtain an $\\alpha'$-cocycle $\\mathbb{W}$ via\n\\[\n[\\mathbb{W}_g\\xi](s) = (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\xi(s),\\quad \\xi\\in L^2(\\mathbb R,\\mathcal H).\n\\]\nGiven how $\\tilde{\\alpha}$ acts on the domain of $\\Theta$, we can observe (using \\cite[Corollary VIII.1.4]{Takesaki03}) for any $g\\in G$, $x \\in M$, $\\xi \\in L^2(\\mathbb{R}, \\mathcal{H})$ and $s \\in \\mathbb{R}$ that\n\\[\n\\begin{array}{ccl}\n[\\Theta(\\tilde{\\alpha}_g(x))\\xi ](s) &=& \\sigma^\\varphi_{-s}(\\alpha_g(x))\\xi(s) \\\\\n&=& \\operatorname{Ad}(D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\Big( \\sigma_{-s}^{\\varphi\\circ\\alpha_g^{-1}}(\\alpha_g(x)) \\Big)\\xi(s) \\\\\n&=& \\Big(\\operatorname{Ad}(D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s}\\circ\\alpha_g\\Big) \\big( \\sigma^\\varphi_{-s}(x) \\big)\\xi(s) \\\\\n&=& \\Big[ \\Big( \\operatorname{Ad}(\\mathbb{W}_g)\\circ\\alpha'_g\\circ\\Theta\\Big)(x)\\, \\xi \\Big](s).\n\\end{array}\n\\]\nMoreover, using the cocycle identity and the chain rule for the Connes cocycle (\\cite[Theorem VIII.3.7]{Takesaki03}), we can see for any $g \\in G$, $\\xi \\in L^2(\\mathbb{R},\\mathcal{H})$ and $s,t \\in \\mathbb{R}$ that\n\\[\n\\begin{array}{ccl}\n[\\Theta(\\tilde{\\alpha}_g(\\lambda^{\\sigma^\\varphi}_t))\\xi](s) &=& [\\Theta\\big( (D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t\\cdot\\lambda^{\\sigma^\\varphi}_t \\big)\\xi](s) \\\\\n&=& \\sigma^{\\varphi}_{-s}(D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t [ \\Theta(\\lambda^{\\sigma^\\varphi}_t)\\xi ](s) \\\\\n&=& \\sigma^{\\varphi}_{-s}(D \\varphi\\circ\\alpha_g^{-1} : D\\varphi)_t \\xi(s-t) \\\\\n&=& (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s} (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{t-s}^* \\xi(s-t) \\\\\n&=& (D\\varphi: D\\varphi\\circ\\alpha_g^{-1})_{-s} \\Big[ \\Theta(\\lambda_t^{\\sigma^\\varphi}) \\mathbb{W}_g^* \\xi \\Big](s) \\\\\n&=& \\Big[ \\mathbb{W}_g \\Theta(\\lambda_t^{\\sigma^\\varphi}) \\mathbb{W}_g^* \\xi \\Big](s) \\\\\n&=& \\Big[ \\Big( \\operatorname{Ad}(\\mathbb{W}_g)\\circ\\alpha'_g\\circ\\Theta\\Big)(\\lambda_t^{\\sigma^\\varphi}) \\xi \\Big](s)\n\\end{array}\n\\]\nHere we used that $\\alpha'_g$ fixes the shift operator given by $\\Theta(\\lambda^{\\sigma^\\varphi}_t)$ for all $g\\in G$ and $t\\in\\mathbb{R}$.\nLastly, it is trivial to see that $\\alpha'$ fixes operators of the form $\\Theta(\\lambda^{\\hat{\\sigma}^\\varphi}_t)$, which in turn also commute pointwise with the cocycle $\\mathbb{W}$.\nIn conclusion, all of these observations culminate in the fact that the isomorphism $\\Theta$ extends to the cocycle conjugacy $(\\Theta,\\mathbb{W})$ between $\\tilde{\\alpha}$ and $\\alpha'$.\n\\end{proof}\n\nThe following represents our most general McDuff-type absorption theorem:\n\n\\begin{theorem} \\label{thm:general-McDuff}\nLet $G$ be a second-countable locally compact group.\nLet $\\alpha: G \\curvearrowright M$ be an action on a von Neumann algebra with separable predual and let $\\delta: G \\curvearrowright \\mathcal{R}$ be a strongly self-absorbing action on the hyperfinite II$_1$-factor.\nThen $\\alpha\\simeq_{\\mathrm{cc}}\\alpha\\otimes\\delta$ if and only if there exists a unital equivariant $*$-homomorphism $(\\mathcal R,\\delta) \\to (M_{\\omega,\\alpha},\\alpha_\\omega)$.\n\\end{theorem}\n\\begin{proof}\nThe ``only if'' part follows exactly as in the proof of Corollary~\\ref{cor:equivalence_equivariant_McDuff}, so we need to show the ``if'' part.\nSince Corollary~\\ref{cor:equivalence_equivariant_McDuff} already covers the case when $M$ is finite, we may split off the largest finite direct summand of $M$ and assume without loss of generality that $M$ is properly infinite.\n\nLet $\\varphi$ be a faithful normal state on $M$.\nAs in Remark~\\ref{rem:tt-basics}, we consider the (semi-finite) continuous core $\\tilde{M}$ and the extended $G$-action $\\tilde{\\alpha}: G\\curvearrowright\\tilde{M}$.\nSince the image of $\\tilde{\\alpha}$ commutes with the dual flow $\\hat{\\sigma}^{\\varphi}$, we have a continuous action $\\beta=\\tilde{\\alpha}\\times\\hat{\\sigma}^\\varphi: G\\times\\mathbb{R}\\curvearrowright\\tilde{M}$ via $\\beta_{(g,t)}=\\tilde{\\alpha}_g\\circ\\hat{\\sigma}^\\varphi_t$ for all $g\\in G$ and $t\\in\\mathbb{R}$.\nLet us also consider the action $\\delta^{\\mathbb R}: G\\times\\mathbb{R}\\curvearrowright\\mathcal R$ given by $\\delta^{\\mathbb R}_{(g,t)}=\\delta_g$ for all $g\\in G$ and $t\\in\\mathbb{R}$, which is evidently also strongly self-absorbing.\n\nWe apply Proposition~\\ref{prop:cp-centralizer} to $\\mathbb{R}$ in place of $G$ and to the modular flow in place of $\\alpha$.\nIn this context, recall that the ultrapower flow $(\\sigma^\\varphi)^\\omega$ already acts continuously on the ultrapower $M^\\omega$ (see \\cite[Theorem 4.1]{AndoHaagerup14}), and its restriction to $M_\\omega$ is trivial.\nSo $M_\\omega=(M_{\\omega,\\sigma^\\varphi})^{(\\sigma^\\varphi)^\\omega}$ and Proposition~\\ref{prop:cp-centralizer} implies that the inclusion $M\\subset\\tilde{M}$ induces an embedding $M_\\omega\\to \\tilde{M}_\\omega$.\nSince by definition, one has $\\tilde{\\alpha}|_M=\\alpha$ as $G$-actions, it is clear that bounded $(\\alpha,\\omega)$-equicontinuous sequences in $M$ become $(\\tilde{\\alpha},\\omega)$-equicontinuous sequences in $\\tilde{M}$.\nKeeping in mind that the dual flow $\\hat{\\sigma}^\\varphi$ acts trivially on $M$ by definition, the aforementioned inclusion therefore induces an equivariant unital $*$-homomorphism\n\\[\n(M_{\\omega,\\alpha},\\alpha_\\omega) \\to ( (\\tilde{M}_{\\omega,\\beta})^{(\\hat{\\sigma}^\\varphi)^\\omega},\\tilde{\\alpha}_\\omega ).\n\\]\nIf we compose this $*$-homomorphism with a given unital equivariant $*$-homomorphism $(\\mathcal R,\\delta)\\to (M_{\\omega,\\alpha},\\alpha_\\omega)$, we can view the resulting map as an $(G\\times\\mathbb{R})$-equivariant unital $*$-homomorphism $(\\mathcal R,\\delta^{\\mathbb R}) \\to (\\tilde{M}_{\\omega,\\beta},\\beta_\\omega)$.\nSince $\\tilde{M}$ is semi-finite, it follows from Corollary~\\ref{cor:equivalence_equivariant_McDuff} that $\\beta$ and $\\beta\\otimes\\delta^{\\mathbb R}$ are cocycle conjugate as $(G\\times\\mathbb R)$-actions, say via $(\\Psi,\\mathbb{V}): (\\tilde{M},\\beta)\\to (\\tilde{M}\\bar{\\otimes}\\mathcal{R},\\beta\\otimes\\delta^{\\mathbb R})$.\nRemembering $\\beta=\\tilde{\\alpha}\\times\\hat{\\sigma}^\\varphi$, we consider the $\\hat{\\sigma}^\\varphi\\otimes \\mathrm{id}_\\mathcal{R}$-cocycle $\\mathbbm{w}_t=\\mathbb{V}_{(1_G,t)}$ and the $\\tilde{\\alpha}\\otimes \\delta$-cocycle $\\mathbbm{v}_g=\\mathbb{V}_{(g,0)}$.\nThe cocycle identity for $\\mathbb{V}$ then implies the relation \n\\begin{equation} \\label{eq:cocycle-interaction}\n\\mathbbm{w}_t(\\hat{\\sigma}^\\varphi_t\\otimes\\operatorname{id}_{\\mathcal R})(\\mathbbm{v}_g)=\\mathbbm{v}_g(\\tilde{\\alpha}\\otimes\\delta)_g(\\mathbbm{w}_t)\n\\end{equation}\nfor all $g\\in G$ and $t\\in\\mathbb R$.\nThe cocycle conjugacy of flows $(\\Psi,\\mathbbm{w})$ induces an isomorphism\n\\[\n\\Lambda:=(\\Psi,\\mathbbm{w})\\rtimes\\mathbb R: \\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R \\to (\\tilde{M}\\bar{\\otimes}\\mathcal R)\\rtimes_{\\hat{\\sigma}^\\varphi\\otimes\\operatorname{id}_{\\mathcal R}}\\mathbb R \\cong \\big(\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R\\big)\\otimes\\mathcal R\n\\]\nvia\n\\[\n\\Lambda|_{\\tilde{M}} = \\Psi \\quad\\text{and}\\quad \\Lambda(\\lambda^{\\hat{\\sigma}^\\varphi}_t)=\\mathbbm{w}_t(\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}),\\quad t\\in\\mathbb R.\n\\]\nWith slight abuse of notation (as in Proposition~\\ref{prop:Takai-duality}), we also denote by $\\tilde{\\alpha}$ the obvious induced $G$-action on the crossed product $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$.\nUsing that $(\\Psi,\\mathbb{V})$ was a cocycle conjugacy, we observe for all $g\\in G$ and $t\\in\\mathbb{R}$ that \n\\[\n\\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\tilde{\\alpha}\\otimes\\delta)_g\\circ\\Lambda|_{\\tilde{M}} = \\operatorname{Ad}(\\mathbb{V}_{(g,0)})\\circ(\\beta\\otimes\\delta^{\\mathbb R})_{(g,0)}\\circ\\Psi=\\Psi\\circ\\beta_{(g,0)}=\\Psi\\circ\\tilde{\\alpha}_g\n\\]\nand\n\\[\n\\begin{array}{ccl}\n\\big( \\operatorname{Ad}(\\mathbbm{v}_g)\\circ(\\tilde{\\alpha}\\otimes\\delta)_g\\circ\\Lambda \\big)(\\lambda^{\\hat{\\sigma}^\\varphi}_t) \n&=& \\mathbbm{v}_g \\big( (\\tilde{\\alpha}\\otimes\\delta)_g(\\mathbbm{w}_t) (\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\big) \\mathbbm{v}_g^* \\\\\n&\\stackrel{\\eqref{eq:cocycle-interaction}}{=}& \\mathbbm{w}_t(\\hat{\\sigma}^\\varphi_t\\otimes\\operatorname{id}_{\\mathcal R})(\\mathbbm{v}_g) (\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\mathbbm{v}_g^* \\\\\n&=& \\mathbbm{w}_t(\\lambda^{\\hat{\\sigma}^\\varphi}_t\\otimes 1_{\\mathcal R}) \\ = \\ \\Lambda(\\lambda^{\\hat{\\sigma}^\\varphi}_t).\n\\end{array}\n\\]\nIn conclusion, the pair $(\\Lambda,\\mathbbm{v})$ defines a cocycle conjugacy between the $G$-actions $\\tilde{\\alpha}$ on $\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R$ and $\\tilde{\\alpha}\\otimes\\delta$ on $\\big(\\tilde{M}\\rtimes_{\\hat{\\sigma}^\\varphi}\\mathbb R\\big)\\otimes\\mathcal R$.\nBy Proposition~\\ref{prop:Takai-duality}, the action $\\tilde{\\alpha}$ is cocycle conjugate to $\\operatorname{id}_{\\mathcal B(\\ell^2(\\mathbb{N}))}\\otimes\\alpha$.\nSince we assumed $M$ to be properly infinite, it furthermore follows that $\\operatorname{id}_{\\mathcal B(\\ell^2(\\mathbb{N}))}\\otimes\\alpha$ is cocycle conjugate to $\\alpha$.\\footnote{Although this appears to be well-known, we could not find a good literature source for this precise claim and in this generality. We note, however, that the recent proof of the analogous C$^*$-algebraic statement \\cite[Proposition 1.4]{GabeSzabo22kp} carries over in an obvious way to this setting.}\nCombining all these cocycle conjugacies yields one between $\\alpha$ and $\\alpha\\otimes\\delta$.\nThis finishes the proof.\n\\end{proof}\n\nThe following consequence is our last main result, which generalizes Corollary~\\ref{cor:McDuff-passes} to actions on arbitrary von Neumann algebras.\n\n\\begin{theorem} \\label{thm:general-amenable-McDuff}\nLet $G$ be a countable discrete group and $M$ a von Neumann algebra with separable predual such that $M \\cong M\\bar{\\otimes} \\mathcal{R}$. \nThen for every amenable action $\\alpha: G \\curvearrowright M$, one has $\\alpha \\simeq_{\\mathrm{cc}} \\alpha \\otimes \\mathrm{id}_\\mathcal{R}$.\n\\end{theorem}\n\\begin{proof}\nChoose a faithful normal state $\\varphi$ on $M$.\nRecall that the induced faithful normal state $\\varphi^\\omega$ on $M^\\omega$ restricts to a tracial state on $M_\\omega$. We denote by $\\|\\cdot\\|_2=\\|\\cdot\\|_{\\varphi^\\omega}$ the induced tracial 2-norm on $M_\\omega$.\nSince we assumed $M$ to be McDuff, it follows that $M_\\omega$ is locally McDuff in the following sense:\nGiven any $\\|\\cdot\\|_2$-separable $*$-subalgebra $B\\subset M_\\omega$, there exists a unital $*$-homomorphism $\\mathcal R\\to M_\\omega\\cap B'$.\n\nNow we choose $N_1=\\mathcal{Z}(M)$ as a subalgebra of $M_\\omega$.\nWe may then choose a unital $*$-homomorphism $\\psi_1: \\mathcal{R}\\to M_\\omega$, and define $N_2$ to be the $\\|\\cdot\\|_2$-closed $*$-subalgebra generated by $N_1$ and the range of $\\alpha_{\\omega,g}\\circ\\psi_1$ for all $g\\in G$.\nAfter that, we may choose a unital $*$-homomorphism $\\psi_2: \\mathcal{R}\\to M_\\omega\\cap N_2'$, and define $N_3$ to be the $\\|\\cdot\\|_2$-closed $*$-subalgebra generated by $N_2$ and the range of $\\alpha_{\\omega,g}\\circ\\psi_2$ for all $g\\in G$.\nCarry on inductively like this and obtain an increasing sequence of $\\alpha_{\\omega}$-invariant separable von Neumann subalgebras $N_k\\subseteq M_\\omega$.\nThe $\\|\\cdot\\|_2$-closure of the union $\\bigcup_{k\\geq 1} N_k$ is then a separably acting finite von Neumann subalgebra, which is clearly McDuff and $\\alpha_{\\omega}$-invariant.\nFurthermore $N$ contains the center of $M$ equivariantly, which implies that the action $\\alpha_\\omega$ is amenable on $N$.\n\nBy Corollary~\\ref{cor:McDuff-passes} (with $H=G$), it follows that $\\alpha_{\\omega}|_N$ is cocycle conjugate to $(\\alpha_\\omega|_N)\\otimes\\operatorname{id}_{\\mathcal R}$.\nIn particular we may find some unital $*$-homomorphism $\\mathcal R\\to (N_\\omega)^{\\alpha_\\omega}$.\nApplying a standard reindexation trick, we may use this to obtain a unital $*$-homomorphism $\\mathcal R\\to (M_\\omega)^{\\alpha_\\omega}$, which finishes the proof by Theorem~\\ref{thm:general-McDuff}.\n\\end{proof}\n\n\n\\addcontentsline{toc}{section}{References}\n\\bibliographystyle{gabor}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}