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+{"text":"\\section*{Abstract}\nThe ability to register image data to a common coordinate system is a\ncritical feature of virtually all imaging studies that\nrequire  multiple subject analysis,\ncombining single subject data from multiple\nmodalities, or both.  However, in spite of the\nabundance of literature on the subject and the existence of\nseveral variants of registration algorithms, their practical\nutility remains\nproblematic, as commonly acknowledged even by developers of these\nmethods because the complexity of the problem\nhas resisted a general, flexible, and robust theoretical and\ncomputational framework.\n\nTo address this issue, we present\na new registration method that is\nsimilar in spirit to the current state-of-the-art technique\nof diffeomorphic mapping, but is more general and flexible. The\nmethod utilizes a Hamiltonian formalism and constructs registration as\na sequence of symplectomorphic maps in\nconjunction with a novel phase space regularization based on\nthe powerful entropy spectrum pathways (ESP)\nframework.\n\nThe main advantage of the ESP\nregularized symplectomorphic approach versus the standard\napproach of coordinates-only diffeomorphic mapping lies in use of\na common metric that remains valid even\nwith image dependent regularization.\nMoreover, the fusion of the\nHamiltonian framework with the ESP theory goes beyond just providing\nan alternative spatially varying smoothing strategy - it provides an\nefficient and straightforward way to combine multiple\nmodalities.\n\nThe method is demonstrated on the three different magnetic resonance\nimaging (MRI) modalities routinely used for human neuroimaging\napplications by mapping between high resolution anatomical\n(HRA) volumes, medium resolution diffusion weighted MRI\n(DW-MRI) and HRA volumes, and low\nresolution functional MRI (fMRI) and HRA volumes.\nThe typical processing time for high\nquality mapping ranges from less than a minute to several minutes on a\nmodern multi core CPU for typical high resolution\nanatomical ($\\sim 256^3$ voxels) MRI volumes.\n\nFor validation of the framework we developed a panel of deformations\nexpressed in analytical form that includes deformations based on known\nphysical processes in MRI that\nreproduces various distortions and artifacts typically present in\nimages collected using these different MRI modalities.  Use\nof this panel allows us to quantify repeatability and reproducibility\nof our method in comparison to several available alternative\napproaches. The panel can be used in future studies especially for\nquantitative clinical validation of\ndifferent registration approaches.\n\nThe registration tool will be available as a part of the\nQUEST suite from the UCSD Center for Scientific Computation in Imaging\n(CSCI).\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nModern imaging systems are increasingly capable of acquiring\ndata sensitive to a wide range of physical parameters at multiple\nresolutions, thus offering greater sensitivity to structural and\ndynamical information in complex biological systems.  However, these\ntechnological advancements present the increasingly important\ntheoretical and computational challenge of how to rigorously and\nefficiently combine, or \\textit{register}, such data in order to be\nable to accurately detect and quantify subtle and complex system\ncharacteristics.\n\n\nThe ability to register image data to a common coordinate system\nis a critical feature of virtually all imaging studies that require\nquantitative statistical analysis of group populations, as well as for\ncombining single subject modalities.  Consequently, this subject has\nbeen the focus of a great deal of research.  This has been a focus in\ncomputational neuroanatomy which has motivated the developed of\n\\textit{diffeomorphic} registrations methods\n\\citep{pmid15551602,pmid17761438,pmid17354694,pmid18979814,pmid22194239}\nfor which faster and more efficient algorithms continue to be\ndeveloped\\citep{pmid26221678, pmid24968094, pmid19709963,\n  pmid18979813, pmid23685032}, as well as various regularizations\n\\citep{pmid24409140, pmid20879371} and additional enhancements such as\nlocal-global mixture, contrast changes, multichannel mapping, etc\n\\citep{pmid24217008, pmid21197460, pmid22972747}, and the use of\nprobabilistic diffeomorphic registration methods \\citep{pmid25320790,\n  pmid20879365}.  These registration advancements are important to\ngroup analyses and the development of standard atlases\n\\citep{pmid20347998,pmid24579121, pmid15501084, pmid23769915,\n  pmid21995026, pmid21276861, pmid17354780} which serve a critical\nrole in the standardization of studies.  The emergence of diffusion\ntensor imaging (DTI) methods and their variants for connectivity\nstudies required the extension of diffeomorphic registration methods\nto accommodate tensor data \\citep{pmid23880040, pmid22941943,\n  pmid20382233, pmid19694253, pmid21134814, pmid19398016,\n  pmid21316463, pmid25433212, pmid25333121, pmid24579120,\n  pmid23286046, pmid22156979, pmid21761677, pmid18390342}.  These\nmethods have had a profound effect on the success of numerous\nscientific studies on important clinical issues such as Alzheimer's\nand traumatic brain injury \\citep{pmid24936424, pmid23333372,\n  pmid23322456, pmid20879457, pmid20211269, pmid17999940}, as well as\nstudies in other organs (cardiac, lungs, etc) \\citep{pmid24505703,\n  pmid22481815, pmid16093505, pmid15508155, pmid20363173}.  Another\nimportant and even more challenging task is a multi-modal registration\n(i.e. registering T1 and T2 images, or T1 and DTI, etc), as the\noptimal choice of an appropriate objective function is unknown.\nDesigning and evaluating a universal algorithm that can fit various\napplications (among subjects, multi-modal within-subject, multi-modal\nacross subjects) is an important problem that needs to be addressed,\nas existing approaches do not currently posses such universality (see,\ne.g., \\citet{pmid23739795} for a comprehensive review).\n\nIn spite of the abundance of literature and the existence of several\nvariants of diffeomorphic algorithms their practical appeal are still\nrather limited (possibly due to an interplay of a variety of reasons\n-- speed, accuracy, robustness, complexity, repeatability, etc), as\ncommonly acknowledged even by developers of these registration\nmethods. For example citing the developer of one of the relatively\nbroadly used approaches -- Large Deformation Diffeomorphic Metric\nMapping \\citep{pmid19398016, pmid24579120, pmid23286046, pmid21761677,\n  pmid17999940, pmid21521665} -- ``applications of the LDDMM framework\non volumetric 3D medical images still remain limited for practical\nreasons'' \\citep{LDDMM}.  Two large and thorough comparison studies\n(i.e. \\citet{pmid19195496, Ribeiro2015}) also confirm that although\ncurrently available methods are in general able to perform the\nregistration task with varying degrees of success (although some are\nexceedingly slow and some are not particular accurate), the practical\nuse limitations seem to drive an interest in improvements at least in\nterms of speed and accuracy.\n\nThe recent review paper \\citep{pmid27427472} \nconducted a retrospective analysis of the past two decades\nof the field of medical image registration since publication of the\noriginal review \\citep{pmid10638851}. It is alarming again that the\nmain conclusion of this twenty years \nretrospective is that in spite of all the progress in the\nfield of registration ``the two major problems mentioned in\n\\citep{pmid10638851} -- validation of registration methods and\ntranslation of these to the clinic -- are major problems still, which\nhave even been aggravated by the elaboration of registration\nmethods.''\n\nTo address these issues we present in this paper a new method that is\nsimilar in spirit to diffeomorphic mapping, but is more general and\nflexible.  The transformation is developed within a Hamiltonian\nformalism \\citep{vialard:tel-00400379,pmid26643025,pmid19059343} in\nwhich not just the spatial coordinates are considered, but the\nentirety of phase spac\n, which is a called a\n\\textit{symplectomorphism}. \nThis theoretical construct enables a novel\nflexible, accurate, and robust computational method based on a\nsequence of energy shell transformations.  The incorporation\nof phase space constraints allows us to use the same simple metric\non the space of diffeomorphisms that remains valid even with image dependent\nregularization, something that is missing in currently available\nmethods.\n\nThe generality of the Hamiltonian framework facilitates the inclusion\nof powerful prior information for spatially varying regularization in\nphase space using our recently developed method of entropy spectrum\npathways (ESP) \\citep{Frank:2014pre}.  This is in contrast with the\ncurrent state-of-the-art approaches that introduce\nregularization as a differential form (almost always with constant\ncoefficients) acting on the map itself (see\ne.g.~\\citet{Beg2005,doi:10.1137\/140984002,5204344}), that effectively\napply regularization as an additional post processing step, thus\ncreating additional problems, especially for validation and comparison\nbetween different approaches and even different regularization\ntechniques.  Even the existence of several techniques for spatially\nvarying smoothing strategies that have been recently proposed\n\\citep{pmid25485406,pmid25333122} do not remediate this validation\nissue (and this is in addition to being of rather limited practical\nutility, possibly adding even more speed--accuracy--complexity issues\nthan providing solutions).  Generally speaking, the Hamiltonian\napproach facilitates validation of different regularizations without\ndestroying or modifying the metric on the space of diffeomorphisms.\n\nThe importance and the main advantage of symplectomorphic approach\nversus coordinates only diffeomorphic mapping can be understood from\nthe fact that use of the same common metric allows quantitative\nassessment of differences between registrations as well as evaluation\nof performance for different regularization schemes. An existence of\nvolumetric\/surface\/line measures allows accurate comparison of\nfeatures between subvolumes, surface areas or linear curves.\n\nWhat is even more important is that this fusion of the Hamiltonian\nframework with the ESP theory goes beyond just providing an\nalternative spatially varying smoothing strategy. It provides an\nefficient and straightforward way to combine multiple modalities, for\nuse in tractography, structural and functional connectivity,\netc. (although the details of implementation go beyond the subject of\nthis paper and will be reported elsewhere).  \n\nOur method also incorporates fast, accurate, and flexible spatial\npreconditioning using our spherical wave decomposition (SWD)\n\\citep{swd}.  The SWD approach uses fast FFT--based algorithms to\nexpand images in spherical wave modes and therefore allows to do image\nresampling, scaling, rotating and filtering with the highest possible\norder of polynomial accuracy, but at a fraction of a time.\n\nThe method is validated on a well characterized numerical phantom and\nthen demonstrated on a set of the ``standard'' neuro-MRI data\nacquisitions (HRA, DTI, rsFMRI) routinely collected at our UCSD Center\nfor FMRI (CFMRI).  We demonstrate the ability to accurately\nco-register the data volumes in computational times significantly\nfaster and more accurately than current state-of-the-art methods.  \nThe resulting image volumes also demonstrate previously unobserved image\ncontrasts that suggest the ability of our method to uncover more\nsubtle and important structural features in the data.\n\nIt is well known that different MRI acquisition schemes and protocols\nmay include a variety of incompatible distortions and artifacts due\nnot only to variations of scanner hardware and\npulse sequence designs but also due to intrinsic\nvariations in individual subject morphology, as well as just due to\nsimple motions.  Thus validation of a registration method's\nability to disentangle the complex interplay of the acquisition\ndetails with the physical effects producing distortions within any\nparticular individuals brain is an exceedingly non-trivial problem.\nTherefore, in order to facilitate a more\nquantitative validation of all these different\nconditions we developed a panel of deformations defined analytically\nand based on well-known physical effects present in the\ndifferent MRI modalities.  The deformations from the panel can be\napplied to images of different modalities and acquisition condition\nand potentially can be appropriate for quick and robust validation in\nclinical settings as well.  This validation approach is somewhat\nsimilar to Gaussian deformations used in \\citet{pmid15896998}, but our\npanel includes deformations that can be attributed to a variety of\nreal physical processes present in different acquisition protocols and\nmodalities (i.e.~twist, whirl, stretch, etc).\n\nTo evaluate the practical aspects of our implementation and to\ndemonstrate the competitiveness of our approach we compared the\naccuracy and speed of phantom registration with several commonly used\nregistration methods that are often reported as top performers\n\\citep{pmid19195496} in either speed or accuracy (ANTs Diffeomorphic\nDemons, ANTs SyN, FSL FNIRT and AFNI 3dQWarp).  While a variety of\nsimilarity metrics are available, for this paper we used a simple\nRoot-Mean-Square Deviation (RMSD) as a metric to evaluate\nthe accuracy of numerical phantom registration and\nwall--clock time (that characterizes the human perception of the\npassage of time from the start to the completion of a task, referred\nto as \\textit{time} afterwards) as a practical and intuitive\nmeasure of the algorithms efficiency.\n\nIn summary, this paper utilizes a Hamiltonian formalism to develop a\nnew approach to non-linear flexible image registration. The method\nbuilds a diffeomorphic mapping as a sequence of symplectomorphic maps\nwith each map embedded in a separate energy shell. The approach adds a\nnovel phase space regularization based on the\npowerful entropy spectrum pathways framework. The framework provides a\nunique opportunity to tailor image details into the\nregularization scheme by choosing an image derived regularization\nkernel.  A spherical wave decomposition is applied as a\npowerful preconditioning tool in the position\ndomain to allow accurate and fast\ninterpolation, resampling and estimation of fixed shape rotation and\nscale.  The result is an efficient and versatile method capable of\nfast and accurate registration of a variety of volumetric images of\ndifferent modalities and resolutions.\n\n\\section{Symplectomorphic mapping}\n\\label{sec:theory}\nWe introduce the Hamiltonian function\n$\\mathcal{H}(\\mvec{q},\\mvec{p})$ on a fixed Cartesian grid $\\mvec{x}$\nas\n\\begin{equation}\\label{eq::hamiltonian}\n\\mathcal{H}(\\mvec{q},\\mvec{p}) = \\frac{1}{2V}\\int\\l[\\mvec{p}^2 +\n  \\l(I_0(\\mvec{x})-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}.\n\\end{equation}\nHere $I_0$ and $I_1$ are two multidimensional images defined on the\nsame fixed Cartesian grid $\\mvec{x}$, $V$ is the measure (volume) of\nthe reference $I_0$ image domain ($V\\equiv \\int d\\mvec{x}$), and\n$(\\mvec{q}(\\mvec{x},t),\\mvec{p}(\\mvec{x},t))$ is a set of canonical\ncoordinates, that define a time dependent mapping from Cartesian grid\n$\\mvec{x}$ to a new curvilinear grid\n$\\mvec{y}\\equiv\\mvec{q}(\\mvec{x},t)$, such that initially at $t=0$ the\ngrids are identical, i.e. ($\\mvec{q}(\\mvec{x},0),\\mvec{p}(\\mvec{x},0))\n\\equiv (\\mvec{x},0$).\n\nThe Hamiltonian \\cref{eq::hamiltonian} defines a flow at each location\non a fixed grid through a system of Hamilton's\nequations\n\\begin{align}\n\\label{eq::flow:q}\n\\dd{\\mvec{q}}{t} &= \\Dv{\\mathcal{H}}{\\mvec{p}} \\equiv\n\\mvec{p}\\\\\n\\label{eq::flow:p}\n\\dd{\\mvec{p}}{t} &=-\\Dv{\\mathcal{H}}{\\mvec{q}} \\equiv\n\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{q}}\n\\end{align}\nwhere $\\delta\\mathcal{H}\/\\delta ...$ denotes variational (or\nfunctional) derivative.\n\nThe flow defined by \\cref{eq::flow:q,eq::flow:p}\nis called a \\textit{Hamiltonian flow} and takes place in the space\nof the coordinates $(\\mvec{q},\\mvec{p})$, which is called\n\\textit{phase space}.  Diffeomorphisms in this phase space are\ncalled \\textit{Hamiltonian diffeomorphisms} or\n\\textit{symplectomorphisms} since a phase space is a symplectic\nmanifold.  Thus symplectomorphisms preserve the symplectic structure\n(including the volume) of phase space.  This is a very important\nfeature that will allow the generation of a shell-like sequence of\ntransformations suitable for volumetric measurements and\nquantifications.\n\nBecause the Hamiltonian function \\cref{eq::hamiltonian} and the\nreference image $I_0$ are defined on a Cartesian grid $\\mvec{x}$ we do\nnot calculate the curvilinear gradient $\\Ds{I_1}{\\mvec{q}}$\ndirectly. Instead we express $I_1(\\mvec{q})$ as a function on\na Cartesian grid $I_1(\\mvec{q}(\\mvec{x},t))$ and use\nthe chain rule to evaluate the curvilinear gradient through\na gradient on Cartesian grid $\\Ds{I_1}{\\mvec{x}}$ and Jacobian\n$J\\equiv \\Ds{\\mvec{q}}{\\mvec{x}}$ as\n$\\Ds{I_1}{\\mvec{x}}(\\Ds{\\mvec{q}}{\\mvec{x}})^{-1}$.\n\nAn evolution of the Jacobian with time can be obtained by differentiating\nthe position equation (\\cref{eq::flow:q}) on a fixed grid, giving a\nclosed set of equations\n\\begin{align}\n\\label{eq::flow1:q}\n\\dd{\\mvec{q}}{t} &= \\mvec{p}\\\\\n\\label{eq::flow1:p}\n\\dd{\\mvec{p}}{t} &=\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{x}}J^{-1}\\\\\n\\label{eq::flow1:J}\n\\dd{J}{t} &=\\D{\\mvec{p}}{\\mvec{x}}\n\\end{align}\nIntegrating these equations with initial conditions\n$\\mvec{q}(\\mvec{x},0)=\\mvec{x}$, $\\mvec{p}(\\mvec{x},0)=0$, and\n$J(\\mvec{x},0) = \\mathds{1}$ generates a symplectomorphic\ntransformation $\\mvec{x} \\rightarrow \\mvec{q}(\\mvec{x},t)$.  A new\nmetric can be defined for the position part $\\mvec{q}$ of the\ncanonical coordinates by introducing the metric tensor $G \\equiv\n\\{g_{ij}\\} = {(J^{-1})}^{T} J^{-1}$, where indices $i$ and $j$\ncorrespond to derivatives over $q_i$ and $q_j$ components of the\ncurvilinear coordinates $\\mvec{q}$ such that in Euclidean space\n$g_{ij}=\\delta_{ij}$ where $\\delta_{ij}$ is the Kronecker delta.  The\nmetric tensor is important for providing accurate measures of line and\nsurface properties using the curvilinear coordinate system\n$\\mvec{q}$. For example, a length of a curve parameterized by\n${\\mvec{x}}(s)$ with a parameter $s$ between zero and one in Cartesian\nspace can be expressed using the metric tensor and curvilinear mapping\nas\n\\begin{equation}\n\\int\\limits^{1}_{0}\\left|\\dd{\\mvec{x}}{s}\\right|ds=\n\\int\\limits^{1}_{0}\\sqrt{g_{ij}\\dd{q^i}{s}\\dd{q^j}{s}}ds,\n\\end{equation}\nwhere repeated indices $i$ and $j$ represent summation.\n\nTo ensure that the transformation is symplectomorphic at every\nlocation on a fixed grid $\\mvec{x}$ during numerical integration we set\na small constant $\\epsilon$ and impose a requirement that both the\nJacobian and the inverse Jacobian are bounded by this constant, i.e.\n\\begin{equation}\n\\label{eq::jacobian:bound}\n\\epsilon < |J(x,t)| < \\epsilon^{-1},\n\\end{equation}\nFor the majority of the results presented in the paper a\nvalue of $\\epsilon=0.01$ was used.\nWhen the Jacobian becomes sufficiently close to zero the further\nintegration does not make sense as it will\nnot be able to guarantee either the symplectomorphic or\ndiffeomorphic properties of the flow (even\nnumerical stability of the solution can be compromised).  Therefore,\nwhen the condition of \\cref{eq::jacobian:bound} is violated we stop\nnumerical integration, freeze the flow, and restart the integration\n(i.e., setting $t=0$) beginning at a new set of phase space\ncoordinate $\\{\\mvec{q}^{(n)}(\\mvec{x},0),\n\\mvec{p}^{(n)}(\\mvec{x},0)\\}$ where $n$ is the number of restart\ntimes.  Since the Hamiltonian is an operator that describes the\n``energy'' of a system, we refer to these $n$ different sets of\ninitial conditions as \\textit{energy shells}.  Each restart of the\nintegration therefore represents the initiation of a new energy\nshell.\n\nThe new initial conditions that define the energy shells are\nrelated to the stopping point of the coordinates in the previous\nenergy shell by the following conditions:\n\\begin{align}\n\\label{eq::embedded:ic}\n\\mvec{q}^{(n)}(\\mvec{x},0) &= \\mvec{q}^{(n-1)}(\\mvec{x},t^{(n)}-t^{(n-1)}),\\\\\n\\mvec{p}^{(n)}(\\mvec{x},0) &= 0,\\\\\nJ^{(n)}(\\mvec{x},0) &= \\mathds{1}\n\\end{align}\nRepeating this sequence of initial conditions therefore\ngenerates a set of shell-embedded symplectomorphic transformations\nsuch that the total transformation is diffeomorphic with the Jacobian\ndefined as a product of $J^{(n)}$\n\\begin{align}\n\\label{eq::jacobian:shell}\nJ\\l(\\mvec{x},t\\r) = &J^{(n)}\\l(\\mvec{x},t-t^{(n)}\\r)\\cdot\n J^{(n-1)}\\l(\\mvec{x},t^{(n)}-t^{(n-1)}\\r) \\cdot \n\\ldots \\cdot\n J^{0}\\l(\\mvec{x},t^{(1)}\\r)\n\\end{align}\nIt is worth noting that this updating\nequation for the Jacobian effectively results in an updating of the\nmetric tensor $G = {(J^{-1})}^{T} J^{-1}$ that\ncharacterizes the local geometry and assures volume preservation.\n\nWe would like to emphasize that our use of Hamiltonian framework\nprovides a major advantage over conventional approaches in both\nefficiency and accuracy. For example, similar considerations for\nlimiting the Jacobian were employed in \\citet{pmid18290061} where\nEuler equations of viscous flow were used to describe the displacement\nfield on a fixed grid. The introduction of fixed Eulerian reference\nframe resulted in frequent use of costly and inaccurate template\nregridding procedure that is completely avoided\nby our formulation.\n\nAn important practical implementation issue is that the\nnumber of shells $n$ does not have to be introduced in advance and can\nbe determined based on overall convergence (or even devised from\nrunning time constraints).  In our numerical implementation the shells\nwere terminated as soon as $I_1 \\rightarrow I_0$ convergence condition\n\\begin{align}\\label{eq::convergence}\n\\int\\l[ \n  \\vphantom{\n    \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n)})\\r)^2 \n    \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n-1)})\\r)^2}\n\\r. & \\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n)})\\r)^2\n-\\l.\n\\l(I_0(\\mvec{x})-I_1(\\mvec{q}^{(n-1)})\\r)^2\\r]d\\mvec{x} <0\n\\end{align}\nwas not satisfied.\n\n\\section{Entropy spectrum pathways as a phase space regularization}\n\\label{sec:esp}\nThe form of Hamiltonian function used in \\cref{eq::hamiltonian}\nassumes only local input from difference between $I_0$ and $I_1$\nimages to the flow momentum $\\mvec{p}$ at every point on the fixed\ngrid $\\mvec{x}$. A more reasonable assumption would be an inclusion of\nsome information relevant to the structure of $I_0$ and $I_1$\nimages. One possible (and by far the most straightforward) way to\nprovide this structure based preconditioning\nis the entropy spectrum pathways (ESP) approach\n\\citep{Frank:2014pre} that takes into account nearest neighbor\ncoupling between adjacent grid locations.\n\nThe ESP approach starts with generating the coupling density\n$Q(\\mvec{x},\\mvec{x}^\\prime)$ which can be as simple and trivial as just the\nadjacency matrix \n\\begin{eqnarray}\nQ(\\mvec{x},\\mvec{x}^\\prime) = \n\\begin{cases}\n1 & \\mbox{if $\\mvec{x}$ and $\\mvec{x}^\\prime$ are connected}\\\\\n0 & \\mbox{if $\\mvec{x}$ and $\\mvec{x}^\\prime$ are not connected}\n\\end{cases}\n\\end{eqnarray}\nor may in general include a strength of coupling\nthrough some kind of coupling potentials that may depend on the grid\npositions. The ESP approach solves the generalized eigenvalue problem\n\\begin{equation}\\label{eq::esp:eigenproblem}\n\\lambda\\psi(\\mvec{x}) = \\int Q(\\mvec{x},\\mvec{x}^\\prime) \\psi(\\mvec{x}^\\prime)\nd\\mvec{x}^\\prime,\n\\end{equation}\nfinding the largest eigenvalue $\\lambda$ and corresponding eigenvector\n$\\psi(\\mvec{x})$ and then constructs the quantity\n\\begin{equation}\\label{eq::esp:tp}\n\\rho(\\mvec{x}^\\prime,\\mvec{x}) = \\frac{Q(\\mvec{x},\\mvec{x}^\\prime)\n  \\psi(\\mvec{x}^\\prime)}{\\lambda\\psi(\\mvec{x})}\n\\end{equation}\ncalling it the transition probability density for transition between\ngrid locations $\\mvec{x}$ and $\\mvec{x}^\\prime$. The square of the\neigenvector $\\psi(\\mvec{x})$ is called the equilibrium probability\n$\\mu(\\mvec{x})$ in the sense that it represents\nthe stationary solution that satisfies the stationary point condition\n\\begin{equation}\\label{eq::esp:ep}\n\\mu(\\mvec{x}^\\prime) = \\int\n\\rho(\\mvec{x}^\\prime,\\mvec{x})\n\\mu(\\mvec{x})\nd \\mvec{x}\n\\end{equation}\n\n\\cref{eq::esp:tp} can be included in \\cref{eq::hamiltonian} to take\ninto account nonlocal effects and provide a way of regularization\nby defining a non-local Hamiltonian\n\\begin{align}\n\\label{eq::hamiltonian:non:local}\n\\mathcal{H}^{nl}(\\mvec{q},\\mvec{p}) =& \\frac{1}{2V}\\int \\int\n\\l[\n\\delta(\\mvec{x},\\mvec{x}^\\prime) \\mvec{p}^2 \\r.\n+ \\l.\n\\rho(\\mvec{x},\\mvec{x}^\\prime)\n\\l(I_0(\\mvec{x}^\\prime)-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}d\\mvec{x}^\\prime,\n\\end{align}\nhere $\\delta(\\mvec{x},\\mvec{x}^\\prime)$ is Dirac delta function,\n$\\mvec{q}\\equiv\\mvec{q}(\\mvec{x}^\\prime,t)$ and\n$\\mvec{p}\\equiv\\mvec{p}(\\mvec{x}^\\prime,t)$. This\nnonlocal expression for the Hamiltonian function produces non-local\nHamilton's equations \n\\begin{align}\n\\label{eq::flowR:q}\n\\dd{\\mvec{q}}{t} &= \\mvec{p}\\\\\n\\label{eq::flowR:p}\n\\dd{\\mvec{p}}{t} &=\\int\\l[\\rho(\\mvec{x},\\mvec{x}^\\prime)\n\\l(I_0-I_1\\r)\\D{I_1}{\\mvec{x}}J^{-1}\n\\r]d\\mvec{x}^\\prime\n\\\\\n\\label{eq::flowR:J}\n\\dd{J}{t} &=\\D{\\mvec{p}}{\\mvec{x}}\n\\end{align}\nwhere the momentum equation (\\cref{eq::flowR:p}) is the\nnon-local version of \\cref{eq::flow1:p} that now includes the\nconvolution of a local potential (gradient of squared image difference\nin our case) with a kernel $\\rho(\\mvec{x},\\mvec{x}^\\prime)$ that\ndepends on the coupling between grid locations.\n\nAlternatively the non-local Hamiltonian function can be specified as\n\\begin{align}\n\\label{eq::hamiltonian:non:local1}\n\\mathcal{H}^{nl}(\\mvec{q},\\mvec{p}) =& \\frac{1}{2V}\\int \\int\n\\rho(\\mvec{x},\\mvec{x}^\\prime) \n\\l[\n\\mvec{p}^2 \n+\n\\l(I_0(\\mvec{x}^\\prime)-I_1(\\mvec{q}))\\r)^2\\r]d\\mvec{x}d\\mvec{x}^\\prime,\n\\end{align}\nproviding alternative non-local form for the coordinate equation (\\cref{eq::flow1:q}) as well\n\\begin{align}\n\\label{eq::flowR:qn}\n\\dd{\\mvec{q}}{t} &= \n\\int\\rho(\\mvec{x},\\mvec{x}^\\prime)\\,\\mvec{p}\\,\nd\\mvec{x}^\\prime\n\\end{align}\n\n\nAssuming that the coupling density $Q(\\mvec{x},\\mvec{x}^\\prime)$ does\nnot depend on position $\\mvec{x}$ but depends only on a difference\nbetween them (i.e. $Q(\\mvec{x},\\mvec{x}^\\prime) \\equiv\nQ(\\mvec{x}-\\mvec{x}^\\prime)$), the ESP scheme can provide a variety of\nposition independent regularization kernels often used as convolution\nfilters in image registration \\citep{pmid17761438}. As a trivial\nexample, an eigenvalue problem (\\cref{eq::esp:eigenproblem}) for\nposition independent Gaussian coupling density\n$Q(\\mvec{x}-\\mvec{x}^\\prime)=\\exp(-(\\mvec{x}-\\mvec{x}^\\prime)^T S\n(\\mvec{x}-\\mvec{x}^\\prime)))$ in infinite $n$-dimensional domain has\nmaximum eigenvalue $\\lambda = \\sqrt{\\pi^n\/\\det{S}}$ and a trivial\neigenvector $\\psi(\\mvec{x})=\\mathrm{const}$, resulting in the commonly\nused Gaussian regularization kernel.  This simple\nillustration is merely meant to demonstrate that the commonly used\nGaussian kernel is naturally derived from our very general\nprocedure.  In practice, more complex coupling schemes can provide\nmore informative prior information, resulting in more robust warping\nschemes.\n\nWe would like to emphasize the significant\nadvantages that ESP regularization provides.  Its general\nformulation \\citep{Frank:2014pre} is probabilistic in nature and\nprovides a framework for the incorporation of available information.\nIn the present context of image registration it naturally provides a\nmechanism to incorporate information from either or both of the\n$I_0$ and $I_1$ images.  The position dependent coupling naturally\ncreates image dependent regularization.  Moreover, the ESP\napproach can also include any information that is not present in\nthe images themselves but known \\textit{a priori} and related to\nimages in some quantitative way can be easily included into the\ncoupling scheme with some sort of linear or nonlinear\nparameterization.  We have recently demonstrated this ability\nto incorporate multiple priors in ESP coupling in the related\nproblem of multi-modal parameter estimation \\citet{quna}, where the\nsymplectomorphic registration method of this paper was used for \nregistration of multiple modalities.\nAdditionally, incorporation of the ESP method into the\nHamiltonian formalism provides a\nsimple and efficient way for introduction of different image matching\nterms by modification of the position--based part of either\nlocal or nonlocal Hamiltonian function.  This provides great\nflexibility for tailoring the method to specific applications.\n\n\\section{Spherical waves decomposition as a position domain preconditioning}\n\\label{sec:swd}\n\nThe set of Hamilton's equations\n(\\cref{eq::flowR:q,eq::flowR:p,eq::flowR:J}) used in the previous\nsections to generate a sequence of energy shell-embedded\nsymplectomorphic transformations (\\cref{eq::jacobian:shell}) requires\nequal dimensionality of images $I_0$ and $I_1$. However, in many cases\nthe images to be registered are of different spatial resolutions so\nthat some form of interpolation is required.  To provide an\neffective way to do position domain resampling, interpolation,\nfiltering and estimation of best orthogonal transform in a single step\nwe used the spherical waves decomposition (SWD) approach \\citep{swd}.\n\nThe SWD approach uses fast algorithms to expand both $I_0$ and $I_1$\nimages in spherical wave modes\n\\begin{align}\\label{eq::swd}\nf_{lmn}^{\\{0,1\\}} =& \\int_0^a\\int_0^\\pi\\int_0^{2\\pi}\nI_{\\{0,1\\}}(r,\\theta,\\phi)R_{nl}(r)\nY_{l}^{m\\star}(\\theta,\\phi) r^2 dr\n\\sin\\theta\nd\\theta d\\phi,\n\\end{align}\nwhere $Y_{l}^{m\\star}(\\theta,\\phi)$ are the spherical harmonics, and\n$R_{nl}(r)$ can be expressed through the spherical Bessel function\n\\begin{equation}\\label{eq::bessels}\nR_{ln}(r)=\\frac{1}{\\sqrt{\\mathcal{N}_{ln}}}j_l(k_{ln}r),\n\\end{equation}\nwith an appropriate choice of normalization constants\n$\\mathcal{N}_{ln}$ and the discrete spectrum wave numbers $k_{ln}$\ndetermined by the boundary conditions.  The number of modes\n($l,m=0\\dots L_{max}$ and $n=1\\dots N_{max}$) are determined by the\nhighest image resolution.  The details of definitions of the spherical\nharmonics $Y_{l}^{m}(\\theta,\\phi)$ and spherical Bessel Functions\n$j_l(r)$ can be found in \\citet{swd}.  The interpolation and\nresampling are then implemented as fast inverse spherical wave\ntransform\n\\begin{equation}\\label{eq::swd:inv}\nI_{\\{0,1\\}}^{NL}(r,\\theta,\\phi) =\n\\sum_{n=1}^{N}\\sum_{l=0}^{L}\\sum_{m=-l}^{l}\n\\mathcal{F}_{lmn}f_{lmn}^{\\{0,1\\}}R_{ln}(r)Y_{l}^{m}(\\theta,\\phi),\n\\end{equation}\nusing appropriate grid locations $(r,\\theta,\\phi)$ and assigning\n$f_{lmn}$ to zeros for modes with $n>N_{max}$ or $l,m>L_{max}$. A\nvariety of low\/band\/high pass filters can be used for frequency domain\nfilter $\\mathcal{F}$ following the standard image processing\ntechniques.\n\nThe scale and the amount of rigid rotation between images can be\neasily and effectively estimated using the decomposition of\nthe radial and spherical parts using the partial transforms\n\\begin{align}\\label{eq::swd:inv:rtp}\nI_{\\{0,1\\}}^{N}(r) &=\\frac{1}{2\\sqrt{\\pi}}\n\\sum_{n=1}^{N}\n\\frac{1}{\\sqrt{\\mathcal{N}_{0n}}}\n\\mathcal{F}_{00n}f_{00n}^{\\{0,1\\}}j_{0}(k_{0n}r),\\\\\nI_{\\{0,1\\}}^{L}(\\theta,\\phi) &=\n\\sum_{l=0}^{L}\n\\frac{1}{\\sqrt{\\mathcal{N}_{l1}}}\n\\sum_{m=-l}^{l}\\mathcal{F}_{lm1}f_{lm1}^{\\{0,1\\}}\nY_{l}^{m}(\\theta,\\phi),\n\\end{align}\nand finding the parameters of the similarity transformation (scale $s_r$ and\nrotation angles $\\theta_r$ and $\\phi_r$) by solving the two (one and two\ndimensional) minimization problems\n\\begin{align}\\label{eq::rigid}\ns_r &= \\arg \\min_{s_r}\n\\int\\limits_{0}^{R_{max}}\n\\l[\\l(I_{0}^{N}(r)\\r)^2-\\l(I_{1}^{N}(s_r r)\\r)^2\\r] dr,\\\\\n(\\theta_r,\\phi_r) &= \\arg\n\\min_{\\theta_r \\phi_r}\n\\int\\limits_{0}^{2\\pi}\n\\int\\limits_{0}^{\\pi}\n \\l[\\l(I_{0}^{L}(\\theta,\\phi)\\r)^2 - \n\\l(I_{1}^{L}(\\theta -\n   \\theta_r,\\phi-\\phi_r)\\r)^2\\r] d\\theta d\\phi,\n\\end{align}\nusing small number of modes ($L<L_{max}$ and $N<N_{max}$) for initial\ncoarse search and increasing them to refine the estimate, thus\navoiding being trapped in local minimums and at the same time creating\ncomputationally efficient approach. Criteria similar to\nthe considerations about optimal order of SWD transform expressed\nin \\citet{swd} can be used as a stopping condition\nfor this scale refinement procedure. A similar scale refinement procedure\nwas also used for the calculation of the symplectomorphic mapping.\n\n\\section{Results and Discussion}\n\\label{sec:results}\n\nAll results shown below were coded in standard C\/C++ and\nparallelized using POSIX threads.\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig1}\n\\caption[]{3D extension of the classical ``toy'' example used for\n  benchmarking of diffeomorphic registration: fitting ``circle'' --\n  sphere in ({\\bf b}) -- to ``C'' -- spherical shell with a hole in\n  ({\\bf a}). Results of direct ({\\bf c}) and inverse ({\\bf d}) maps\n  obtained in 8 embedded energy shells. Subset of curvilinear grid\n  lines plotted for three neighboring layers selected from three\n  orthogonal planes for direct ({\\bf e}) and inverse ({\\bf f})\n  maps. The different colors were used to distinguish between the\n  anterior-posterior grid lines (blue), the dorsal-ventral grid lines\n  (green) and the right-left lateral grid lines (red). Both inverse\n  and direct maps were obtained in a single run, the processing time\n  for 200x200x200 volumetric datasets was just above 30 seconds on 12\n  cores Intel\\textregistered\\ Core\\textsuperscript{TM} i7-4930K CPU \\@\n  3.40GHz.\n\\label{fig::toy}\n}\n\\end{figure*}\n\n\\subsection{Phantom data registration}\n\\label{sec:phantom}\nTo test the approach we first applied it to 3D extension of\na classical ``toy'' example commonly used to show the\nperformance of non-linear registration approaches --\nthe registration of the ``C'' shape to\nthe ''circle'' shape. The original 200x200x200 3D ``C'' and\n''circle'' volumes are shown in \\cref{fig::toy} ({\\bf a}) and ({\\bf\n  b}). Panels ({\\bf c}) and ({\\bf d}) show 3D view and central slices\nfor the forward and inverse maps. Panels ({\\bf c}) and ({\\bf d}) show\ngrid lines for a subset of points selected from three groups of\northogonal planes. The overall performance of our approach seems to be\nvery good, with nearly perfect forward map of the ``circle'' to the\n``C'' shape, and only a slight signature of the original\nhole in the inverse map of ``C'' shape to the ``circle''. No SWD\npreconditioning was used in this example and simple adjacency type\nmatrix was used for phase space coupling.\n\nThe second stage of our evaluation procedure was the\ncomparison with some of the existing state-of-the-art non-linear\nregistration methods commonly used.\nAs it is clearly beyond the scope of our paper to conduct\na comprehensive evaluation using all possible\nvariations of nonlinear deformation\nalgorithms versus all possible variations\nof accuracy metrics (such as is\ndone in evaluation studies, like e.g. \\citet{pmid19195496}), we\ndecided to report here just the simplest and the most straightforward\ntype of metrics and have restricted our choice of registration tools\nto those commonly used at our institution.\nFollowing the recommendation from \\citet{pmid19195496} concerning the\nspeed and accuracy, we processed the phantom registration using\nDiffeomorphic Demons (which is reported as one of the fastest) and\nsymmetric diffeomorphic image registration SyN (reported as one of the\nmost accurate), both from the Advanced Normalization Tools (ANTs)\npackage \\citep{pmid17659998}, as well as using FNIRT non-linear\nregistration utility from the FSL \\citep{FNIRT} and 3dQwarp non-linear\nwarping utility from AFNI\\citep{pmid8812068}.\n\nAs noted above, our quantitative comparison of accuracy and efficiency\nare based on two simple and well-understood metrics: The time is used\nas a practical measure of efficiency and a simple root mean square\ndeviation between the reference and the registered image is used as a\nmeasure of accuracy.  The evaluation results are summarized in\n\\cref{_TABLE_} and demonstrate the enhanced accuracy and efficiency of\nour symplectomorphic registration approach in comparison with some\nwell established non-linear registration techniques for these two\nsimple metrics and a well-defined standard numerical phantom.\n\n\\begin{table*}[!htb]\n\\renewcommand{\\arraystretch}{1.3}\n  {\\footnotesize\n    \\caption{Comparison of accuracy and efficiency of phantom\n      registration between three commonly used packages (ANTs,\n      FSL and AFNI) and our symplectomorphic registration (SYM-REG)\n      approach. Simple Root-Mean-Square Deviation (RMSD) between\n      reference and registered images are used as an accuracy metric\n      and time is reported to characterize an\n      efficiency. One of the center slices of registered images is\n      also included to provide visual insights into correspondent\n      registration artifacts. \\label{_TABLE_}}\n \n    \\begin{threeparttable}\n     \n      {\\scriptsize\n        \\bgroup\\def\\arraystretch{1.0\n        \\begin{tabular}{l||ccccc}\\hline\n         \n         \n         \n         \n         \n          & \\parbox{0.75cm}{\\centering \\bfseries ANTs\\tnote{a,c}} \n          & \\parbox{0.75cm}{\\centering \\bfseries ANTs\\tnote{b,c}}\n          & \\parbox{0.75cm}{\\centering \\bfseries FSL\\tnote{d}} \n          & \\parbox{0.75cm}{\\centering \\bfseries AFNI\\tnote{e}} &\n          \\bfseries SYM-REG\\\\\n          \\hline\\hline \\bfseries RMSD & 156.4 &\n          81.0 & 448.2 & 234.3 & 54.0 \\\\\\hline \\bfseries Time & 10min\n          15sec & 1h 21min & 12min 43sec & 45min 50sec & 38sec \\\\\\hline \\bfseries\n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-8pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-ns}\n            \\end{minipage}\n                        & \n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-6pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-ants-greedyexp}\n            \\end{minipage}\n                        & \n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-6pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-ants-syn}\n            \\end{minipage}\n                        & \n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-6pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-fsl-fnirt}\n            \\end{minipage}\n                        &\n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-6pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-afni}\n            \\end{minipage}\n                        &\n            \\begin{minipage}{.13\\textwidth}\n              \\hspace*{-6pt}\n              \\includegraphics[height=20mm,clip,trim=100 100 100 100]\n                {toy-sym-reg}\n            \\end{minipage}\n                        \\\\\\hline\n        \\end{tabular}\n        \\egroup\n      }\n    \\end{threeparttable}\n    \\begin{tablenotes}\n    \\item {$^a$ {\\bfseries GreedyExp} -} We used -t GreedyExp option to produce Diffeomorphic\n      Demons style exponential mapping. It was required to increase\n      number of iterations using -i 400x160x80 as the default (-i\n      10x10x5) choice produced unacceptable results with RMSD=326.\n    \\item {$^b$ {\\bfseries SyN[0.3]} -} This was the best result we were able to achieve\n      using native ANTs diffeomorphic registration with -t SyN[0.3]\n      parameter. The number of iterations was again increased to\n      400x160x80. It may be possible to improve this results by\n      further increasing the number of iterations, but we decided that\n      going beyond 1h 30min for this simple $200^3$ phantom would not\n      justify the improvement.\n    \\item {$^c$ {\\bfseries MSD} -} We used -m MSD (mean square difference) similarity\n      model for both of ANTs methods as the rest of the models\n      (cross-correlation -- CC, mutual information -- MI, and\n      probability mapping -- PR) appeared to be both less accurate and\n      more slow for this numerical phantom registration.\n    \\item {$^d$ {\\bfseries FNIRT} -} We used the default (--miter=5,5,5,5) number of\n      iterations and it produced the unacceptable results shown\n      here. Increasing the number of iterations to 100,100,20,5\n      increased the runtime to over an hour with virtually no\n      improvement in accuracy.\n    \\item {$^e$ {\\bfseries 3dQWarp} -} We used -penfac 0.02 -workhard options for 3dQwarp to\n      allow larger grid deformations and extra iterations to find\n      better alignment (the default settings produced unacceptable\n      results after 30min with RMSD=418).\n    \\end{tablenotes}\n    \\hfill{}\n  }\n\\end{table*}\n\n\\subsection{High resolution anatomical MR data registration}\n\\label{sec:hra}\n\n\\subsubsection{Multi-subject registration}\n\nFor our first test of the method on actual data, we addressed\nthe most common usage of registration algorithms: to register\na set\nof high resolution anatomical (HRA) images to a common reference\nimage.  This is a typical multi subject analysis task appearing in a\nvariety of group studies that involve morphometry based comparison\nbetween different subjects or subject groups.  We utilized HRA data\ncollected on the 3T GE Discovery MR750 whole body system at the UCSD\nFunctional MRI (CFMRI) using a 32 channel head coil for ten different\nsubjects previously collected in a study to determine the effects of\ncaffeine on the resting state brain activity\n\\citep{Wong:2013}. However, only high resolution T1 data (all having\n290x262x262 voxel resolution) were used for registration test\ndescribed in the current subsection.  Further details are available in\n\\citet{Wong:2013}. Normalized T1 intensities were used as matching\nterms $I_0$ and $I_1$ in symplectomorphic registration.\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig2}\n\\caption[]{Results of high resolution anatomical (HRA) mapping to the\n  same anatomical reference volume (shown in {\\bf a}). ({\\bf b})\n  Central planes for four volumes out of ten subjects used for\n  mapping. ({\\bf d}) Residual images of SWD preconditioning (fitted with\n  orthogonal transform) for the same four volumes, ({\\bf c}) all ten\n  volumes averaged. Residual images of symplectomorphic transforms using 5\n  ({\\bf f}) and 15 ({\\bf h}) embedded shells with all ten subject\n  averages in ({\\bf e}) and ({\\bf g}) respectively. \n  ({\\bf i}) Convergence plots of the same four subjects as a function of\n  a number of shells.\n  ({\\bf j})\n  Illustrative plots of curvilinear grids for the same four subjects,\n  using blue color for the anterior-posterior grid lines, green for\n  the dorsal-ventral grid lines, and red for the right-left lateral\n  grid lines.\n\\label{fig::anat}\n}\n\\end{figure*}\n\n\\cref{fig::anat} shows the collage of images related to this\nregistration test.  The central plane from anatomical volume used as\nreference is shown in ({\\bf a}) panel. The same location planes for\nrandomly selected four volumes out of ten subjects are shown in ({\\bf\n  b}). Panel ({\\bf d}) shows the result of SWD preconditioning step\nequivalent to rigidly fitting each volume to reference with orthogonal\ntransform that includes rotation and uniform scaling for the same four\nvolumes. Panel ({\\bf c}) shows image obtained by averaging of SWD\npreconditioned volumes for all ten subjects. The next four panels show\nresults of symplectomorphic transforms using 5 ({\\bf f}) and 15 ({\\bf\n  h}) embedded energy shells again with correspondent all subject\naverages in ({\\bf e}) and ({\\bf g}) respectively. And finally, panel\n({\\bf i}) show illustrative plots of curvilinear grids for the same\nrandomly chosen four subjects.\n\nOverall, as would be expected, the symplectomorphic registration shows\nsignificant improvement over rigidly fitted volumes, with additional\nimprovement due to increase of a number of energy shells used in\nregistration.  In general, there is no obvious relationship between\nthe number of shells and the accuracy, although practically, as the\nnumber of shells is determined by selected limits of Jacobian range\n($\\epsilon$) (and indirectly can be affected by a selected policy of\ntime step adjustments), symplectomorphic registration with increased\nnumber of shells may allow to obtain better overall accuracy.  The\ntotal processing time for all ten subject fitting ranges from 15 to 40\nminutes based on the selected quality (this is time\nmeasured by running the registration on 12 cores\nIntel\\textregistered\\ Core\\textsuperscript{TM} i7-4930K CPU \\@\n3.40GHz).\n\nWe note, while the quality of the fit is measured here by\nRMSD, the most advantageous implementation in any particular\nclinical or research scenario of course depends on on several\nparameters, including the desired quality of the fit, the type of\nregularization, the type of coupling in ESP step, aggressiveness of\ntime step updating, etc.  These trade-offs necessarily depend on the\nspecific desires of the user in any particular application.\n\n\\subsubsection{Synthetic deformation maps}\n\nAn important practical aspect of non-linear registration\nmethods in their application to MRI is their robustness to image\ndistortions produced in the MRI procedure.  The dependence of\nthe spatial encoding process in MRI on magnet field gradient linearity\nresults, in practice, in a wide range of complex non-linear\ndistortions due to gradient non-linearities induced by both machine\ndependent factors (e.g. imperfect gradients) and subject\nmorphological variations (e.g. susceptibility effects).  To add to\nthe complexity of this problem, the machine dependent variations can\ndepend not only on the scanner vendor, but on the scanner software\nrevision as well.  And the subject morphological variations\ncertainly differ between subjects. \n\nTherefore, in order to provide a quantitative\nassessment of the symplectomorphic registration using HRA data\nunder more realistic conditions encountered in practical\napplications, we took several T1 brains acquired on different\nhardware (Siemens and GE scanners) at different resolutions using\ndifferent acquisition sequences and subjected them to\nartificially generated distortions designed to mimic some of the\nmost prominent non-linear distortions common to MRI acquisitions.\n\nWe utilized 5 different high resolution datasets:\n(1) MNI152 T1 2mm with 91x109x91 voxels, (2) T1\nMPRAGE 1.2mm with 160x200x200 voxels, (3) T1 MPRAGE 1mm with\n212x240x256 voxels, (4) T1 1mm with 290x262x262 voxels, and (5) T1 1mm\nwith 256x176x176 voxels. All subjects were resampled into MNI152 space\n(to provide the same accuracy and required workload for all subjects)\nand then were distorted several times using five different types of\nwarpage (see \\cref{fig::5x5}). The different types of warpage include\nnonlinear or differential rotation, nonlinear stretching, nonlinear\ncompression, etc, and \\cref{fig::5x5} shows one of the subjects with\nall five warpage types applied to this subject.\n\nThe idea of generation and use of synthetic deformation maps for\nvalidation is not new. A somewhat similar procedure of synthetic\nGaussian 2D deformations was used recently for registration validation\nin \\citet{pmid15896998}.  The important difference of our deformation\npanel is that it includes a variety of more complex deformation modes\nobserved and frequently present in various MRI acquisition protocols\nand modalities.\n\nThe 25 warped volumes\nwere then processed using the two non-linear\nregistration methods most commonly used at our institution -- ANTs\nSyN and AFNI 3dQwarp, in addition to\nSYM-REG, and compared the restored volumes with\nthe original unwarped datasets. The results are summarized in\n\\cref{_TABLE_HRA_} showing mean values for the RMSD as well as mean\nexecution times for all subjects (top) and for all warpage types\n(bottom). The complete processing took 976, 3414 and 1223 seconds for\nSYM-REG, AFNI and ANTs respectively giving 304.6, 358.5 and 584.0 for\nthe RMSD values with SYM-REG providing both the lowest RMSD\nerror and the lowest execution time (the default processing options\nwere used for all packages).\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig2-1}\n\\caption[]{Different nonlinear warpage types applied to 5 different\n  subjects (only one subject is shown for each warpage type): ({\\bf\n    a}) an original subject; ({\\bf b}) differential rotation with the\n  amount of rotation proportional to the distance from the center in\n  the axial plane (whirl); ({\\bf c}) differential stretch in the\n  anterior direction; ({\\bf d}) differential\n  rotation with the amount of rotation proportional to the distance\n  from the axial plane in the longitudinal direction (twist); ({\\bf\n    e}) nonuniform compression in the axial plane proportional to the\n  the longitudinal distance; ({\\bf f}) nonuniform compression in the\n  longitudinal direction relative to the position in the axial plane.\n\\label{fig::5x5}\n}\n\\end{figure*}\n\n\\begin{table*}[!htb]\n\\renewcommand{\\arraystretch}{1.3}\n  {\\footnotesize\n    \\caption{Comparison of accuracy and efficiency of HRA registration\n      between ANTs SyN, AFNI and our symplectomorphic registration\n      (SYM-REG) approach. Five different subjects and five different\n      warpage types were used for the comparison. Two subtables\n      include mean RMSD values and mean execution times calculated\n      across subject (top) and across warpage types (bottom) with the\n      best values marked in red. The average of all subjects and\n      warpage types gives RMSD of 304.6, 348.5 and 584.0 for SYM-REG,\n      AFNI and ANTs respectively with 976, 3414 and 1223 seconds\n      required to finish the processing for each method.\n      \\label{_TABLE_HRA_}}\n \n    \\begin{threeparttable}\n     \n      {\\scriptsize\n        \\bgroup\\def\\arraystretch{1.5\n        \\begin{tabular}{l||C{1.55cm}C{1.55cm}C{1.55cm}C{1.55cm}C{1.55cm}}\\hline\n          \\bfseries Method & \n          \\multicolumn{5}{c}{\n          \\parbox{0.5\\textwidth}{\n          \\bfseries Mean RMSD \/ time (s) across subjects}}\\\\\\hline\\hline\n          \\bfseries AFNI~3dQwarp &402.2\/130.8  &363.8\/139.4  &371.6\/130.0  &316.5\/137.8  &338.1\/144.8 \\\\\\hline \n          \\bfseries ANTs SyN     &578.2\/\\cell{38.2}  &606.0\/47.0  &586.2\/43.2  &581.1\/60.0  &568.5\/56.2  \\\\\\hline \n          \\bfseries SYM-REG      &\\cell{298.2}\/54.8  &\\cell{312.4}\/\\cell{37.0}  &\\cell{311.8}\/\\cell{34.2}  &\\cell{300.7}\/\\cell{33.4}  &\\cell{299.8}\/\\cell{35.8}  \\\\\\hline\\hline\n          & \n          \\multicolumn{5}{c}{\n          \\parbox{0.5\\textwidth}{\\hspace*{-10pt}\n          \\bfseries Mean RMSD \/ time (s) across warpage types}}\\\\\\hline\\hline\n          \\bfseries AFNI~3dQwarp &162.9\/134.0  &373.5\/137.2  &482.7\/135.6  &404.6\/130.0  &368.6\/146.0  \\\\\\hline \n          \\bfseries ANTs SyN     &140.8\/50.6  &705.0\/47.4  &602.9\/53.6  &781.9\/51.0  &689.4\/42.0  \\\\\\hline \n          \\bfseries SYM-REG      &\\cell{87.0}\/\\cell{38.4}  &\\cell{331.6}\/\\cell{46.8}  &\\cell{438.7}\/\\cell{36.2}  &\\cell{351.8}\/\\cell{38.6}  &\\cell{313.9}\/\\cell{35.2}  \\\\\\hline \n        \\end{tabular}\n        \\egroup\n      }\n    \\end{threeparttable}\n   \n   \n    \\hfill{}\n  }\n\\end{table*}\n\n\\subsection{Diffusion weighed MR image registration}\n\\label{sec:dwi}\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig3}\n\\caption[]{Medium resolution (100x100x72) diffusion weighted (DWI)\n  volume registration to high resolution (168x256x256) T1\n  reference. ({\\bf a}) Reference T1 MRI image (2D center slice top and\n  3D view bottom), ({\\bf b}) DWI b0 MRI image, ({\\bf c}) equilibrium\n  probability DWI image (same resolution as b0 image), ({\\bf d}) DWI\n  image SWD preconditioned and registered to T1 image (same resolution\n  as T1 image). Side by side comparison of reference ({\\bf e}) and\n  symplectomorphic registration of DWI volume ({\\bf f}) demonstrates\n  previously unobserved image contrasts additionally differentiating\n  white matter and probably emphasizing areas of the highest\n  concentration of fibers that suggest the ability of our method to\n  uncover more subtle and important structural features in the data.\n\\label{fig::dwi}\n}\n\\end{figure*}\n\nThe second important application we investigated was the\nregistration of diffusion-weighted images (DWI) to the HRA image of\nthe same subject.  This is a critical step \nrequired for collocation of diffusion tractography\nbased quantities with high resolution anatomical morphometry.\nRegistration of DWI to HRA is generally a more complicated\nproblem than registering HRA to HRA because DWI images are typically\nacquired with echo-planar imaging (EPI) acquisitions that are not\nonly prone to more severe non-linear susceptibility distortions than\nHRA images but are invariably acquired at lower spatial resolution\nthan the HRA for the same subject.\n\nThe data used for this example were again\ncollected at the UCSD Center for Functional MRI\n(CFMRI) using 3T GE Discovery MR750 whole body system to study the\neffects of traumatic brain injuries (TBI).  The HRA T1 volume has\n168x256x256 voxel size with 1.2x0.9375x0.9375mm resolution.\n\nA multiband DTI EPI acquisition \\citep{Setsompop:2011} using three\nsimultaneous slice excitations was used to acquire data with three\ndiffusion sensitizations (at b-values b=1000\/2000\/3000\ns\/mm\\textsuperscript{2}) for 30, 45 and 65 different diffusion\ngradients (respectively) uniformly distributed over a unit sphere.\nSeveral baseline (b=0) images were also recorded.  The data were\nreconstructed offline using the CFMRI's multiband reconstruction\nroutines.  The DWI data has 100x100x72 voxel size with\n2mm\\textsuperscript{3} resolution.  The normalized HRA T1 intensity\nwas used for the reference image matching term $I_0$ and the\nequilibrium probability map (see \\citet{dwi-esp}) was used as a moving\nimage matching term $I_1$.\n\n\\cref{fig::dwi} shows a central slice and a 3D view of the reference\nvolume ({\\bf a}), DWI b=0 volume ({\\bf b}), DWI equilibrium\nprobability volume that has the same resolution as b=0 volume ({\\bf\n  c}), and the final symplectomorphic registration of the DWI\nequilibrium probability volume ({\\bf d}). The details about the\nequilibrium probability and how it is obtained can be found in\n\\citet{dwi-esp}. The last two panels ({\\bf e}-{\\bf f}) show enlarged\nside by side comparison of the HRA reference and transformed DWI with\nthe same resolution.\n\nTwo tractography examples\n-- one that applies symplectomorphic registration \nto a set of tracts as a postprocessing step, and the other\nthat includes high resolution symplectomorphic\ngrid together with the HRA data during the tracking\nstage -- are shown in \\cref{fig::dwi:trac}.\nThe figure\nshows that the symplectomorphic registration method allows\nvery accurate localization of diffusion derived tracts with\nthe high resolution anatomical features.  More details that include a\nSYM-REG based tractography implementation as well as multi-modal\nestimation in general are not relevant to this paper and are reported\nin \\citet{quna}.\n\nWe would like to mention one important consideration here.  A\nconsiderable amount of work has been spent recently not just on\nspatial registration of diffusion imaged volumes, but also on devising\ntechniques for local reorientation of diffusion tensors that would be\nconsistent with the new deformed spatial grid [see e.g\n  \\citet{pmid21134814, pmid23880040, pmid22941943, pmid20382233,\n    pmid19398016}].  These methods are both time consuming and an\nunnecessary intermediate step from our viewpoint.  An important\nfeature of our method is that we can directly import the diffeomorphic\nmaps together with the high resolution data into our diffusion\nestimation and tractography technique GO-ESP \\citep{dwi-esp},\nso that both the estimation of local diffusion properties and\nthe generation of tracts are performed in the locally warped space\ncharacterized by the spatially dependent metric tensor, thereby\nobviating the need to use any \\textit{ad hoc} proceed to impose\ngeometric consistency, which in our method is guaranteed by the\nsymplectomorphic nature of transformation.  The result is\na method that provides a  fast and effective way\nof adding a new level of details to relatively low resolution output\navailable from diffusion weighted tractography.\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig4}\n\\caption[]{Two examples of diffusion weighted tractography that\n  (left) applied symplectomorphic registration to a set of \n  diffusion weighted tracts, and (right) \n  used\n  the GO-ESP \\citep{dwi-esp} technique modified by addition\n  of HRA dataset through inverse symplectomorphic mapping to the\n  diffusion weighted data \\citep{quna}.\n\\label{fig::dwi:trac}\n}\n\\end{figure*}\n\n\\subsection{Registration of functional MR images}\n\\label{sec:fmri}\n\nThe third and final important application we investigated was\nthe registration of low spatial\nresolution functional resting state FMRI (rs-FMRI) data\nto HRA data from a single subject.  As in DWI, FMRI data is\ntypically acquired using EPI acquisitions that, as previously\nmentioned, are more prone to non-linear geometric distortions than\nHRA acquisitions, and are of lower spatial resolution.\n\nThe data used for this test were from the same caffeine\nstudy dataset \\citep{Wong:2013} used in \\cref{sec:hra}. Registration\nof functional rs-FMRI data to anatomical images is required to\nestablish an accurate localization of activation regions in the high\nresolution maps of gray matter.  Only the data collected prior to\ncaffeine administration were used. Whole brain BOLD resting-state data\nwere acquired over thirty axial slices using an echo planar imaging\n(EPI) sequence (flip angle = $70\\degree$, slice thickness=4mm, slice\ngap=1mm, FOV=24cm, TE= 30 ms,TR = 1.8 s, matrix size = $64 \\times 64\n\\times 30$).  Further details are available in \\citet{Wong:2013}.  All\ndata were pre-processed using the standard pre-processing analysis\npathway at the CFMRI (as described in \\citet{Wong:2013}).  Nuisance\nterms were removed from the resting-state BOLD time series through\nmultiple linear regression.  These nuisance regressors included: i)\nlinear and quadratic trends, ii) six motion parameters estimated\nduring image co-registration and their first derivatives, iii)\nRETROICOR (2nd order Fourier series) \\citep{Glover:2000} and RVHRCOR\n\\citep{Chang:2009B} physiological noise terms calculated from the\ncardiac and respiratory signals, and iv) the mean BOLD signals\ncalculated from WM and CSF regions and their first respective\nderivatives, where these regions were defined using partial volume\nthresholds of 0.99 for each tissue type and morphological erosion of\ntwo voxels in each direction to minimize partial voluming with gray\nmatter.  The normalized HRA T1 intensity was used for the reference\nimage matching term $I_0$ and the temporal mean of functional\nactivation was used as a moving image matching term $I_1$.\n\n\\cref{fig::fmri} shows side by side comparison for 3D views of rs-FMRI\n({\\bf a}), T1 ({\\bf b}) and rs-FMRI mapped to T1 ({\\bf c})\nvolumes. The processing was carried through 30 energy embedded shells\nand required about 5 minutes of waiting time from the start to the\nfinish, with a subset of the final grid shown in ({\\bf d}).  The work\nis currently underway to include flexible mapping grids directly to\nour rs-FMRI mode detection approach \\citep{2016JPhA...49M5001F,fmri-efd}.\n\n\\cref{fig::fmri:modes} shows comparison of original and registered\nimages for several of the resting state functional modes obtained\nusing our entropy field decomposition (EFD) technique\n\\citep{2016JPhA...49M5001F,fmri-efd}. The default mode ({\\bf a}) and ({\\bf d}), the\nvisual lateral ({\\bf b}) and ({\\bf e}), and the visual occipital ({\\bf\n  c}) and ({\\bf f}) modes are shown for some of the subjects from\n\\cref{fig::anat}.  The symplectomorphically mapped images overlayed\nover correspondent HRA slices ({\\bf d}, {\\bf e}, and {\\bf f}) show\nvery accurate localization of functional modes in the appropriate\nregions of HRA volumes.\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig5}\n\\caption[]{3D view of low resolution (64x64x30) rs-FMRI volume ({\\bf\n    a}) vs T1 high resolution (290x262x262) anatomical volume ({\\bf\n    b}). SWD preconditioned rs-FMRI volume after registration to high\n  resolution T1 template ({\\bf c}).  The final mapping grid used 30\n  shells ({\\bf d}) and took about 5 minutes on 12 cores\n  Intel\\textregistered\\ Core\\textsuperscript{TM} i7-4930K CPU \\@\n  3.40GHz. The same color scheme is used for the displacement field\n  with blue corresponding to the anterior-posterior grid lines, green\n  to the dorsal-ventral grid lines, and red to the right-left lateral\n  grid lines.\n\\label{fig::fmri}\n}\n\\end{figure*}\n\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics[width=1.0\\fw]{Fig6}\n\\caption[]{Several randomly selected resting state modes obtained\n  using low resolution (64x64x30) rs-FMRI volume registered to T1 high\n  resolution (290x262x262) anatomical volume -- default mode ({\\bf a})\n  and ({\\bf d}), visual lateral ({\\bf b}) and ({\\bf e}), and visual\n  occipital ({\\bf c}) and ({\\bf e}) -- for some of the subjects from\n  \\cref{fig::anat}.  The upper panels ({\\bf a}, {\\bf b}, and {\\bf c})\n  show the original low resolution rs-FMRI modes. The symplectomorphic\n  maps in lower panels ({\\bf d}, {\\bf e}, and {\\bf f}) show accurate\n  localizations of functional modes in the appropriate regions of HRA\n  volumes.\n\\label{fig::fmri:modes}\n}\n\\end{figure*}\n\n\\section{Conclusions}\nIn this paper we have presented a new flexible\nmultidimensional image registration approach that is based on the\nHamiltonian formalism. The method generates a set of Hamilton's\nequations capable of producing a symplectomorphic transformation for\nmapping between Cartesian and curvilinear grids\nthat minimizes some predefined image difference metric.  The final\ndiffeomorphic mapping is constructed as a multiplicative sequence of\nsymplectomorphic transforms with gradually diminishing levels of total\nenergy, thus providing a sequence of energy embedded symplectomorphic\nshells.  For demonstration purposes, we used both a simple local\nsquared difference, as well as a more complicated non-local image\nsquared difference, as a Hamilton function.\n\nAn application of the powerful and versatile ESP approach\n\\citep{Frank:2014pre} to the phase space domain resulted in a\nnon-local form of Hamilton's equations. The non-local form represents\nan efficient and relatively straight forward way to introduce\nregularization that is capable of taking\ninto account some image specific details or even additional knowledge\nbased parameterizations.  More generally, the Hamiltonian formalism\nallows easy adaptation of custom and possibly more complex forms of\nimage difference metrics, and at the same time allows the metric in\nthe diffeomorphism space to be kept the same, which facilitates the\ncomparison and validation between different techniques and\nregularization approaches.\n\nThe resolution differences between images as well as rigid shape\nalignment is addressed using the preconditioning step based on the SWD\ntechnique \\citep{swd}. This efficient volumetric decomposition computes\na set of fast spherical harmonics and spherical Bessel transforms and\nis able to produce accurate interpolation, filtering and fitting of\nrigid shapes.\n\nOne of the major hurdles in the translation of registration\nmethods to clinical practice \\citep{pmid27427472} is the difficulty\nin validation of these methods in the face of the exceedingly\ncomplex interplay of the many pulse sequence details, the scanner\nhardware, and the anatomical variations that interact with the\nmagnetic fields. To this end, we have developed a panel\nof analytically defined deformations based on known physical\neffects present in MR acquisitions.  While these certainly do not\nconstitute an exhaustive list of possible effects, they are a fair\nrepresentation of the major distortions in the major MR image\nacquisition schemes.  This deformation panel was used to\nvalidate our method and quantitatively demonstrate its robustness to a\nwide range of different types of distortions that regularly plaque MR\nstudies.  The deformations from the panel can be applied to images\nof different modalities and acquisition conditions and potentially can\nbe appropriate for quick and robust validation in clinical settings.\nThis validation approach is somewhat similar to 2D Gaussian\ndeformations used in \\citet{pmid15896998}, but our panel includes\ndeformations that can be attributed to a variety of real physical\nprocesses present in different acquisition protocols and modalities\n(i.e.~twist, whirl, stretch, etc).\n\nOverall the symplectomorphic registration approach is both accurate\nand fast and is capable of processing of a variety of volumetric\nimages of different modalities and resolutions.  In the tests reported\nin this paper we were able to handle all three of the major neuro-MRI\nmodalities routinely used for human neuroimaging applications,\nincluding mapping between high-resolution anatomical volumes, medium\nresolution diffusion weighted volumes and high-resolution\nanatomicals, and low resolution functional MRI images and\nhigh-resolution anatomicals. The typical processing time for\nhigh quality mapping ranges from less than a minute to several minutes\non a modern multi-core CPU for a typical high resolution anatomical\nMRI volumes.  The speed, accuracy, and flexibility of this new method\nhas the potential to play an important\nrole in the quantitative assessment of neuroimaging data in a wide\nrange of both basic research and clinical applications.\n\n\\section*{Acknowledgments}\nThe authors thank Dr Alec Wong and Dr Tom Liu at the UCSD CFMRI for\nproviding the resting state data and Dr Scott Sorg at the VA San Diego\nHealth Care System for providing the diffusion weighted imaging data.\nLRF and VLG were supported by NSF grants DBI-1143389, DBI-1147260,\nEF-0850369, PHY-1201238, ACI-1440412, ACI-1550405 and NIH grant R01\nMH096100.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \\label{sec:intro}\n\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime.\nOne variant of the prime number theorem states that\n\\[\n \\frac{1}{x}\n  \\sum_{n \\le x}\n   \\frac{d_n}{\\log p_n}\n    \\sim \n     1\n      \\quad \n       (x \\to \\infty),\n\\]\nthat is, $d_n\/\\log p_n$ is approximately $1$ on average over \n$n \\le x$.\nAs to the finer questions pertaining to the distribution of \nprimes, we have little more than conjecture in the way of answers.\nHeuristics based on Cram\\'er's model%\n\\footnote{%\nFor details, we highly recommend the insightful expository article \n\\cite{SOUND} of Soundararajan. \n}\nsuggest that for any given real numbers $b > a \\ge 0$, \n\\[\n   \\frac{1}{x}\n  \\#\\big\\{ n \\le x : a < d_n\/\\log p_n \\le b \\big\\} \n     \\sim\n      \\int_a^b\n       \\e^{-t}\n        \\dd t\n         \\quad \n       (x \\to \\infty).\n\\]\nHowever, we do not even know of any specific limit point of the \nsequence $(d_n\/\\log p_n)$, except for $0$ and $\\infty$, \nthe former having been known for just a decade, thanks to the \ngroundbreaking work of Goldston--Pintz--Y{\\i}ld{\\i}r{\\i}m \n\\cite{GPY}.\n(The latter follows from a 1931 result of Westzynthius \n\\cite{WES}.)  \n\nThis limit point lacuna notwithstanding, Hildebrand and Maier \n\\cite{HM} showed in 1988 that a positive (but unspecified) \nproportion of nonnegative real numbers are limit points of \n$(d_n\/\\log p_n)$.\nMore recently, the second author, Banks and Maynard \\cite{BFM} \nhave shown that in fact at least $12.5\\%$ of nonnegative real \nnumbers are limit points of $(d_n\/\\log p_n)$.\nThe proof strategy in \\cite{BFM} incorporates an \n``Erd{\\H o}s--Rankin'' type construction for producing long gaps \nbetween consecutive primes into the celebrated Maynard--Tao sieve, \nwhich was originally developed to produce short gaps between \nprimes.\nMore recently still, Ford, Green, Konyagin, and Tao \\cite{FGKT},  \nand (independently) Maynard \\cite{MAY2}, have settled the \nnotorious ``Erd{\\H o}s--Rankin problem'' by showing that $\\infty$ \nis a limit point of $(d_n\/R(p_n))$, where%\n\\footnote{%\nWe define $\\log_2 T \\defeq \\log\\log T$,  \n$\\log_3 T \\defeq \\log\\log\\log T$ and so on. \n}\n$\n R(T)\n  \\defeq \n   \\log T \\log_2 T \\log_4 T\/(\\log_3 T)^2\n$.\n\nWe are therefore motivated to study limit points of \n$(d_n\/R(p_n))$.\nUsing basically the same strategy as in \\cite{BFM}, and the work \nof Ford, Green, Konyagin, and Tao \\cite{FGKT}, \nPintz \\cite{PIN2,PIN3} has shown that at least $25\\%$ of \nnonnegative real numbers are limit points of $(d_n\/R(p_n))$.\nIn fact, Pintz's result is that the same statement holds if the \nnormalizing function $R(T)$ is replaced by any function --- \nsubject to certain technical conditions --- that tends to infinity \nno faster than $R(T)$, for example $\\log T\\log_2 T\/(\\log_3 T)^2$.\n\nFord, Green, Konyagin, Maynard and Tao \\cite{FGKMT} have actually \nshown that, for infinitely many $n$, $d_n \\gg R(p_n)\\log_3 p_n$.\nThe purpose of this paper is to fully integrate the work of the  \nfive-author paper \\cite{FGKMT} into the study of limit points of \nnormalized prime gaps initiated in \\cite{BFM, PIN2, PIN3}.\nIn so doing, we extend the aforementioned result of Pintz in three \nways. \n\nFirst, we show that the normalizing function $R(T)$ may be \nreplaced by any ``reasonable'' function that tends to infinity \nmore slowly than $R(T)\\log_3 T$, for example\n$R_1(T) = \\log T\\log_2 T\/\\log_3 T$.\nSecond, we show that the $25\\%$ may conditionally be improved to \n$33\\frac{1}{3}\\%$ or even $50\\%$ on a certain conjecture \nconcerning the level of distribution of the primes.\nThird, we also consider ``chains'' of normalized, consecutive gaps \nbetween primes (cf.\\ Theorem \\ref{thm:chains}).\n\nPrecisely what we mean by a ``reasonable'' function is best \nexplained in context, so we defer the statement of our main result \nto \\S\\ref{sec:BFM} (cf.\\ Theorem \\ref{thm:general}).\nExamples of ``reasonable'' functions are $\\log_6 T$, \n$\\sqrt{\\log T}$, $\\log_2 T\/\\sqrt{\\log_3 T}$, $(\\log T)^{7\/9}$, \n$\\log T$, $R(T)$, $R_1(T)$ and $R_1(T)\\log_5 T$.\nAny one of these could replace $R_1(T)$ in the following special \ncase of Theorem \\ref{thm:general}, which will serve as a \nplaceholder.\n\n\\begin{theorem}\n \\label{thm:main}\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[R_1]$ denote the set of limit points \nin $[0,\\infty]$ of the sequence $(d_n\/R_1(p_n))_{p_n \\ge T_0}$, \nwhere \n\\[\n R_1(T)\n  \\defeq \n   \\log T \\log_2 T\/\\log_3 T\n\\]\nand $T_0$ is large enough so that $\\log_3 T_0 \\ge 1$.\nGiven any five nonnegative real numbers $\\alpha_1,\\ldots,\\alpha_5$ \nwith $\\alpha_1 \\le \\cdots \\le \\alpha_5$, we have \n$\n \\{\\alpha_j - \\alpha_i : 1 \\le i < j \\le 5\\}  \n  \\cap \n   \\LP[R_1]\n    \\ne \n     \\emptyset.\n$\n\\end{theorem}\n\nAs in \\cite[Corollary 1.2]{BFM}, one may deduce from Theorem 1.1 \nthat, with $\\lambda$ denoting the Lebesgue measure on $\\RR$, \n\\begin{equation}\n\\label{eq:BFM1.4}\n  \\lambda([0,X] \\cap \\LP[R_1])\n   \\ge \n    X\/(4(1 + 1\/2 + 1\/3 + 1\/4))\n     \\quad (X \\ge 0), \n\\end{equation} \nand%\n\\footnote{%\nHere, by $o(1)$ we mean a positive quantity that tends to zero as \n$X$ tends to infinity.\n} \n(with an ineffective $o(1)$),\n\\begin{equation}\n\\label{eq:BFM1.3}\n  \\lambda([0,X] \\cap \\LP[R_1])\n   \\ge \n    (1 - o(1))\n     X\/4\n      \\quad (X \\to \\infty).\n\\end{equation}\nAs we will see, assuming a certain variant of the \nElliott--Halberstam conjecture (cf.\\ Hypothesis \\ref{hyp:EH} \nbelow), one has\n$\n\\{\\alpha_2 - \\alpha_1, \\alpha_3 - \\alpha_1, \\alpha_3 - \\alpha_2\\} \n \\cap \\LP[R_1] \n  \\ne \n   \\emptyset\n$\nfor any {\\em three} nonnegative real numbers \n$\\alpha_1 \\le \\alpha_2 \\le \\alpha_3$, with corresponding \nimprovements to \\eqref{eq:BFM1.4} and \\eqref{eq:BFM1.3}\n(viz.\\ $2 = 3 - 1$ replaces $4 = 5 - 1$).\n\n\\subsection*{Acknowledgments}\n\nThe second author gratefully acknowledges the hospitality of \nBrigham Young University, where the work on this paper commenced.\n\n\n\\section{Notation and terminology}\n \\label{sec:notation}\n\nWe rely heavily on the paper \\cite{FGKMT} of Ford, Green, \nKonyagin, Maynard and Tao, and we follow their notation and \nconventions.\nWe explain these conventions here, among others, for completeness' \nsake.\n\n\\begin{enumerate}[label=---]\n \\item The set of all primes is denoted by $\\bP$; $p,q,s$ stand \n       for primes; $p_n$ denotes the $n$th smallest prime.\n \\item For $a,b \\in \\ZZ$, we define \n       $a \\pod{b} \\defeq \\{a + bc : c \\in \\ZZ\\}$.\n       %\n       Thus, $a_1 \\equiv a_2 \\pod{b}$ if and only if \n       $a_1 \\pod{b} = a_2 \\pod{b}$.         \n \\item A finite set $\\cH$ of integers is {\\em admissible} if and \n       only if $\\cH$ is not a complete set of residues modulo $p$, \n       for any prime $p$.\n \\item We say an integer is {\\em $x$-smooth} ($x \\in \\RR$) if and \n       only if its prime divisors are all less than or equal to \n       $x$.    \n \\item For $n \\in \\ZZ$ and $\\cH \\subseteq \\ZZ$, we define   \n       $n + \\cH \\defeq \\{n + q : q \\in \\cH\\}$.       \n \\item For statements $S$, $\\ind{S} \\defeq 1$ if $S$ is true and \n       $\\ind{S} \\defeq 0$ if $S$ is false.\n \\item The cardinality of a set $\\cS$ is denoted by $\\#\\cS$ or \n       $\\#(\\cS)$.\n       %\n       The indicator function for $\\cS \\subseteq \\cT$ (with $\\cT$ \n       clear in context) is denoted $\\ind{\\cS}$.\n       %\n       That is, for $t \\in \\cT$, \n       $\\ind{\\cS}(t) \\defeq \\ind{t \\in \\cS}$. \n \\item We write $\\PP$ for probability and $\\EE$ for expectation.\n \\item Boldface symbols such as $\\bX$ or $\\ba$ denote random \n       variables, while non-boldface symbols such as $X$ or $a$ \n       denote their deterministic counterparts.\n       %\n       Vector-valued random variables are indicated in arrowed \n       boldface, for instance \n       $\\vec{\\ba} = (\\vec{\\ba}_s)_{s \\in \\cS}$ denotes a random \n       tuple of random variables indexed by the set $\\cS$.\n \\item If $\\bX$ takes at most countably many values, we define the \n       {\\em essential range} of $\\bX$ to be the set of all $X$ \n       such that $\\PP(\\bX = X) \\ne 0$.\n \\item If $E$ is an event of nonzero probability, \n       \\[\n        \\PP(F \\mid E)\n         \\defeq \n          \\frac{\\PP(F \\land E)}{\\PP(E)}\n       \\]\n       for any event $F$, and \n       \\[\n        \\EE(\\bX \\mid E)\n         \\defeq \n          \\frac{\\EE(\\bX \\ind{E})}{\\PP(E)}\n       \\]\n       for any absolutely integrable real-valued random variable \n       $\\bX$.\n       %\n       If $\\bY$ is another random variable taking at most \n       countably many values, we define the conditional \n       probability $\\PP(F \\mid \\bY)$ to be the random variable \n       that equals $\\PP(F \\mid \\bY = Y)$ on the event $\\bY = Y$ \n       for each $Y$ in the essential range of $\\bY$, and similarly \n       define the conditional expectation $\\EE(\\bX \\mid \\bY)$ to \n       be the random variable that equals $\\EE(\\bX \\mid \\bY = Y)$ \n       on the event $\\bY = Y$.\n \\item Throughout, $x$ denotes a parameter to be thought of as \n       tending to infinity.\n       %\n \\item Thus, $o(1)$ signifies a quantity that tends to zero as \n       $x \\to \\infty$ and $X \\sim Y$ denotes that \n       $X = (1 + o(1))Y$.\n \\item Expressions of the form $X = O(Y)$, $X \\ll Y$ and $Y \\gg X$ \n       all denote that $|X| \\le c|Y|$ throughout the domain of \n       $X$, for some constant $c > 0$.\n \\item The constant $c$ is to be taken as independent of any \n       parameter unless indicated otherwise, as in \n       $X \\ll_{\\delta,A} Y$ for instance, in which $c$ depends on \n       $\\delta$ and $A$.\n \\item We write $X = O_{\\le}(Y)$ to denote that one can take \n       $c = 1$.\n \\item We write $X \\asymp Y$ to denote that $X \\ll Y \\ll X$.\n\\end{enumerate}\n\n\n\\section{Proof strategy}\n \\label{sec:outline}\n \nLet $x$ be a large number and set \n$\n y \\defeq cx\\log x\\log_2 x\/\\log_3 x\n$, \nwhere $c > 0$ is a certain small constant.\nFord, Green, Konyagin, Maynard and Tao \\cite{FGKMT} show that if \n$C$ is large enough, then there exists a vector \n$(c_p \\pod{p})_{p \\le Cx}$ of residue classes for which \n\\[\n \\big((x,y] \\cap \\ZZ \\big) \n  \\setminus \\, {\\textstyle \\bigcup_{p \\le Cx} c_p \\pod{p}}\n =\n   \\emptyset.\n\\]\nThus, if $b \\pod{W}$ is the residue class modulo \n$W \\defeq \\prod_{p \\le Cx} p = \\e^{(1 + o(1))Cx}$ for which \n$b \\equiv -c_p \\pod{p}$ for each $p \\le Cx$, then for \n$n \\equiv b \\pod{W}$ with $n + x > Cx$, \n\\[\n \\bP \\cap (n + x,n + y] = \\emptyset.\n\\]\n\nWe generalize this slightly by proving that if $\\cH$ is any set of \n$K$ primes in $(x,y]$ with $K \\le \\log x$ (say), the residue \nclasses may be chosen so that \n\\[\n \\big((x,y] \\cap \\ZZ \\big) \n  \\setminus \\, {\\textstyle \\bigcup_{p \\le Cx} c_p \\pod{p}}\n  =\n   \\cH\n\\]\nand hence \n\\[\n \\bP \\cap (n + x,n + y] = \\bP \\cap n + \\cH.\n\\]\nNote that $\\cH$, being a set of $K \\le \\log x$ primes greater than \n$p_K = O(K\\log K)$, is admissible.  \n\nNow let $M \\ge 2$ be an integer with $M \\mid K$ and \nlet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be a partition of $\\cH$ \ninto $M$ subsets of equal size.\nAs was shown in \\cite{BFM}, with $M = 9$, a smaller choice of $y$ \nand a minor technical condition on $\\cH$, the Maynard--Tao sieve \nmethod establishes that for large $N$, there exists \n$n \\in (N,2N] \\cap b \\pod{W}$ and a pair $i < j$ for which \n\\[\n \\#(\\bP \\cap n + \\cH_{i}), \\#(\\bP \\cap n + \\cH_{j}) \\ge 1,\n\\] \nprovided $K$ is sufficiently large and $W \\le N^{\\eta}$ for some \nsmall $\\eta$.\n\nChoosing $j - i$ to be minimal, we obtain a pair of consecutive \nprimes in $n + \\cH$.\nWe may carefully choose our primes in $\\cH$ so that the spacings \nbetween them grow faster than $x\/\\log x$ but slower than $y$.\n\nActually, for reasons related to level of distribution and \n``Siegel'' zeros, we require that $W$ not be a multiple of a \ncertain putative ``exceptional'' modulus less than\n$N^{O(\\eta)}$.\nThe largest prime divisor $p'$ of this exceptional modulus, if it \nexists, satisfies $p' \\gg \\log_2 N^{\\eta} \\gg \\log x$.\nFor this reason, we introduce a set $\\cZ$ of ``unusable'' primes, \nwhich has the properties of $\\{p'\\}$.\nTheir effect is negligible.\n\nPintz \\cite{PIN3} has very recently given an elegant \nsimplification of part of this argument, which we take advantage \nof in this paper, and which shows that one can take $M = 5$.\n\n\n\\section{A modification of Ford--Green--Konyagin--Maynard--Tao}\n \\label{sec:FGKMT}\n \n\\subsection{Main results}\n \\label{subsec:fgkmtmain}\n\nGiven a large number $x$ we define  \n\\begin{equation}\n \\label{eq:fgkmt3.1}\n  y \\defeq cx\\frac{\\log x \\log_3 x}{\\log_2 x},\n\\end{equation}\nwhere $c$ is a certain (small) fixed positive constant, and \n\\begin{equation}\n \\label{eq:fgkmt3.2}\n  z \\defeq x^{\\log_3 x\/(4\\log_2 x)}.\n\\end{equation}\nWe then define\n\\begin{align}\n \\cS & \\defeq \\{\\text{$s$ prime} : (\\log x)^{20} < s \\le z\\}, \\label{eq:fgkmt3.3} \\\\\n \\cP & \\defeq \\{\\text{$p$ prime} : x\/2 < p \\le x\\},           \\label{eq:fgkmt3.4} \\\\\n \\cQ & \\defeq \\{\\text{$q$ prime} : x < q \\le y\\}.             \\label{eq:fgkmt3.5} \n\\end{align}\nFor vectors of residue classes \n$\\vec{a} \\defeq (a_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{b} \\defeq (b_p \\pod{p})_{p \\, \\in \\, \\cP}$, \nwe define sifted sets\n\\[\n  S(\\vec{a})\n  \\defeq \n   \\ZZ \n    \\, \\setminus \\, \n     \\textstyle{\\bigcup_{s \\, \\in \\, \\cS}} \\,\n      a_s \\pod{s}\n\\quad \n \\text{and}\n  \\quad \n  S(\\vec{b})\n  \\defeq \n   \\ZZ \n    \\, \\setminus \\, \n     \\textstyle{\\bigcup_{p \\, \\in \\, \\cP}} \\,\n      b_p \\pod{p}.\n\\]      \nWe note that in view of the prime number theorem (with suitably \nstrong error term) and \\eqref{eq:fgkmt3.1},   \n\\begin{equation}\n \\label{eq:Qsize}\n \\#\\cQ \n  = \n   \\frac{y}{\\log x}\n    \\bigg(\n     1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\n    \\bigg).\n\\end{equation}\nFinally, let $K$ be any natural number satisfying \n\\begin{equation}\n \\label{eq:Kbnd}\n  K \\le \\log x.\n\\end{equation}\nBy \\eqref{eq:Qsize}, we may suppose $x$ is large enough so that \n$\\cQ$ contains at least $K$ primes.\nSince $p_K \\ll K\\log K$, we may also suppose that $p_K \\le x$.\nWe fix any \n\\begin{equation}\n \\label{eq:cHdef}\n \\cH \n  \\defeq \\{q_1,\\ldots,q_K\\}\n   \\subseteq \\cQ \n  \\quad \n   \\text{with}\n    \\quad \n   \\#\\cH = K.\n\\end{equation}\nNote that $\\cH$, being a set of $K$ primes larger than $p_K$, is \nan admissible set.\n\n\\begin{theorem}[Sieving for primes]\n \\label{thm:fgkmt2}\nFor all sufficiently large $x$, there exist vectors of residue \nclasses \n$\\vec{a} = (a_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{b} = (b_p \\pod{p})_{p \\, \\in \\, \\cP}$ \nsuch that $\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$ and \n\\begin{equation}\n \\label{eq:fgkmt3.6}\n  \\#(\\cQ \\cap S(\\vec{a}) \\cap S(\\vec{b}))\n   \\ll\n    \\frac{x}{\\log x}, \n\\end{equation}\nwhere the implied constant is absolute.\n\\end{theorem}\n\nThe only difference between Theorem \\ref{thm:fgkmt2} and \n\\cite[Theorem 2]{FGKMT} is our additional requirement that \n$\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$.\nUnsurprisingly, the proof of Theorem \\ref{thm:fgkmt2} follows that \nof \\cite[Theorem 2]{FGKMT} very closely, even verbatim in many \nparts.\nNevertheless, the details must be checked, and by including them \nhere we are also able to point out the minor differences between \nthe two proofs.\n\nWe now introduce a set $\\cZ$ of ``unusable'' primes with the \nproperty that for any $p' \\in \\cZ$,\n\\begin{equation}\n \\label{eq:Zsparse}\n  \\sums[p \\ge p'][p \\in \\cZ]\n   \\frac{1}{p}\n    \\ll\n     \\frac{1}{p'}\n      \\ll\n       \\frac{1}{\\log x}.\n\\end{equation}\n\n\\begin{corollary}\n \\label{cor:thm2}\nLet $C$ be a sufficiently large but fixed positive constant.\nFor all sufficiently large $x$, there exists a vector of residue \nclasses \n$(c_p \\pod{p})_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}$ such that \n$\n \\cH\n  =\n   \\big(\\ZZ \\cap (x,y]\\big)\n    \\setminus \\, \n     {\\textstyle \\bigcup_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}} \n       \\, c_p \\pod{p}.\n$ \n\\end{corollary}\n\nWe deduce Corollary \\ref{cor:thm2} from Theorem \\ref{thm:fgkmt2} \nwith the aid of Lemma 5.1 of \\cite{BFM}, which is as follows.\n\n\\begin{lemma}\n \\label{lem:BFM5.1}\nLet $\\sH,\\sT$ be sets of integers, $\\sP$ a set of \nprimes, such that for some $x \\ge 2$, \n$\\sH \\subseteq \\sT \\subseteq [0,x^2]$ and \n$\n \\#\\{p \\in \\sP : p > x\\} \n  > \n   \\#\\sH + \\#\\sT\n$.\nIf $\\sH$ is admissible then there exists a vector of residue \nclasses $(\\gamma_p \\pod{p})_{p \\, \\in \\, \\sP}$ such that \n$\n \\sH\n  = \n   \\sT \n    \\, \\setminus \\,\n     {\\textstyle \\bigcup_{p \\, \\in \\, \\sP}}\\, \\gamma_p \\pod{p}.\n$\n\\end{lemma}\n\n\\begin{proof}[Deduction of Corollary \\ref{cor:thm2}]\nWe choose $x$, $\\vec{a}$ and $\\vec{b}$ so that the conclusions \nof Theorem \\ref{thm:fgkmt2} hold, and work with the enlarged \nsifted sets\n\\[\n  S_{\\cZ}(\\vec{a})\n  \\defeq \n   \\ZZ \n    \\, \\setminus \\, \n     \\textstyle{\\bigcup_{s \\, \\in \\, \\cS \\, \\setminus \\, \\cZ}} \\,\n      a_s \\pod{s}\n\\quad \n \\text{and}\n  \\quad \n  S_{\\cZ}(\\vec{b})\n  \\defeq \n   \\ZZ \n    \\, \\setminus \\, \n     \\textstyle{\\bigcup_{p \\, \\in \\, \\cP \\, \\setminus \\, \\cZ}} \\,\n      b_p \\pod{p}.\n\\]   \nNote that if $n \\in S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$, then \neither $n \\in S(\\vec{a}) \\cap S(\\vec{b})$, \n$n \\equiv a_s \\pod{s}$ for some $s \\in \\cS \\cap \\cZ$ \nor \n$n \\equiv b_p \\pod{p}$ for some $p \\in \\cP \\cap \\cZ$.\nNow,  \n\\[\n \\sum_{s \\, \\in \\, \\cS \\cap \\cZ}\n  \\sums[n \\le y][n \\equiv a_{s} \\pod{s}]\n   1\n \\, +\n    \\sum_{p \\, \\in \\, \\cP \\cap \\cZ}\n     \\sums[n \\le y][n \\equiv b_{p} \\pod{p}]\n      1\n    \\le \n     y\n      \\sums[p \\, \\in \\, \\cZ][p > (\\log x)^{20}]\n       \\frac{1}{p'}\n        \\ll\n          \\frac{y}{(\\log x)^{20}}\n\\]\nby \\eqref{eq:fgkmt3.3}, \\eqref{eq:fgkmt3.4} and \n\\eqref{eq:Zsparse}.\nThus, the elements of \n$\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$\nthat are not in \n$\\cQ \\cap S(\\vec{a}) \\cap S(\\vec{b})$ \nnumber at most $y\/(\\log x)^{20} \\ll x\/(\\log x)^{19}$ \nby \\eqref{eq:fgkmt3.1}.\nWe conclude from \\eqref{eq:fgkmt3.6} that \n\\begin{equation}\n \\label{eq:QZbnd}\n  \\#(\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b}))\n   \\ll\n    \\frac{x}{\\log x}.\n\\end{equation}\n\nLet \n\\[\n \\cL \n  \\defeq \n   \\{\n     \\text{$\\ell$ prime} : \n      \\ell \\in [2,(\\log x)^{20}] \\cup (z,x\/2] \n   \\}\n\\]\nso that $\\cS \\cup \\cL \\cup \\cP$ is a partition of the primes less \nthan or equal to $x$.\nWe define a vector of residue classes \n$(c_p \\pod{p})_{p \\le x, \\, p \\not\\in \\cZ}$ \nby setting \n\\begin{align*}\n c_p  \n  \\defeq \n   \\begin{cases}\n    a_p  & p \\in \\cS \\, \\setminus \\, \\cZ \\\\\n    b_p  & p \\in \\cP \\, \\setminus \\, \\cZ \\\\\n    0    & p \\in \\cL \\, \\setminus \\, \\cZ. \n   \\end{cases}\n\\end{align*}\nRecalling that Theorem \\ref{thm:fgkmt2} gives \n$\\cH \\subseteq S(\\vec{a}) \\cap S(\\vec{b})$, and noting that $\\cH$ \nconsists of primes larger than $x$ by definition, we see that\n\\[\n \\cH\n  \\subseteq\n  \\cT\n  \\defeq \n   \\big(\\ZZ \\cap (x,y]\\big)\n    \\setminus \\, \n     {\\textstyle \\bigcup_{p \\le x, \\, p \\not\\in \\cZ}} \\,\n      c_p \\pod{p}.\n\\]\n\nNow, if $n \\in \\cT$ then $x < n \\le y$ and either \n\\begin{enumerate}[label=(\\arabic*)]\n \\item $n$ is divisible by a prime $p' > x\/2$, \n \\item $n$ is divisible by a prime $p' \\in (z,x\/2]$, or\n \\item $n$ is $z$-smooth.\n\\end{enumerate}\n\nIn case (1), $n = mp'$ for some \n$m \\le y\/p' < 2y\/x = o(\\log x)$ by \\eqref{eq:fgkmt3.1} and so, by \n\\eqref{eq:Zsparse}, $m$ is not divisible by any prime in $\\cZ$ \n(provided $x$ is sufficiently large, as we assume).\nNor is $m$ divisible by any other prime $p < 2y\/x$, since \nsuch primes are in $\\cL \\, \\setminus \\, \\cZ$ and $c_{p} = 0$ for \nsuch primes.\nHence $m = 1$, and $n$ must belong to \n$\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b})$.\nThus,\n\\begin{equation}\n \\label{eq:(1)bnd}\n  \\#\\{n \\in \\cT : \\text{(1) holds} \\} \n   \\le \n    \\#(\\cQ \\cap S_{\\cZ}(\\vec{a}) \\cap S_{\\cZ}(\\vec{b}))\n     \\ll\n      \\frac{x}{\\log x}\n\\end{equation}\nby \\eqref{eq:QZbnd}.\nIn case (2), the prime $p'$ must belong to $\\cZ$, for otherwise \n$c_{p'} = 0$. \nThus, \n\\begin{equation}\n \\label{eq:(2)bnd}\n  \\#\\{n \\in \\cT : \\text{(2) holds} \\}\n \\ll\n y\n  \\sums[p > z, \\, p \\in \\cZ] \n   \\frac{1}{p}\n    \\ll\n     \\frac{y}{z}\n    = \n     o\\Big(\\frac{x}{\\log x}\\Big)\n\\end{equation}\nby \\eqref{eq:fgkmt3.2} and \\eqref{eq:fgkmt3.1}.\nAs shown in \\cite[Theorem 2 et seq.]{FGKMT}, smooth number \nestimates give \n\\begin{equation} \n \\label{eq:(3)bnd}\n \\#\\{n \\in \\cT : \\text{(3) holds} \\}\n = \n  o\\Big(\\frac{x}{\\log x}\\Big).\n\\end{equation}\n\nCombining \\eqref{eq:(1)bnd}, \\eqref{eq:(2)bnd} and \n\\eqref{eq:(3)bnd}, we obtain $K + \\#\\cT \\ll x\/\\log x$ in view of \n\\eqref{eq:Kbnd}.\nWe may therefore choose our constant $C$ to be large enough so \nthat\n\\[\n \\#\\{\\text{$p$ prime}: p \\in (x,Cx], \\, p \\not\\in \\cZ \\} \n  >\n   K + \\#\\cT.\n\\] \n(The number of primes in $\\cZ$ that belong to $(x,Cx]$ is \nnegligible, for \n\\[\n  \\#\\{p \\in \\cZ : p \\le Cx\\} \\ll \\log x,\n\\]\nas can be seen from \\eqref{eq:Zsparse} [write $1 = p\/p$ and sum \ndyadically].)\nAs $\\cH$ is an admissible subset of $\\cT \\subseteq (x,y]$, we \nmust conclude, in view of Lemma \\ref{lem:BFM5.1}, that for \nsufficiently large $x$ there exist residue classes $c_p \\pod{p}$ \nfor $p \\in (x,Cx]$, $p \\not\\in \\cZ$, such that \n\\[\n \\cH\n  =\n   \\cT\n    \\, \\setminus \\,\n    {\\textstyle \\bigcup_{p \\, \\in \\, (x,Cx], \\, p \\, \\not\\in \\, \\cZ}} \n      \\, c_p \\pod{p}\n  =\n   \\big(\\ZZ \\cap (x,y]\\big)\n    \\setminus \\, \n     {\\textstyle \\bigcup_{p \\le Cx, \\, p \\, \\not\\in \\, \\cZ}} \n       \\, c_p \\pod{p}.\n\\] \n\\end{proof}\n\nIn order to prove Theorem \\ref{thm:fgkmt2} we must first establish \nthe following result, which is analogous to \n\\cite[Theorem 4]{FGKMT}. \n\n\\begin{theorem}[Random construction]\n \\label{thm:fgkmt4}\nLet $x$ be sufficiently large.\nThere exists a positive number $C$ with \n\\begin{equation}\n \\label{eq:fgkmt4.29}\n  C \\asymp \\frac{1}{c},\n\\end{equation}\nthe implied constants being independent of $c$, a set of \npositive integers $\\{h_1,\\ldots,h_r\\}$ with $r \\le \\sqrt{\\log x}$, \nand random vectors  \n$\\vec{\\ba} = (\\ba_s \\pod{s})_{s \\, \\in \\, \\cS}$ \nand \n$\\vec{\\bn} = (\\bn_p)_{p \\, \\in \\, \\cP}$\nof residue classes $\\ba_s \\pod{s}$ and integers $\\bn_p$ \nrespectively, satisfying the following.\n\\begin{enumerate}\n \\item For every $\\vec{a} = (a_s \\pod{s})_{s \\in \\cS}$ in the \n       essential range of $\\vec{\\ba}$, we have \n       \\[\n        \\cH \\cap a_s \\pod{s} = \\emptyset \\quad  (s \\in \\cS).\n       \\]\n \\item For every $\\vec{n} = (n_p \\pod{p})_{p \\in \\cP}$ in the \n       essential range of $\\vec{\\bn}$, we have \n       \\[\n        \\cH \\cap n_p \\pod{p} = \\emptyset \\quad (p \\in \\cP).\n       \\]\n \\item For every $\\vec{a}$ in the essential range of $\\vec{\\ba}$, \n       we have\n       \\[\n        \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n         \\le \n          x^{-3\/5}\n           \\quad \n            (p \\in \\cP),\n       \\]\n       where \n       $\n        \\be_p(\\vec{a}) \\defeq \\{\\bn_p + h_ip : i \\le r\\}\n         \\cap \n          \\cQ \n           \\cap \n            S(\\vec{a})\n       $.\n  \\item With probability $1 - o(1)$, we have  \n        \\begin{equation}\n         \\label{eq:fgkmt4.30}\n          \\#(\\cQ \\cap S(\\vec{\\ba}))\n           \\sim \n            80cx\\frac{\\log_2 x}{\\log x}.\n        \\end{equation}\n  \\item Call an element $\\vec{a}$ in the essential range of \n        $\\vec{\\ba}$ ``good'' if, for all but at most \n        $\\frac{x}{\\log x \\log_2 x}$ elements \n        $q \\in \\cQ \\cap S(\\vec{a})$, one has \n        \\begin{equation}\n         \\label{eq:fgkmt4.31}\n          \\sum_{p \\in \\cP}\n           \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n            =\n             C + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^{2}}\\bigg).\n        \\end{equation}\n        %\n        Then $\\vec{\\ba}$ is good with probability $1 - o(1)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\subsection{Proof of Theorem \\ref{thm:fgkmt4}}\n \\label{subsec:thm4pf}\n\nLet $x,c,y,z,\\cS,\\cP,\\cQ,K$ and $\\cH$ be as in \nTheorem \\ref{thm:fgkmt4}.\nWe set\n\\begin{equation}\n \\label{eq:fgkmt6.8}\n  r \\defeq \\lfloor (\\log x)^{1\/5} \\rfloor\n\\end{equation}\nand let $\\{h_1,\\ldots,h_r\\}$ be the admissible set with \n$h_i \\defeq (2i-1)^2$ for $i \\le r$.\nOur first lemma is a special case of \\cite[Theorem 5]{FGKMT}.\n\\begin{lemma}[Existence of a good sieve weight]\n \\label{thm:fgkmt5}\nThere exist positive quantities\n\\begin{equation}\n \\label{eq:fgkmt6.2}\n  \\tau \\ge x^{o(1)} \n   \\quad \n    \\text{and}\n     \\quad \n    u \\asymp \\log_2 x,\n\\end{equation}\nand a function $w : \\cP \\times \\ZZ \\to \\RR^+$ supported on \n$\\cP \\times [-y,y]$, satisfying the following.\n\\begin{enumerate} \n \\item Uniformly for $p \\in \\cP$,  \n\\begin{equation}\n \\label{eq:fgkmt6.4}\n  \\sum_{n \\in \\ZZ} w(p,n)\n   =\n    \\bigg(\n       1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n    \\bigg)\n     \\tau\n      \\frac{y}{(\\log x)^r}.\n\\end{equation}\n \\item Uniformly for $q \\in \\cQ$ and $i \\le r$,   \n\\begin{equation}\n \\label{eq:fgkmt6.5}\n  \\sum_{p \\in \\cP}\n   w(p,q - h_ip)\n    =\n    \\bigg(\n      1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n    \\bigg)\n     \\tau\n      \\frac{u}{r} \n       \\frac{x}{2(\\log x)^r}.\n\\end{equation}  \n \\item Uniformly for $(p,n) \\in \\cP\\times \\ZZ$,   \n\\begin{equation}\n \\label{eq:fgkmt6.7}\n  w(p,n) = O\\big(x^{1\/3 + o(1)}\\big).\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\nWe choose $\\tau$, $u$ and $w : \\cP \\times \\ZZ \\to \\RR^+$ \naccording to Lemma \\ref{thm:fgkmt5}, and define \n$w_{\\cH} : \\cP \\times \\ZZ \\to \\RR^+$ by setting \n\\[\n w_{\\cH}(p,n) \n  \\defeq \n   \\begin{cases}\n    w(p,n) & \\text{if $\\cH \\cap n \\pod{p} = \\emptyset$} \\\\\n    0      & \\text{otherwise.}  \n   \\end{cases}\n\\]\n\n\\begin{lemma}\n \\label{lem:rcb2}\nStatements \\textup{(}i\\textup{)}, \\textup{(}ii\\textup{)} and \n\\textup{(}iii\\textup{)} of Lemma \\ref{thm:fgkmt5} all hold with \n$w_{\\cH}$ in place of $w$, provided the hypothesis $q \\in \\cQ$ in \n\\textup{(}ii\\textup{)} is replaced by the hypothesis \n$q \\in \\cQ\\setminus \\cH$.\n\\end{lemma}\n\n\\begin{proof}\nWe only need to consider (i) and (ii).\nFor every $p \\in \\cP$ we have \n\\begin{align*}\n 0 \\le \\sum_{n \\in \\ZZ} (w(p,n) - w_{\\cH}(p,n))\n    \\le \\sum_{j=1}^K \\sums[|n| \\le y][n \\equiv q_j \\pod{p}] w(p,n)\n     \\ll x^{1\/3 + o(1)}y\/p\n      \\ll x^{-2\/3 + o(1)}y,\n\\end{align*}\nwhich gives the analog of \\eqref{eq:fgkmt6.4} for $w_{\\cH}$ in \nview of \\eqref{eq:fgkmt6.2}.\nFor every $q \\in \\cQ\\setminus \\cH$ and $i \\le r$, we see \nsimilarly that \n\\begin{align*}\n 0 \n  & \\le \n     \\sum_{p \\in \\cP} (w(p,q - h_ip) - w_{\\cH}(p,q - h_ip)) \n  \\\\\n  & \\hspace{45pt}\n    \\le\n     x^{1\/3 + o(1)}\n      \\sum_{j=1}^K\n       \\sum_{q - h_ip \\equiv q_j \\pod{p}} 1 \n    \\le \n     x^{1\/3 + o(1)}\n      \\sum_{j=1}^K\n       \\sum_{p \\mid q - q_j} 1 \n    \\ll\n     x^{1\/3 + o(1)},\n\\end{align*}\nwhich gives the analog of \\eqref{eq:fgkmt6.5}.\n\\end{proof}\n\nFor each $p \\in \\cP$, let $\\tilde{\\bn}_p$ denote the random \ninteger with probability density \n\\[\n  \\PP(\\tilde{\\bn}_p = n) \n   \\defeq \n    \\frac{w_{\\cH}(p,n)}{\\sum_{n' \\in \\ZZ} w_{\\cH}(p,n')}\n\\]\nfor all $n \\in \\ZZ$.\nUsing Lemma \\ref{lem:rcb2}, we verify that  \n\\begin{equation}\n \\label{eq:fgkmt6.9}\n  \\sum_{p \\in \\cP}\n   \\PP(q = \\tilde{\\bn}_p + h_ip)\n   =\n    \\ind{\\cQ\\setminus\\cH}(q)\n    \\bigg(\n     1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n    \\bigg)\n      \\frac{u}{r}\n       \\frac{x}{2y}\n  \\quad \n   (q \\in \\cQ, i \\le r),\n\\end{equation}\n\\begin{equation}\n \\label{eq:fgkmt6.10}\n  \\PP(\\tilde{\\bn}_p = n) \\ll x^{-2\/3 + o(1)}\n   \\quad (p \\in \\cP, n \\in \\ZZ)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:probnph0}\n  \\PP(\\cH \\cap \\tilde{\\bn}_p \\pod{p} \\ne \\emptyset) = 0\n   \\quad (p \\in \\cP).\n\\end{equation}\nBy \\eqref{eq:fgkmt6.10}, the analog of \\eqref{eq:fgkmt6.9} holds \nwith a single prime deleted from $\\cP$.\n\nWe choose the random vector \n$\\vec{\\ba} \\defeq (\\ba_s \\pod{s})_{s \\in \\cS}$ by selecting each \n$\\ba_s \\pod{s}$ uniformly at random from \n\\begin{equation}\n \\label{eq:defOmega}\n  \\Omega_{\\cH}(s)\n   \\defeq \n    (\\ZZ\/s\\ZZ)\\setminus\\{q \\pod{s} : q \\in \\cH\\},\n\\end{equation}\nindependently in $s$ and independently of the $\\tilde{\\bn}_p$.\nNote, then, that for any random vector $\\vec{\\ba}$, \n$\\cH \\subseteq S(\\vec{\\ba})$.\n\nThe sifted set $S(\\vec{\\ba})$ is a random periodic subset of $\\ZZ$ \nwith density \n\\[\n \\sigma_{\\cH}\n  \\defeq \n   \\prod_{s \\in \\cS}\n    \\bigg(\n     1 - \\frac{1}{\\#\\Omega_{\\cH}(s)}\n    \\bigg).\n\\]\nLet us compare $\\sigma_{\\cH}$ with the quantity \n\\[\n \\sigma \n  \\defeq \n   \\prod_{s \\in \\cS}\n    \\bigg(\n     1 - \\frac{1}{s}\n    \\bigg)\n\\]\ndefined in \\cite{FGKMT}. \nAs $(\\log x)^{20} - \\log x \\le s - K \\le \\#\\Omega_{\\cH}(s) \\le s$ \n(cf.\\ \\eqref{eq:fgkmt3.3} and \\eqref{eq:Kbnd}), one may verify, in \na straightforward manner, that \n\\begin{equation}\n \\label{eq:sigmahatsigma}\n  \\sigma_{\\cH} \n   = \n    \\sigma\\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{19}}\\bigg)\\bigg). \n\\end{equation}\nConsequently, in the estimates that follow, $\\sigma_{\\cH}$ and \n$\\sigma$ are interchangeable. \n\nAs noted in \\cite{FGKMT}, by the prime number theorem (with \nsuitably strong error term), \\eqref{eq:fgkmt3.2} and \n\\eqref{eq:fgkmt3.3}, \n\\[\n \\sigma \n  = \n   \\bigg(\n    1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n   \\bigg)\n    \\frac{80\\log_2 x}{\\log x \\log_3 x\/\\log_2 x},\n\\]\nso by \\eqref{eq:fgkmt3.1} we have \n\\begin{equation}\n \\label{eq:fgkmt6.11}\n  \\sigma y\n   =\n   \\bigg(\n    1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\n   \\bigg)    \n    80c x\\log_2 x.\n\\end{equation}\nAlso, by \\eqref{eq:fgkmt6.8} we have \n\\begin{equation}\n \\label{eq:fgkmt6.12}\n  \\sigma^r = x^{o(1)}.\n\\end{equation}\n\nLet\n\\begin{equation}\n \\label{eq:fgkmt6.14}\n X_p(\\vec{a})\n  \\defeq \n   \\PP(\\tilde{\\bn}_p + h_ip \\in S(\\vec{a}) \\, \\, \\hbox{for all} \\, \\, i \\le r)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:rcb11a}\n  \\cP(\\vec{a})\n     \\defeq \n      \\bigg\\{\n       p \\in \\cP : \n        X_p(\\vec{a})\n   =\n    \\bigg(\n     1 + O_{\\le}\\bigg(\\frac{1}{(\\log x)^{6}}\\bigg)\n    \\bigg)\n     \\sigma^r\n      \\bigg\\}.\n\\end{equation}\nIt will transpire that with probability $1 - o(1)$, most primes in \n$\\cP$ lie in $\\cP(\\vec{\\ba})$.\n\nWe now define $\\bn_p$ in a slightly complicated way.\nLet \n\\[\n Z_p(\\vec{a};n)\n  \\defeq \n   \\ind{\n    (n + h_j p \\, \\in \\, S(\\vec{a}) \\, \\, \\forall j \\le r)\n       }\n        \\PP(\\tilde{\\bn}_p = n).\n\\]\nSuppose we are in the event that $\\vec{\\ba} = \\vec{a}$.\nIf $p \\in \\cP \\, \\setminus \\, \\cP(\\vec{a})$, we set \n$\\bn_p = 0$.\nOtherwise, let $\\bn_p$ be the random integer with \n\\begin{equation}\n \\label{eq:rcb11}\n  \\PP(\\bn_p = n \\mid \\vec{\\ba} = \\vec{a})\n   =\n    \\frac{Z_p(\\vec{a};n)}{X_p(\\vec{a})},\n\\end{equation}\nwith the $\\bn_p$ jointly conditionally independent on the \nevent $\\vec{\\ba} = \\vec{a}$. \n(We easily verify that \n$\n \\sum_{n \\in \\ZZ} Z_p(\\vec{a};n) = X_p(\\vec{a})\n$,\nso that \\eqref{eq:rcb11} makes sense.)\n\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \n \\textup{(}i\\textup{)} -- \\textup{(}iii\\textup{)}]\nLet $p \\in \\cP$.\nWe claim that \n\\[\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset) = 0.\n\\]\nTo prove the claim it suffices to show \n$\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset \\mid \\vec{\\ba} = \\vec{a}) \n  = 0\n$\nfor every $\\vec{a}$.\nThis is easily checked if $p \\not\\in \\cP(\\vec{a})$, since \n$\\cH \\cap 0 \\pod{p} \\subseteq \\cQ \\cap 0 \\pod{p} = \\emptyset$; \notherwise \n\\[\n \\PP(\\cH \\cap \\bn_p \\pod{p} \\ne \\emptyset \\mid \\vec{\\ba} = \\vec{a})\n  \\le \n   \\frac{\\PP(\\cH \\cap \\tilde{\\bn}_p \\pod{p} \\ne \\emptyset)}\n        {X_p(\\vec{a})}\n    =\n     0.\n\\]\n\nWe see that Theorem \\ref{thm:fgkmt4} (i) and (ii) hold and we can \nnow prove Theorem \\ref{thm:fgkmt4} (iii).\nGiven $\\vec{a}$ in the essential range of $\\vec{\\ba}$, and \n$q \\in \\cQ \\cap S(\\vec{a})$, we have \n\\[\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n  \\le \n   \\PP( \\bn_p + h_ip = q  \\, \\, \\hbox{for some} \\, \\, i \\le r\\mid \\vec{\\ba} = \\vec{a}).\n\\]\nThe right-hand side is $0$ if $p \\not\\in \\cP(\\vec{a})$.\nOtherwise,\n\\begin{align*}\n \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a})\n  & \\le \n   r \\max_{n \\in \\ZZ} \\PP(\\bn_p = n \\mid \\vec{\\ba} = \\vec{a})  \n    \\ll\n     r\\sigma^{-r} \\max_{n \\in \\ZZ} \\PP(\\tilde{\\bn}_p = n) \\\\\n   &   \\ll\n       x^{-2\/3 + o(1)}.\n\\end{align*}\n(Here, we have used \\eqref{eq:fgkmt6.8}, \\eqref{eq:fgkmt6.10} and \n\\eqref{eq:fgkmt6.12}.)\n\\end{proof}\n\nThe following lemma is analogous to \\cite[Lemma 6.1]{FGKMT}.\n\n\\begin{lemma}\n \\label{lem:rcb4}\nLet $n_1,\\ldots,n_t$ be distinct integers of magnitude $x^{O(1)}$, \n$t \\le \\log x$, such that \n$\\cH \\cap \\{n_1,\\ldots,n_t\\} = \\emptyset$.\nThen for all sufficiently large $x$, \n\\[\n \\PP(n_1,\\ldots,n_t \\in S(\\vec{\\ba}))\n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n    \\sigma^t,\n\\]\nwhere the implied constant is absolute.\n\\end{lemma}\n\\begin{proof}\nFor $s \\in \\cS$, let \n$\n t_{s}\n  \\defeq \n  \\#(\\Omega_{\\cH}(s) \\cap \\{n_1 \\pod{s},\\ldots,n_t \\pod{s}\\}) \n$.\nWe have \n\\[\n \\PP(n_1,\\ldots,n_t \\in S(\\vec{\\ba}))\n  =\n  \\prod_{s \\in \\cS}\n   \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg).\n\\]\nNote that for $s \\in \\cS$, \n$\n 1 - t_{s}\/\\#\\Omega_{\\cH}(s)\n  = \n   1 + O\\br{t\/s}\n    =\n    1 + O\\br{1\/(\\log x)^{19}}\n$.\n\nLet $\\cS'$ be the set of primes $s \\in \\cS$ such that either \n$n_i \\equiv n_j \\pod{s}$ for some $i \\ne j$ or \n$\\cH \\cap n_i \\pod{s} \\ne \\emptyset$ for some $i$.\nFor $i \\ne j$ we have $1 \\le |n_i - n_j| \\ll x^{O(1)}$, so \n$n_i - n_j$ has at most $O(\\log x)$ prime divisors.\nSimilarly, for each $i$ and $j$, $n_i - q_j$ has at most \n$O(\\log x)$ prime divisors.\nWe see that $|\\cS'| \\ll (t^2 + tK)\\log x \\ll (\\log x)^3$ \n(cf.\\ \\eqref{eq:Kbnd}), and \n\\begin{align*}\n & \n \\prod_{s \\in \\cS'}\n  \\bigg(1 - \\frac{t}{\\#\\Omega_{\\cH}(s)}\\bigg)^{-1}\n   \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg)\n \\\\\n & \\hspace{30pt} \n  =\n   \\prod_{s \\in \\cS'}\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{19}}\\bigg)\\bigg)^{|\\cS'|}\n  =\n   1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg).\n\\end{align*}\nFor $s \\in \\cS \\, \\setminus \\, \\cS'$ we have $t_{s} = t$.\nThus, \n\\begin{align*}\n \\prod_{s \\in \\cS}\n  \\bigg(1 - \\frac{t_{s}}{\\#\\Omega_{\\cH}(s)}\\bigg)\n & = \n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n     \\prod_{s \\in \\cS}\n      \\bigg(1 - \\frac{t}{\\#\\Omega_{\\cH}(s)}\\bigg)\n \\\\\n & =\n     \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n      \\sigma^t\n      \\prod_{s \\in \\cS}\n       \\bigg(1 + O\\bigg(\\frac{t^2}{s^2}\\bigg)\\bigg) \n \\\\\n & = \n  \\sigma^t\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\n\\end{proof}\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \\textup{(}iv\\textup{)}]\nLet $\\cR \\defeq \\cQ\\setminus \\cH$.\nRecalling \\eqref{eq:cHdef} and \\eqref{eq:Kbnd}, we have   \n\\begin{equation}\n \\label{eq:rcb13}\n \\#(\\cQ \\cap S(\\vec{\\ba})) \n  = \n  \\#(\\cR \\cap S(\\vec{\\ba})) + K \n   =\n    \\#(\\cR \\cap S(\\vec{\\ba})) + O_{\\le}(\\log x).\n\\end{equation}\nLet \n$\n X \n  \\defeq \n   \\sum_{q \\in \\cR} \\ind{q \\in S(\\vec{\\ba})}.  \n$\nWe have\n\\[\n  \\EE X \n   =\n    \\sum_{q \\in \\cR}\n     \\PP(q \\in S(\\vec{\\ba}))\n   =\n    (\\#\\cR)\\sigma\n      \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n\\]\nfrom Lemma \\ref{lem:rcb4}.\nNote that by \\eqref{eq:Qsize} we have \n\\[\n (\\#\\cQ)\\sigma\n  = \n    \\frac{\\sigma y}{\\log x}\n     \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg).\n\\]\nBy \\eqref{eq:rcb13}, the same estimate holds for $(\\#\\cR)\\sigma$,\nand\n\\begin{equation}\n \\label{eq:rcb14}\n  \\EE \\#(\\cQ \\cap S(\\vec{\\ba}))\n   =\n    \\frac{\\sigma y}{\\log x}\n     \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg).\n\\end{equation}\nWe similarly have \n\\begin{align*}\n \\EE X^2\n  & =\n   \\sum_{q_1 \\in \\cR}\n    \\sum_{q_2 \\in \\cR}\n     \\ind{q_1,q_2 \\in S(\\vec{\\ba})}\n \\\\\n  & = \n   \\sums[(q_1,q_2) \\in \\cR^2][q_1 \\ne q_2]\n    \\sigma^2\n     \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n   + \n     \\sum_{q \\in \\cR}\n      \\sigma\n       \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n  \\\\\n  & = \n   \\sigma^2(\\#\\cR)^2\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\nThus,  \n\\begin{equation}\n \\label{eq:rcb15}\n  \\EE (X - \\EE X)^2 \n   =\n    \\EE X^2 - (\\EE X)^2\n   \\ll\n    \\frac{\\sigma^2(\\#\\cR)^2}{(\\log x)^{16}}.\n\\end{equation}\n\nNow we use Chebyshev's inequality: \n\\begin{align}\n \\begin{split}\n  \\label{eq:rcb16}\n  \\PP\\big(|X - \\EE X| > (\\#\\cR)\\sigma(\\log x)^{-3}\\big)\n   & \n    \\le \n    (\\#\\cR)^{-2}\\sigma^{-2}(\\log x)^6 \\, \\EE (X - \\EE X)^2\n  \\\\\n  & \n   \\ll\n    \\frac{1}{(\\log x)^{10}}.\n \\end{split}\n\\end{align}\nRecalling \\eqref{eq:rcb13} again we get \n\\[\n \\#(\\cQ \\cap S(\\vec{\\ba}))\n  =\n   \\frac{\\sigma y}{\\log x}\n    \\bigg(1 + O\\bigg(\\frac{\\log_2 x}{\\log x}\\bigg)\\bigg)\n\\]\nwith probability $1 - O((\\log x)^{-10})$, and \nTheorem \\ref{thm:fgkmt4} (iv) follows on recalling that, by \n\\eqref{eq:fgkmt6.11}, \n$\n \\sigma y\/\\log x \\sim 80cx \\log_2 x \/\\log x.\n$\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb5}\n\\textup{(}i\\textup{)}\nWith probability \n$1 - O\\big(\\frac{1}{\\log x}\\big)$, \n$\\cP(\\vec{\\ba})$ contains all but \n$O\\big(\\frac{\\#\\cP}{(\\log x)^3}\\big)$ \nof the primes in $\\cP$.\n\\textup{(}ii\\textup{)}\nWe have \n\\[\n \\EE \\, \\#\\cP(\\vec{\\ba}) \n  = (\\#\\cP)\n     \\bigg(\n      1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\n     \\bigg).\n\\]\n\\end{lemma}\n\\begin{proof}\nWe have \n\\begin{align*}\n \\EE \\, X_p(\\vec{\\ba})\n  & = \n   \\sum_{n \\in \\ZZ}\n    \\PP(\\tilde{\\bn}_p = n)\n     \\PP(n + h_i p \\in S(\\vec{\\ba}) \\,\\, \\forall  i \\le r)\n \\\\\n & = \n  \\sum_{n \\in \\ZZ} \n   \\PP(\\tilde{\\bn}_p = p)\n    \\sigma^r\n     \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n \\\\\n & = \n  \\sigma^r\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\n(For the second step, we supplement Lemma \\ref{lem:rcb4} with \nthe observation that $\\PP(\\bn_p = n) = 0$ whenever \n$n + h_ip \\in \\cH$ for some $i \\le r$.) \n\nLet $\\tilde{\\bn}_p^{(1)}$ and $\\tilde{\\bn}_p^{(2)}$ be independent \nrandom variables having the same probability distribution as \n$\\tilde{\\bn}_p$.\nThen \n\\begin{align*}\n X_p(\\vec{\\ba})^2\n  & = \n   \\PP\\big(\\tilde{\\bn}_p^{(1)} \\in S(\\vec{\\ba}) \\,\\, \\forall  i \\le r\\big)\n    \\PP\\big(\\tilde{\\bn}_p^{(2)} \\in S(\\vec{\\ba}) \\,\\, \\forall  i \\le r\\big)\n  \\\\\n  & = \n   \\PP\\big(\\tilde{\\bn}_p^{(l)} \\in S(\\vec{\\ba}) \\,\\, \\forall l \\le 2, i \\le r\\big).\n\\end{align*}\nArguing as above, \n\\[\n \\EE X_p(\\vec{\\ba})^2\n  = \n   \\sum_{n_1 \\in \\ZZ}\n    \\sum_{n_2 \\in \\ZZ} \n     \\PP\\big(\\tilde{\\bn}_p^{(1)} = n_1\\big)\n      \\PP\\big(\\tilde{\\bn}_p^{(2)} = n_2\\big)\n       \\sigma^{t(n_1,n_2)}\n         \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg),\n\\]\nwhere $t(n_1,n_2)$ is the number of distinct integers \n$n_l + h_ip$ ($l \\le 2$, $i \\le r$).\n\nNow fix $n_1$.\nThere are less than $r^2$ values of $n_2$ for which \n$t(n_1,n_2) \\ne 2r$.\nSince $\\PP(\\bn_p^{(2)} = n_2) \\ll x^{-2\/3 + o(1)}$ (cf.\\ \n\\eqref{eq:fgkmt6.10}), we obtain\n\\begin{align*}\n &\n  \\EE X_p(\\vec{\\ba})^2\n \\\\\n & \\hspace{15pt}\n =\n   \\sum_{n_1 \\in \\ZZ}\n    \\PP(\\tilde{\\bn}_p^{(1)} = n_1)\n     \\bigg\\{\n         \\sigma^r \n          \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n           \\bigg(1 - O\\bigg(\\frac{1}{x^{1\/2}}\\bigg)\\bigg)\n          + O\\bigg(\\frac{1}{x^{1\/2}}\\bigg)\n     \\bigg\\}\n \\\\\n & \\hspace{15pt}\n  = \n   \\sigma^{2r}\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg).\n\\end{align*}\nArguing as in \\eqref{eq:rcb15}, \\eqref{eq:rcb16}, \n\\[\n \\PP\n  \\Big(\n      |\n       X_p(\\vec{\\ba}) \n        - \\big(\n           1 + O\\big({\\textstyle \\frac{1}{(\\log x)^{16}}}\\big)\n          \\big) \n           \\sigma^r\n      |\n       > \n         {\\textstyle \n           \\frac{1}{2} \n            \\frac{{\\displaystyle \\sigma^r}}{(\\log x)^6}}\n  \\Big)\n   \\ll\n    \\frac{1}{(\\log x)^4}.\n\\]\nThus, with probability \n$1 - O\\big(1\/(\\log x)^4\\big)$, \nwe have \n\\[\n X_p(\\vec{\\ba})\n  =\n   \\bigg(\n    1 + O_{\\le}\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\n   \\bigg) \n    \\sigma^r,\n\\]\nthat is, $p \\in \\cP(\\vec{\\ba})$.\nMoreover, \n\\begin{align*}\n & \n \\frac{\\#\\cP}{(\\log x)^3}\n  \\PP\n   \\Big(\n        {\\textstyle \\sum_{p \\in \\cP} } \n         \\ind{p \\not\\in \\cP(\\vec{\\ba})}\n        >\n          {\\textstyle\\frac{\\#\\cP}{(\\log x)^3}}\n   \\Big)\n \\\\\n & \\hspace{60pt}\n  \\le\n   \\EE \\, \n    \\Big(\n     \\sum_{p \\in \\cP}  \n      \\ind{p \\not\\in \\cP(\\vec{\\ba})}\n    \\Big)\n  = \n   \\sum_{p \\in \\cP}\n    \\PP(p \\not\\in \\cP(\\vec{\\ba}))\n  \\ll\n   \\frac{\\#\\cP}{(\\log x)^4}.\n\\end{align*}\nSo with probability $1 - \\big(1\/\\log x\\big)$, \n$\\cP(\\vec{\\ba})$ contains all but \n$O\\big(\\frac{\\#\\cP}{(\\log x)^3}\\big)$ of the primes $p \\in \\cP$.\nFinally, \n\\[\n \\EE \\, \\#\\cP(\\vec{\\ba})\n  = \n   \\sum_{p \\in \\cP}\n    \\PP(p \\in \\cP(\\vec{\\ba}))\n  =\n   (\\#\\cP)\n     \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg).\n\\]\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb6}\nLet $q \\in \\cQ$ and let $\\vec{a}$ be in the essential range of \n$\\vec{\\ba}$.\nThen\n\\begin{equation}\n \\label{eq:rcb17}\n  \\sigma^{-r}\n   \\sum_{i=1}^r \n    \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n     Z_p(\\vec{a};q - h_ip)\n    =\n     \\big(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n      \\sum_{p \\, \\in \\, \\cP}\n       \\PP(q \\in \\be_p(\\vec{a}) \\mid \\vec{\\ba} = \\vec{a}).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nRecalling \\eqref{eq:rcb11}, the left-hand side of \n\\eqref{eq:rcb17} is \n\\begin{align*}\n & \n \\sigma^{-r}\n  \\sum_{i=1}^r \n   \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n    X_p(\\vec{a}) \n     \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a})\n \\\\\n  & \\hspace{60pt}\n   = \n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n     \\sigma^{-r}\n      \\sum_{i=1}^r \n       \\sum_{p \\, \\in \\, \\cP(\\vec{a})}\n        \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a}).  \n\\end{align*}\nSince $q - h_ip \\ne \\bn_p$ if $p \\not\\in \\cP(\\vec{a})$, we may \nrewrite this as \n\\[\n \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n  \\sigma^{-r}\n   \\sum_{i=1}^r \n    \\sum_{p \\, \\in \\, \\cP}\n     \\PP(\\bn_p = q - h_ip \\mid \\vec{\\ba} = \\vec{a}),  \n\\]\nand the lemma follows.\n\\end{proof}\n\nIt is convenient to write \n\\[\n U(q,\\vec{a})\n  \\defeq \n  \\sigma^{-r}\n   \\sum_{i=1}^r \n    \\sum_{p \\, \\in \\, \\cP}\n     Z_p(\\vec{a};q - h_ip).\n\\]\n\n\\begin{lemma}\n \\label{lem:rcb7}\nWe have\n\\begin{equation}\n \\label{eq:rcb18}\n  \\EE \\,\n   \\Big(\n    \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n     \\sigma^r U(q,\\vec{\\ba})\n   \\Big)\n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n    \\frac{\\sigma y}{\\log x}\n     \\frac{\\sigma^{r-1}ux}{2y}\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:rcb19}\n  \\EE \\,\n   \\Big(\n    \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n     \\sigma^r U(q,\\vec{\\ba})^2\n   \\Big)\n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n    \\frac{\\sigma y}{\\log x}\n     \\bigg(\\frac{\\sigma^{r-1}ux}{2y}\\bigg)^2.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe begin with \\eqref{eq:rcb19}.\nLet $\\tilde{\\bn}_p^{(1)}$ and $\\tilde{\\bn}_p^{(2)}$ be \nindependent copies of $\\tilde{\\bn}_p$ that are also independent of \n$\\vec{\\ba}$.\nWe observe that for any $n_1,n_2$, and $p_1,p_2 \\in \\cP$, \n\\[\n Z_{p_1}(\\vec{\\ba};n_1)\n  Z_{p_2}(\\vec{\\ba};n_2)\n =\n  \\ind{\n      (n_l + h_jp_l \\, \\in \\, S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, \\, j \\le r)\n      }\n   \\,\n    \\PP(\\tilde{\\bn}_p^{(1)} = n_1)\n     \\PP(\\tilde{\\bn}_p^{(2)} = n_2).\n\\]\nThe left-hand side of \\eqref{eq:rcb19} is thus\n\\begin{align}\n \\label{eq:rcb20}\n  \\begin{split}\n   & \n   \\EE \\,\n    \\Big(\n     \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n      \\,\n       \\sum_{i_1,i_2 = 1}^r\n        \\,\n         \\sum_{p_1,p_2 \\in \\cP}\n         Z_{p_1}(\\vec{\\ba};q - h_{i_1}p_1)\n          Z_{p_2}(\\vec{\\ba};q - h_{i_2}p_2)\n    \\Big)\n   \\\\\n    &  \n     =\n      \\sum_{q \\in \\cQ}\n       \\,\n       \\sum_{i_1,i_2 = 1}^r\n        \\big( \n         \\PP(q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r)\n  \\\\\n   & \\hspace{90pt} \n      \\times    \n       \\PP(\\tilde{\\bn}_{p_1}^{(1)} = q - h_{i_1}p_1)\n        \\PP(\\tilde{\\bn}_{p_2}^{(2)} = q - h_{i_2}p_2)\n       \\big).\n  \\end{split}\n\\end{align}\n(For $q \\in \\cQ$, the event \n$q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r$ \nis identical to the event \n$q \\in \\cQ \\cap S(\\vec{\\ba})$, \n$q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r$.) \n\nLet $\\Sigma_1,\\Sigma_2$ be the contributions to the right-hand \nside of \\eqref{eq:rcb20} from $p_1 \\ne p_2$, respectively \n$p_1 = p_2$.\n\nFix $p_1,p_2$ in $\\cP$ with $p_1 \\ne p_2$, and fix \n$i_1,i_2 \\le r$.\nThe number of distinct integers $q + (h_j - h_{i_l})p_l$ \n($l \\le 2, j \\le r$) is $2r - 1$ since \n$\n (h_j - h_{i_1})p_1 \\ne (h_j - h_{i_2})p_2\n$\nfor $i_1 \\ne j$.\nHence \n\\begin{equation}\n \\label{eq:rcb21}\n  \\begin{split}\n   & \n   \\PP(q + (h_j - h_{i_l})p_l \\in S(\\vec{\\ba}) \\, \\, \\forall l \\le 2, j \\le r)\n    \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n     \\PP(\\tilde{\\bn}_p^{(2)} = q - h_{i_2}p_2)\n   \\\\ \n   & \\hspace{30pt}\n   =\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg) \n     \\sigma^{2r-1}\n      \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n       \\PP(\\tilde{\\bn}_p^{(2)} = q - h_{i_2}p_2).      \n  \\end{split}\n\\end{equation}\n(Both sides are $0$ if there are $j,i_l$ with \n$q + (h_j - h_{i_l})p_l \\in \\cH$; otherwise \nLemma \\ref{lem:rcb4} applies.)\n\nWe combine \\eqref{eq:rcb21} with the remark after \n\\eqref{eq:probnph0} to obtain \n\\begin{align}\n \\label{eq:rcb22}\n  \\begin{split}\n   \\sumsstxt[][][1]\n    & \n     =\n      \\sums[q \\in \\cQ \\setminus \\cH][i_1,i_2 \\le r]\n       \\sigma^{2r-1}\n        \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n         \\bigg(\\frac{ux}{2ry}\\bigg)^2\n   \\\\\n   & = \n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n     \\frac{\\sigma y}{\\log x}\n      \\bigg(\\frac{\\sigma^{r-1}ux}{2y}\\bigg)^2.\n  \\end{split}\n\\end{align}\nSimilarly, \n\\begin{align*}\n \\sumsstxt[][][2]\n  & \n   =\n    \\sum_{q \\in \\cQ}\n     \\sum_{i_1 = 1}^r\n      \\sum_{p_1 \\in \\cP}\n       \\PP(q + (h_j - h_{i_1})p_1 \\in S(\\vec{\\ba}) \\, \\, \\forall j \\le r)\n        \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)^2\n  \\\\\n   & \n    \\ll\n     x^{-3\/5}\n      \\sum_{q \\in \\cQ}\n       \\sum_{i_1 = 1}^r \n        \\sum_{p_1 \\in \\cP}\n         \\PP(\\tilde{\\bn}_p^{(1)} = q - h_{i_1}p_1)\n  \\\\\n   & \n    \\ll\n     x^{-3\/5}(\\#\\cQ),\n\\end{align*}\nwhich together with \\eqref{eq:rcb22} yields \\eqref{eq:rcb19}.\n\nMuch the same argument gives for the left-hand side of \n\\eqref{eq:rcb18} the expression \n\\begin{align*}\n & \n \\sum_{q \\in \\cQ}\n  \\sum_{i = 1}^r\n   \\sum_{p \\in \\cP}\n    \\PP(q + (h_j - h_i)p \\in S(\\vec{\\ba}) \\, \\, \\forall j \\le r)\n     \\PP(\\tilde{\\bn}_p = q - h_ip)\n \\\\\n & \\hspace{30pt}\n  = \n   \\sum_{q \\in \\cQ}\n    \\sum_{i=1}^r\n     \\sum_{p \\in \\cP}\n      \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n       \\sigma^r\n        \\PP(\\tilde{\\bn}_p = q - h_ip)\n \\\\\n & \\hspace{30pt}\n  =\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n     \\frac{\\sigma y}{\\log x}\n      \\frac{\\sigma^{r-1}ux}{2y}.\n\\end{align*}\n\\end{proof}\n\nWe now specify that the quantity $C$ in Theorem \\ref{thm:fgkmt4} \nis \n\\[\n C \\defeq \\frac{ux}{2\\sigma y},\n\\]\nso that $C \\asymp 1\/c$.\n\n\\begin{lemma}\n \\label{lem:rcb8}\nWith probability $1 - o(1)$, we have \n\\[\n U(q,\\vec{\\ba}) \n  = \n   \\bigg(\n    1 + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^3}\\bigg)\n   \\bigg)\n    C\n\\]\nfor all but at most $\\frac{x}{2\\log x\\log_2 x}$ of the primes \n$q \\in \\cQ \\cap S(\\vec{\\ba})$.\n\\end{lemma}\n\n\\begin{proof}\nUsing \\eqref{eq:rcb14} and Lemma \\ref{lem:rcb7}, we find that \n\\begin{align} \n \\label{eq:rcb23}\n  \\begin{split}\n & \n \\EE \\,\n  \\Big(\n   \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n    (U(q,\\vec{\\ba}) - C)^2\n   \\Big)\n \\\\\n & \n  =\n   \\EE \\,\n    \\Big(\n     \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} U(q,\\vec{\\ba})^2    \n    \\Big)\n    -\n      2C \\,\n       \\EE \\, \n        \\Big(\n         \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} U(q,\\vec{\\ba})\n        \\Big)\n       + \n         C^2 \\,\n          \\EE \\,\n           \\Big(\n            \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})} 1\n           \\Big) \n  \\\\\n  & \n   =\n    \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n     \\frac{u^2x^2}{4\\sigma y\\log x}\n \\\\\n  & \\hspace{45pt}    \n    -\n      \\bigg(1 + O\\bigg(\\frac{1}{(\\log_2 x)^{10}}\\bigg)\\bigg)\n       \\frac{2ux}{2\\sigma y}\n        \\frac{ux}{2\\log x}\n   +\n    \\big(1 + O\\big({\\textstyle \\frac{1}{\\log x}}\\big)\\big)\n     \\frac{u^2x^2}{4\\sigma^2y^2}\n      \\frac{\\sigma y}{\\log x}\n \\\\\n  & \n   \\ll\n    \\frac{u^2x^2}{\\sigma y(\\log x)(\\log_2 x)^{10}}\n \\\\\n & \n  \\ll\n   \\frac{C^2\\sigma y}{(\\log x)(\\log_2 x)^{10}}.\n  \\end{split}\n\\end{align}\nLet $V$ be the event \n\\[\n \\#\\big\\{\n     q \\in \\cQ \\cap S(\\vec{\\ba}) : \n      |U(q,\\vec{\\ba}) - C| \n       >\n        {\\textstyle \\frac{C}{(\\log_2 x)^3}}\n    \\big\\}\n    >\n     \\frac{x}{2\\log x \\log_2 x}.\n\\]\nEvidently \n\\[\n \\EE \\,\n  \\Big(\n   \\sum_{q \\in \\cQ \\cap S(\\vec{\\ba})}\n    (U(q,\\vec{\\ba}) - C)^2\n  \\Big)\n   \\ge \n    \\PP(V)\n     \\frac{x}{2\\log x\\log_2 x}\n      \\frac{C^2}{(\\log_2 x)^6}.\n\\]\nCombining this with \\eqref{eq:rcb23}, and recalling that \n$\\sigma y \\asymp x\\log_2 x$ (cf.\\ \\eqref{eq:fgkmt6.11}), we obtain \n\\[\n \\PP(V) \\ll \\frac{1}{(\\log_2 x)^2}.\n\\]\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:rcb9}\nWe have \n\\begin{equation}\n \\label{eq:rcb24}\n  \\EE \\, \n   \\Big(\n    \\sum_{n \\in \\ZZ}\n     \\sigma^{-r}\n      \\sum_{p \\in \\cP(\\vec{\\ba})} \n       Z_p(\\vec{\\ba};n)\n   \\Big)\n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg)\n    (\\#\\cP)\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:rcb25}\n  \\EE \\, \n   \\Big(\n    \\sum_{n \\in \\ZZ}\n     \\sigma^{-r}\n      \\sum_{p \\in \\cP \\setminus \\cP(\\vec{\\ba})} \n       Z_p(\\vec{\\ba};n)\n   \\Big)\n  \\ll\n   \\frac{\\#\\cP}{(\\log x)^4}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nThe left-hand side of \\eqref{eq:rcb24} is \n\\begin{align*}\n & \n  \\sigma^{-r}\n   \\sum_{\\vec{\\ba}}\n    \\PP(\\vec{\\ba} = \\vec{a})\n     \\sum_{p \\in \\cP(\\vec{a})}\n      \\sum_{n \\in \\ZZ}\n       \\PP(n + h_ip \\in S(\\vec{a}) \\, \\, \\forall i \\le r)\n        \\PP(\\tilde{\\bn}_p = n)\n \\\\\n  & \\hspace{30pt} = \n   \\sigma^{-r}\n    \\sum_{\\vec{\\ba}}\n     \\PP(\\vec{\\ba} = \\vec{a})\n      \\sum_{p \\in \\cP(\\vec{a})}\n       X_p(\\vec{a})\n \\\\\n & \\hspace{30pt}  = \n  \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n   \\sum_{\\vec{\\ba}}\n    \\PP(\\vec{\\ba} = \\vec{a})\n     \\sum_{p \\in \\cP(\\vec{a})} 1\n \\\\\n & \\hspace{30pt}  = \n  \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^6}\\bigg)\\bigg)\n   \\EE \\, \\#\\cP(\\vec{a})\n \\\\\n & \\hspace{30pt}  = \n  \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^4}\\bigg)\\bigg)\n   (\\# \\cP)\n\\end{align*}\nby Lemma \\ref{lem:rcb5}.\nThis proves \\eqref{eq:rcb24}.\n\nNow, \n\\begin{align}\n \\label{eq:rcb26}\n  \\begin{split}\n   \\EE \\,\n    \\Big(\n     \\sum_{n \\in \\ZZ} \\sigma^{-r}\n      \\sum_{p \\in \\cP} Z_p(\\vec{\\ba};n)\n    \\Big)\n  & \n   =\n    \\sigma^{-r}\n     \\sum_{p \\in \\cP}\n      \\sum_{n \\in \\ZZ}\n       \\PP(\\tilde{\\bn}_p = n)\n        \\PP(n + h_jp \\in S(\\vec{a}) \\,\\, \\forall j \\le r)\n \\\\\n & \n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n    \\sum_{p \\in \\cP}\n     \\sum_{n \\in \\ZZ} \n      \\PP(\\tilde{\\bn}_p = n)\n \\\\\n & \n  =\n   \\bigg(1 + O\\bigg(\\frac{1}{(\\log x)^{16}}\\bigg)\\bigg)\n    (\\# \\cP).\n  \\end{split}\n\\end{align}\n(Here we have used Lemma \\ref{lem:rcb4} with a familiar argument.)\nWe obtain \\eqref{eq:rcb25} on subtracting \\eqref{eq:rcb24} from \n\\eqref{eq:rcb26}.\n\\end{proof}\n\n\\begin{proof}%\n[Deduction of Theorem \\ref{thm:fgkmt4} \\textup{(}v\\textup{)}]\nIn view of Lemmas \\ref{lem:rcb6} and \\ref{lem:rcb8} it suffices to \nshow that with probability $1 - O\\big(1\/(\\log x)^3\\big)$, the \nnumber of $q$ in $\\cQ \\cap S(\\vec{\\ba})$ with \n\\begin{equation}\n \\label{eq:rcb27}\n  \\sum_{i = 1}^r\n   \\sum_{p \\in \\cP \\setminus \\cP(\\vec{\\ba})}\n    \\sigma^{-r} Z_p(a; q - h_ip)\n     >\n      \\frac{ux}{\\sigma y(\\log_2 x)^3}\n\\end{equation}\nis at most $\\frac{x}{2\\log x\\log_2 x}$.\n\nLet $W$ be the event that \\eqref{eq:rcb27} holds for more than \n$\\frac{x}{2\\log x\\log_2 x}$ primes in \n\\linebreak \n$\\cQ \\cap S(\\vec{\\ba})$.\nThen \n\\[\n \\PP(W)\n  \\le \n   \\PP\n    \\Big(\n     \\sum_{q \\in \\cQ}\n      \\sum_{i=1}^r\n       \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n        \\sigma^{-r}\n         Z_p(\\vec{\\ba};q - h_ip) > v\n    \\Big),\n\\]\nwhere \n\\[\n v \n  \\defeq \n   \\frac{ux}{\\sigma y(\\log_2 x)^3}\n    \\cdot \n     \\frac{x}{2\\log x \\log_2 x}\n   =\n    \\frac{x}{(\\log x)^{2 - o(1)}}.\n\\]\nThus, \n\\begin{multline*}\n  \\PP(W) \n   \\le \n    \\frac{1}{v} \n     \\EE \\, \n      \\Big(\n       \\sum_{q \\in \\cQ}\n        \\sum_{i=1}^r \n         \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n          \\sigma^{-r}\n           Z_p(q - h_ip)\n      \\Big)\n \\\\\n  \\le \n   \\frac{r}{v}\n    \\EE \\,\n     \\Big(\n      \\sum_{n \\in \\ZZ}\n       \\sigma^{-r} \n        \\sum_{p \\in \\cP\\setminus \\cP(\\vec{\\ba})}\n         Z_p(\\vec{\\ba};n)\n     \\Big)\n  \\ll\n   \\frac{r}{v}\n    \\frac{x}{(\\log x)^5}\n  \\ll\n   \\frac{1}{(\\log x)^2}.\n\\end{multline*}\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:fgkmt2}}\n \\label{subsec:thm2pf}\nWe require one further lemma for the proof of \nTheorem \\ref{thm:fgkmt2}, viz.\\ the following, which is a special \ncase of \\cite[Corollary 3]{FGKMT}.\n\n\\begin{lemma}\n \\label{lem:rcb10}\nLet $\\cQ'$ be a set of primes with $\\#\\cQ' > (\\log_2 x)^3$.\nFor each $p \\in \\cP$, let $\\be_p$ be a random subset of $\\cQ'$ \nwith \n\\[\n \\# \\be_p \\le r, \n  \\quad \n   \\PP(q \\in \\be_p) \\le x^{-3\/5} \\quad (q \\in \\cQ').\n\\]\nSuppose that for all but at most $\\frac{\\#\\cQ'}{(\\log_2 x)^2}$ \nelements $q \\in \\cQ'$, we have \n\\[\n \\sum_{p \\in \\cP} \\PP(q \\in \\be_p)\n  =\n   C + O_{\\le}\\bigg(\\frac{1}{(\\log_2 x)^2}\\bigg),\n\\]\nwhere $C$ is independent of $q$ and \n\\begin{equation}\n \\label{eq:rcb28}\n  {\\textstyle \\frac{5}{4}}\\log 5 \\le C \\ll 1.\n\\end{equation}\nSuppose that for any distinct $q_1,q_2 \\in \\cQ'$, \n\\begin{equation}\n \\label{eq:rcb29}\n  \\sum_{p \\in \\cP'} \\PP(q_1,q_2 \\in \\be_p)\n   \\le \n    x^{-1\/20}.\n\\end{equation}\nThen for any positive integer $m$ with \n\\[\n m \\le \\frac{\\log_3 x}{\\log 5},\n\\]\nwe can find random sets $\\be_p' \\subseteq \\cQ'$ for each \n$p \\in \\cP$ such that $\\be_p'$ is either empty or is in the \nessential range of $\\be_p$, and \n\\begin{equation}\n \\label{eq:rcb30}\n  \\#\\{q \\in \\cQ' : q \\not\\in \\be_p' \\,\\, \\textup{for all} \\,\\, p \\in \\cP \\}\n   \\sim \n    5^{-m}(\\#\\cQ'),\n\\end{equation}\nwith probability $1 - o(1)$.\n\\end{lemma}\n\n\\begin{proof}[Deduction of Theorem \\ref{thm:fgkmt2}]\nBy \\eqref{eq:fgkmt4.29}, we may choose $c$ small enough so that \n\\eqref{eq:rcb28} holds.\nTake \n\\[\n m = \\Big\\lfloor \\frac{\\log_3 x}{\\log 5} \\Big\\rfloor.\n\\]\nLet $\\vec{\\ba}$ and $\\vec{\\bn}$ be as in Theorem \\ref{thm:fgkmt4}.\nSuppose that we are in the probability $1 - o(1)$ event that \n$\\vec{\\ba}$ takes a value $\\vec{a}$ for which \\eqref{eq:fgkmt4.31} \nholds.\nFix some $\\vec{a}$ within this event.\nWe apply Lemma \\ref{lem:rcb10} with $\\cQ' = \\cQ \\cap S(\\vec{a})$, \n$\\be_p = \\be_p(\\vec{a})$.\nWe need only check the hypothesis \\eqref{eq:rcb29}.\nWe have \n\\[\n \\sum_{p \\in \\cP}\n  \\PP(q_1,q_2 \\in e_p(\\vec{a})\n   \\le \n    \\sums[p \\mid q_1 - q_2][p \\in \\cP]\n     \\PP(q_1 \\in e_p(\\vec{a}))\n      \\le \n       x^{-3\/5}\n\\]\n(the sum has at most one term).\n\nLet $\\be_p'(\\vec{a})$ be the random variables provided by Lemma \n\\ref{lem:rcb10}.\nRecalling \\eqref{eq:fgkmt4.30}, \n\\[\n \\#\\{q \\in \\cQ' : q \\not\\in \\be_p' \\,\\, \\text{for all} \\,\\, p \\in \\cP \\}\n  \\sim \n   5^{-m}\\#(\\cQ \\cap S(\\vec{a}))\n    \\ll\n     \\frac{x}{\\log x}\n\\]\nwith probability $1 - o(1)$.\nSince $e_p'(\\vec{a})$ is either empty or \n\\[\n e_p'(\\vec{a})\n  =\n   \\{\\tilde{\\bn}_p' + h_ip : i \\le r\\}\n    \\cap \n     \\cQ \\cap S(\\vec{a})\n\\]\nfor some random integer $\\tilde{\\bn}_p'$, it follows that \n\\[\n \\#\\{\n     q \\in \\cQ \\cap S(\\vec{a}) : \n      q \\not\\equiv \\tilde{\\bn}_p' \\pod{p} \n         \\,\\, \\text{for all} \\,\\, p \\in \\cP\n   \\}\n    \\ll\n     \\frac{x}{\\log x}\n\\]\nwith probability $1 - o(1)$.\nThe bound \\eqref{eq:fgkmt3.6} follows on setting $b_p = n_p'$ for \na specific $\\vec{n}' = (\\tilde{\\bn}_p')$ for which this bound \nholds.\nThat $\\cH$ is contained in $S(\\vec{a}) \\cap S(\\vec{b})$ follows \nfrom parts (i) and (ii) of Theorem \\ref{thm:fgkmt4}.\n\\end{proof}\n\n\n\\section{A modification of Maynard--Tao}\n \\label{sec:MT}\n \n\\begin{definition} \n \\label{def:w}\nWe consider functions of the form \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$, with $T_1,x_1 \\ge 1$.\nLet us say that such a function $f_1$ is ``of the first kind'' if \nand only if \n(i) it is a strictly increasing bijection, \n(ii) $f_1(T) \\le \\log T$ for $T \\ge T_1$,\n(iii) $f_1(2T)\/f_1(T) \\to 1$ as $T \\to \\infty$\nand\n(iv) for $0 < \\eta \\le 1$, there exists $L_{\\eta} \\ge 1$ such that \n$f_1(T)\/f_1(T^{\\eta}) \\to L_{\\eta}$ as \n$T \\to \\infty$.\n\\end{definition}\n\n\\begin{definition}\n \\label{def:sparse}\nWe consider (possibly empty) sets $\\cZ(T)$, $T \\ge 2$, of primes \nless than or equal to $T$.\nLet us that such a set is ``repulsive'' if and only if for any \n$p' \\in \\cZ(T)$,  \n$\\sum_{p \\in \\cZ(T), \\, p \\ge p'} 1\/p \\ll 1\/p' \\ll 1\/\\log_2 T$.\n\\end{definition}\n\nGiven a function $\\upsilon : \\NN \\to \\RR$ with finite support and \nany arithmetic progression $a \\pod{D}$ with $(a,D) = 1$, we \ndefine  \n\\[\n \\Delta(\\upsilon;a \\pod{D})\n  \\defeq \n    \\sum_{n \\equiv a \\pod{D}}\n     \\upsilon(n)\n      -\n       \\frac{1}{\\phi(D)}\n        \\sum_{(n,D) = 1}\n         \\upsilon(n),\n\\]\nwhere $\\phi$ is Euler's totient function.\n\n\\begin{hypothesis} \n \\label{hyp:EH}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nFor any given $A > 0$ and $\\delta \\in (0,\\theta)$, if \n$\\eta = \\eta(A,\\delta) \\in (0,\\theta - \\delta)$ is a \nsufficiently small, fixed number then, for $N \\ge T_1^{1\/\\eta}$, \nthere is a repulsive subset $\\cZ \\defeq \\cZ(N^{4\\eta})$ of the \nprimes less than or equal to $N^{4\\eta}$ such that, with \n$W \\defeq \\prod_{p \\le N^{\\eta}, \\, p \\not\\in \\cZ} p$ and \n$Z \\defeq \\prod_{p \\in \\cZ} p$, we have\n\\[\n  \\sums[r \\le N^{\\theta}\/(N^{\\delta}W)]\n       [(r,WZ) = 1]\n       [\\text{$r$ squarefree}]\n     \\max_{N \\le M \\le 2N}\n      \\max_{(rW,a) = 1}     \n      |\\Delta(\\ind{\\bP}\\ind{(M,M + N]}; a \\pod{rW})|\n      \\ll_{\\delta,A}\n       \\frac{N}{\\phi(W)(\\log N)^A}.\n\\]\n\\end{hypothesis}\n\n\\begin{theorem}\n \\label{thm:BFM4.3}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nSuppose that Hypothesis \\ref{hyp:EH} holds.\nFix a positive integer $a$.\nIn the notation of Hypothesis \\ref{hyp:EH}, if \n$K = K_{\\theta,a}$ is a sufficiently large integer multiple of \n$\\lceil (2\/\\theta) a \\rceil + 1$, if $A = A_K$ is sufficiently \nlarge and if $\\delta = \\delta_{\\theta,a}$ is sufficiently small, \nthen the following holds for $N \\ge N(T_1,K,\\eta)$.\nLet $\\cH \\defeq \\{H_1,\\ldots,H_K\\} \\subseteq [0,N]$ be an \nadmissible set of $K$ distinct integers for which \n$\\prod_{1 \\le i < j \\le K}(H_j - H_i)$ is \n$f_1(N^{\\eta})$-smooth, and let $b$ be an integer such that  \n\\[\n \\textstyle (\\prod_{i = 1}^K (b + H_i),W) = 1.\n\\]\nThen for any partition \n\\[\n \\cH  \n  = \n   \\cH_1 \\cup \\cdots \\cup \\cH_{\\lceil (2\/\\theta) a \\rceil + 1}   \n\\]\nof $\\cH$ into $\\lceil (2\/\\theta) a \\rceil + 1$ sets of equal \nsize, there exists some $n \\in (N,2N] \\cap b \\pod{W}$, and $a + 1$ \ndistinct indices \n$\n i_1,\\ldots,i_{a+1} \n  \\in \n   \\{1,\\ldots,\\lceil (2\/\\theta) a \\rceil + 1\\}\n$, \nsuch that \n\\[\n \\#(\\bP \\cap n + \\cH_{i_1}),\n  \\ldots,\n   \\#(\\bP \\cap n + \\cH_{i_{a + 1}}) \\ge 1.\n\\]\n\\end{theorem}\n\n\n\\begin{theorem}\n \\label{thm:BFM4.2}\nHypothesis \\ref{hyp:EH}, and therefore the statement of \nTheorem \\ref{thm:BFM4.3}, holds with $\\theta = 1\/2$ and any \nfunction $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\n\\end{theorem}\n\n\\begin{proof}%\n[Proof of Theorems \\ref{thm:BFM4.2} and \\ref{thm:BFM4.3}]\nThat Hypothesis \\ref{hyp:EH} holds with $\\theta = 1\/2$ and any \nfunction $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind \nis a consequence of Lemma 4.1 and Theorem 4.2 of \\cite{BFM}.\n\nWe prove Theorem \\ref{thm:BFM4.3} by following Pintz's \\cite{PIN3} \nmodification to the proof of Theorem 4.3 (i) in \\cite{BFM}.\nThere are many parameters involved and it is important to keep \ntrack of their interdependencies.\nIt is also important to note that the implicit constants in all \n$O$-terms are absolute, that is, independent of all \nparameters.\n\nOnly the unconditional case $\\theta = 1\/2$ is considered in \n\\cite{BFM,PIN3}, whereas here we are considering $\\theta \\le 1$.\nTo do this, we need to note that on Hypothesis \\ref{hyp:EH}, the \nterm $4 + O(\\delta)$ may be replaced by $(2\/\\theta) + O(\\delta)$ \non the right-hand side of the inequality in \n\\cite[Lemma 4.5 (iii)]{BFM}.\nIn the proof of this lemma in \\cite[\\S 4.2]{BFM}, the support of \nthe smooth function $G : [0,\\infty) \\to \\RR$, which is \n$[0,1\/4 - 2\\delta]$, may be replaced by \n$[0,(\\theta\/2) - 2\\delta]$, and the rest of the proof may be \ncarried out, mutatis mutandis.\n\nAs in \\cite{PIN3}, we begin with the following observation.\nSuppose $K$ and $M$ are positive integers with $M \\mid K$, and \nlet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be a partition of a set \n$\\cH$ of integers into $M$ subsets of equal size.\nSuppose also that $\\mu'$ and $\\mu$ are positive real numbers \nwith \n\\[\n \\mu' \n  \\defeq\n   \\max_{v \\in \\NN}\n    \\bigg(v - \\mu \\binom{v}{2}\\bigg).\n\\]\n\nGiven an integer $n$, consider the expression  \n\\[\n \\sum_{j=1}^M\n  \\Big\\{ \n   \\sum_{H \\in \\cH_j} \\ind{\\bP}(n + H)\n    -\n     \\mu \n      \\sums[H,H' \\in \\cH_j][H \\ne H']\n       \\ind{\\bP}(n + H)\\ind{\\bP}(n + H')   \n   \\Big\\},\n\\]\nwhere in the double sum each unordered pair \n$\\{H,H'\\} \\subseteq \\cH_j$ with $H \\ne H'$ is counted once only. \nSuppose $\\#(\\bP \\cap n + \\cH_j) = 0$ for all but at most $a$ of \nthe subsets $\\cH_j$.\nThen the above expression is at most $\\mu' a$.\nConsequently, if \n\\[\n \\sum_{H \\in \\cH}\n  \\ind{\\bP}(n + H)\n   -\n    \\mu' a\n     - \n      \\mu\n       \\sum_{j=1}^M\n        \\sums[H,H' \\in \\cH_j][H \\ne H']\n         \\ind{\\bP}(n + H)\n          \\ind{\\bP}(n + H')\n\\]\nis positive then $\\#(\\bP \\cap n + \\cH_j) \\ge 1$ for at least \n$a + 1$ of the subsets $\\cH_j$.\n\nNote that when $\\mu$ is the reciprocal of a positive integer, \nwe have \n\\[\n \\mu' \n  = \n    \\textstyle \n     \\frac{1}{2}\\big(1 + \\frac{1}{\\mu}\\big),\n\\]\nthe maximum being attained when $v = 1\/\\mu$ and \n$v = 1 + 1\/\\mu$.\n\nNow, fix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ of the first kind.\nSuppose that Hypothesis \\ref{hyp:EH} holds.\nFix any positive integer $a$ and let $M = M_{\\theta,a}$ be \nthe integer satisfying \n\\[\n M - 2 < (2\/\\theta)a \\le M - 1.\n\\]\nLet $\\iota = \\iota_{\\theta,a}$ be a small, fixed quantity to \nbe specified.\nSet \n\\[\n \\delta = \\delta_{\\theta,a} \\defeq \\iota^2\/(aM).\n\\]\nLet $K = K_{\\theta,a}$ be the integer satisfying\n\\[\n \\e^{aM^2\/(\\delta(M-1))} < K \\le \\e^{aM^2\/(\\delta(M-1))} + M\n  \\quad \n   \\text{and}\n    \\quad \n     M \\mid K.\n\\]\nFinally, let\n\\[\n \\rho \\defeq \\frac{aM^2\/(M-1)}{\\delta\\log K} < 1.\n\\]\n\nNow let $A = A(K)$, $\\eta = \\eta(A,\\delta)$, \n$N \\ge N(T_1,K,\\eta)$, $\\cH = \\{H_1,\\ldots,H_K\\}$ and $b \\pod{W}$ \nbe as in the statement of the theorem.\nLet $\\cH = \\cH_1 \\cup \\cdots \\cup \\cH_M$ be any partition of $\\cH$ \ninto $M$ subsets of equal size.\nConsider the expression\n\\[\n S \n  \\defeq \n  \\hspace{-7pt}  \n   \\sums[N < n \\le 2N][n \\equiv b \\pod{W}]\n    \\hspace{-3pt}\n    \\Big\\{\n     \\sum_{H \\in \\cH} \\ind{\\bP}(n + H)\n     -\n      \\frac{1 + M}{2}a\n       -\n         \\frac{1}{M}\n          \\sum_{j=1}^M\n           \\sums[H,H' \\in \\cH_j][H \\ne H']\n            \\hspace{-5pt}\n             \\ind{\\bP}(n + H)\\ind{\\bP}(n + H')\n    \\Big\\}\n     \\nu_{\\cH}(n),\n\\]\nwhere $\\nu_{\\cH} : \\NN \\to [0,\\infty)$ is the nonnegative weight \ngiven by \n\\[\n \\nu_{\\cH}(n)\n  \\defeq \n   \\Big( \n    \\sums[d_1,\\ldots,d_K][d_i \\mid n + H_i \\,\\, \\forall i \\le K]\n     \\lambda_{d_1,\\ldots,d_K}\n   \\Big)^2,\n\\]\nand where $(\\lambda_{d_1,\\ldots,d_K})$ is the Maynard--Tao sieve \nas used in \\cite[\\S 4]{BFM}.\nThe aim is to show that $S > 0$, for in that case, by the \nobservation made at the beginning of the proof, there must exist \nsome $n \\in (N,2N] \\cap b \\pod{W}$ and $a + 1$ subsets $\\cH_j$ \nfor which $\\#(\\bP \\cap n + \\cH_{j}) \\ge 1$.\n\nAt this point we invoke Lemmas 4.5 and 4.6 in \\cite{BFM} (with \n$4 + O(\\delta)$ in the latter replaced by \n$2\/\\theta + O(\\delta) \\le (M - 1)\/a + O(\\delta)$).\nTo ease notation define $\\mathfrak{S}$ by the relation \n$\n S = \\mathfrak{S}NW^{-1}B^{-K}I_K(F),\n$\nwith $B$ and $I_K(F)$ as defined in \\cite[\\S 4.2]{BFM}.\nAlso let $\\xi = (\\log K)^{-1\/2}$.\n\nAs in \\cite[\\S 4.2]{BFM} and \\cite[(3.13)]{PIN3}, we find that the \nrelevant estimates yield \n\\begin{align*}\n \\mathfrak{S}\n  & \n   \\ge\n    \\sum_{H \\in \\cH} \\frac{aM^2\/(M-1)}{K}(1 + O(\\xi))\n   - \\frac{1 + M}{2}a \n  \\\\\n  & \\hspace{30pt}      \n       - \\frac{1}{M}\n          \\sum_{j=1}^M \n           \\sums[H,H' \\in \\cH_j][H \\ne H']\n            \\frac{M-1}{a}\\cdot \n             \\frac{(aM^2\/(M-1))^2}{K^2}\n             (1 + O(\\delta + \\xi))\n \\\\\n  & \n   =\n    \\frac{aM^2}{M - 1}(1 + O(\\xi))\n     - \n      \\frac{1 + M}{2}a \n       -\n          \\binom{K\/M}{2}\n           \\frac{aM^4(1 + O(\\delta + \\xi))}{K^2(M-1)}\n            .\n\\end{align*}\nRecalling that $\\e^{aM^2\/(\\delta(M-1))} < K$, we see that \n$M\/K < \\delta$ and hence \n\\[\n \\binom{K\/M}{2}\n  = \n   \\frac{K^2}{2M^2}\\Big(1 - \\frac{M}{K}\\Big)\n    =\n       \\frac{K^2}{2M^2}(1 + O(\\delta)).\n\\]\nWe also have $\\xi^2 < \\delta\/(aM) = \\iota^2\/(a^2M^2)$, so \n$\\delta + \\xi = O(\\iota\/(aM))$.\nWe therefore have\n\\begin{align*}\n \\mathfrak{S}\n  & \\ge \n   \\frac{aM^2}{M-1}\\Big(1 + O\\Big(\\frac{\\iota}{aM}\\Big)\\Big) \n  - \\frac{1 + M}{2}a \n   - \\frac{aM^2}{2(M-1)}\\Big(1 + O\\Big(\\frac{\\iota}{aM}\\Big)\\Big)\n  \\\\\n  & =\n    \\frac{a(1 + O(\\iota))}{2(M - 1)}.\n\\end{align*}\nTaking $\\iota$ sufficiently small gives $\\mathfrak{S} > 0$, and \nhence $S > 0$, as desired.\n\\end{proof}\n\n\n\n\n\\section{Main Theorem and Deduction of Theorem \\ref{thm:main}}\n \\label{sec:BFM}\n\nRecall Definition \\ref{def:w}, in which functions ``of the first \nkind'' are introduced.\nWe now define a second kind of function.\n\n\\begin{definition} \n \\label{def:2ndkind}\nWe consider functions $f_2 : [x_2,\\infty) \\to [z_2,\\infty)$, with \n$x_2,z_2 \\ge 1$. \\linebreak\nLet us say that such a function $f_2$ is ``of the second kind'' \nif and only if\n(i) it is a strictly increasing bijection, \n(ii) \n\\[\n (x\/\\log x)\/f_2(x) \\to 0\n  \\quad \n   \\text{and} \n    \\quad \n   f_2(x)\/(x\\log x \\log_3 x\/\\log_2 x) \\to 0\n\\]\nas $x \\to \\infty$ and \n(iii) for any $C > 0$, $f_2(Cx)\/(Cf(x)) \\to 1$ as $x \\to \\infty$.\n\\end{definition}\n\n\\begin{theorem}\n \\label{thm:general}\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ \nof the first kind, and suppose Hypothesis \\ref{hyp:EH} holds.\nFix a function  \n$f_2 : [x_1,\\infty) \\to [z_2,\\infty)$ \nof the second kind and let \n$f \\defeq f_2 \\circ f_1$.\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[f]$ denote the set of limit \npoints in $[0,\\infty]$ of the sequence \n$(d_n\/f(p_n))_{p_n \\ge T_1}$.\nThen given any $\\lceil (2\/\\theta)  \\rceil + 1$ nonnegative real numbers \n$\n\\alpha_1,\\ldots,\\alpha_{\\lceil (2\/\\theta)  \\rceil + 1}\n$ \nwith \n\\[\n \\alpha_1 \\le \\cdots \\le \\alpha_{\\lceil (2\/\\theta)  \\rceil + 1},\n\\] \nwe have \n\\begin{equation}\n \\label{eq:genthm1}\n \\{\\alpha_j - \\alpha_i : 1 \\le i < j \\le \\lceil (2\/\\theta)  \\rceil + 1\\}  \n  \\cap \n   \\LP[f]\n    \\ne \n     \\emptyset.\n\\end{equation}\nConsequently, letting $\\lambda$ denote the Lebesgue measure on \n$\\RR$, we have \n\\begin{equation}\n \\label{eq:genthm2}\n \\lambda([0,X] \\cap \\LP[f])\n  \\ge \n   c_1(\\theta)X\n    \\quad (X \\ge 0)\n\\end{equation}\nand \n\\begin{equation}\n \\label{eq:genthm3}\n \\lambda([0,X] \\cap \\LP[f])\n  \\ge \n   (1 - o(1))\n   c_2(\\theta)X\n    \\quad (X \\to \\infty),\n\\end{equation}\nwhere \n\\begin{equation}\n \\label{eq:genthm4}\n c_1(\\theta)\n  \\defeq\n   (\\lceil (2\/\\theta)  \\rceil(1 + 1\/2 + \\cdots + 1\/\\lceil (2\/\\theta)  \\rceil))^{-1}\n    \\quad\n     \\text{and}\n      \\quad  \n     c_2(\\theta) \\defeq 1\/\\lceil (2\/\\theta)  \\rceil.\n\\end{equation}\n\\end{theorem}\n\n\\vspace*{1em}\n\n\\begin{center}\n \\label{tab:1}\n\\begin{tabular}{|c|c|c|c|} \n \\hline \n $\\theta$                           & $\\lceil (2\/\\theta)  \\rceil + 1$     & $c_1(\\theta)$    & $c_2(\\theta)$  \\\\ \\hline \\hline\n $1\/2 \\le \\theta < 2\/3$             & $5$                                 & $3\/25$           & $1\/4$          \\\\ \\hline \n $2\/3 \\le \\theta < 1\\phantom{\/1}$   & $4$                                 & $2\/11$           & $1\/3$          \\\\ \\hline \n $        \\theta = 1$               & $3$                                 & $1\/3\\phantom{0}$ & $1\/2$          \\\\ \\hline \n\\end{tabular}\n \\vspace*{1em}\n \\captionof{table}{Possible values of $\\lceil (2\/\\theta)  \\rceil + 1$, $c_1(\\theta)$ and $c_2(\\theta)$.}\n\\end{center}\n\n\\begin{proof}[Deduction of Theorem \\ref{thm:main}]\nIn view of Theorem \\ref{thm:BFM4.2}, we may unconditionally apply \nTheorem \\ref{thm:general} with $\\theta = 1\/2$ and any function \n$\nf_1 : [T_1,\\infty) \\to [x_1,\\infty)\n$ \nof the first kind.\nLet \n$\nf_1 : [\\e^{\\e^{\\e}},\\infty) \\to [\\e^{\\e},\\infty)\n$\nbe given by $f_1(T) = \\log T$, and let  \n$\nf_2 : [\\e^{\\e},\\infty) \\to [\\e^{\\e + 1},\\infty)\n$\nbe given by $f_2(x) = x\\log x\/\\log_2 x$.\nThen $f_1$ is of the first kind, $f_2$ is of the \nsecond kind and \n$\n f_2 \\circ f_1(T) \n =\n  R_1(T)\n   =\n    \\log T \\log_2 T\/\\log_3 T\n$\nfor $T \\ge \\e^{\\e^{\\e}}$.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:goodktuple}\nLet $K$ be a natural number and let $K = K_1 + \\cdots + K_M$ be a \npartition of $K$. \nLet $x$ and $y$ be real numbers such that $K \\le y\/x \\le \\log x$.\nIf $x$ is sufficiently large, then for any $M$ \n\\textup{(}possibly overlapping\\textup{)} subintervals \n$(v_i,v_i + x\/\\log x] \\subseteq (x,y]$, $i \\le M$, there exist $M$ \npairwise disjoint sets of primes  \n$\\cH_i \\subseteq (v_i,v_i + x\/\\log x]$ with $|\\cH_i| = K_i$, such \nthat if $\\cH_1 \\cup \\cdots \\cup \\cH_M = \\{q_1,\\ldots,q_K\\}$, then \n$\\prod_{1 \\le i < j \\le K}(q_j - q_i)$ is $x$-smooth.\n\\end{lemma}\n\n\\begin{proof}\nFor any $M$ sets $\\cJ_i$ with $|\\cJ_i| \\ge K$, \n$i \\le M$, there exist $M$ pairwise disjoint sets \n$\\cH_i$ such that $\\cH_i \\subseteq \\cJ_i$ and $|\\cH_i| = K_i$, \n$i \\le M$.\nFor the sets $\\cJ_i$, let $D$ be the integer satisfying \n$y\/x \\le D < 1 + y\/x$.\nAs $D < 1 + \\log x$ and $y \\le x\\log x$, a suitably strong \nversion of the prime number theorem for arithmetic progressions \n(cf.\\ \\cite[\\S22 (4)]{DAV}) yields, for \n$(v_i,v_i + x\/\\log x] \\subseteq (x,y]$, \n\\[\n \\sums[v < p \\le v + x\/\\log x][p \\equiv 1 \\pod{D}] 1\n  =\n   \\frac{x}{\\phi(D)(\\log x)^2} + O\\bigg(\\frac{y}{(\\log y)^5}\\bigg)\n    \\ge \n     \\frac{x}{(\\log x)^3} + O\\bigg(\\frac{x}{(\\log x)^4}\\bigg).\n\\]\nAs $K \\le \\log x$, we see that if $x$ is sufficiently large, \nthen there are at least $K$ primes $p \\equiv 1 \\pod{D}$ in \n$(v_i,v_i + x\/\\log x]$.\nIf $q < q'$ are any two such primes, as $q' - q \\le y$ \nand $q' \\equiv q \\pod{D}$, any prime divisor $p$ of $q' - q$ \nmust either divide $D < 1 + \\log x$ or be less than or equal to \n$y\/D \\le x$.\nHence $q' - q$ is $x$-smooth.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:general}]\nFix $\\theta \\in (0,1]$ and a function \n$f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ \nof the first kind, and suppose Hypothesis \\ref{hyp:EH} holds.\nIn accordance with Theorem \\ref{thm:BFM4.3} (in which we take \n$a = 1$), let $K = K_{\\theta}$ be a sufficiently large integer \nmultiple of $\\lceil (2\/\\theta)  \\rceil + 1$, let $A = A_K$ be \nsufficiently large and  $\\delta = \\delta_{\\theta}$ and \n$\\eta = \\eta(A,K)$ be sufficiently small.\n\nIn accordance with Corollary \\ref{cor:thm2}, let $C$ be a \nsufficiently large but fixed positive constant and let $x$ be a \nsufficiently large number.\nSuppose, as we may, that $Cx \\ge x_1$, and \n(cf.\\ Definition \\ref{def:w} (i)) set \n\\[\n   N \\defeq (f_1^{-1}(Cx))^{1\/\\eta}.\n\\]\nThus, $Cx = f_1(N^{\\eta})$ and $N$ tends to infinity with $x$.\nSuppose $x$ is large enough so that in accordance with \nTheorem \\ref{thm:BFM4.3}, $N \\ge N(T_1,K,\\eta)$.\n\nBy Definition \\ref{def:w} (iv) there exists $L_{\\eta} \\ge 1$ such \nthat $f_1(N)\/x \\to CL_{\\eta}$ as $x \\to \\infty$.\nFix nonnegative real numbers \n$\\alpha_1,\\ldots,\\alpha_{\\lceil (2\/\\theta)  \\rceil + 1}$ with \n$\\alpha_1 \\le \\cdots \\le \\alpha_{\\lceil (2\/\\theta)  \\rceil + 1}$ \nand set \n\\begin{equation}\n \\label{eq:betaeta}\n \\beta_i \\defeq \\alpha_iC L_{\\eta}, \n  \\quad i \\le \\lceil (2\/\\theta)  \\rceil + 1.\n\\end{equation}\n\nFix a function $f_2 : [x_2,\\infty) \\to [z_2,\\infty)$ of the second \nkind and consider the intervals\n\\begin{equation}\n \\label{eq:intervals}\n (x + \\beta_i f_2(x),x + \\beta_i f_2(x) + x\/\\log x],\n  \\quad \n   i \\le \\lceil (2\/\\theta)  \\rceil + 1.\n\\end{equation}\nRecall that $y \\defeq cx\\log x \\log_3 x\/\\log_2 x$, where $c > 0$ \nis a certain constant (cf.\\ \\eqref{eq:fgkmt3.1}). \nBy Definition \\ref{def:2ndkind} (ii) we have \n$x\/\\log x = o(f_2(x))$ and $f_2(x) = o(y)$.\nSuppose, then, that $x$ is large enough (in terms of \n$\\beta_{\\lceil (2\/\\theta)  \\rceil+1}$) so that the intervals in \n\\eqref{eq:intervals} are all contained in $(x,y]$. \n\nIn accordance with Lemma \\ref{lem:goodktuple}, choose \n$\\lceil (2\/\\theta)  \\rceil + 1$ pairwise disjoint sets of primes \n$\\cH_i$ of equal size, with \n\\begin{equation}\n \\label{eq:rightsize}\n \\cH_i \n  \\subseteq  \n   (x + \\beta_i f_2(x),x + \\beta_i f_2(x) + x\/\\log x],\n  \\quad \n   i \\le \\lceil (2\/\\theta)  \\rceil + 1. \n\\end{equation}\nThus, letting \n\\[\n \\cH \n  \\defeq \n   \\cH_1 \\cup \\cdots \\cup \\cH_{\\lceil (2\/\\theta)  \\rceil + 1}\n    \\eqdef \\{q_1,\\ldots,q_K\\},\n\\]\nwe have that $\\prod_{1 \\le i < j \\le K}(q_j - q_i)$ is $x$-smooth, \nand hence $f_1(N^{\\eta})$-smooth (we may suppose that $C \\ge 1$).\n\nAs $K$ is fixed we may of course suppose that $K \\le \\log x$ and \n$p_K \\le x$, so that \\eqref{eq:Kbnd} is satisfied and $\\cH$, being \na set of $K$ primes larger than $p_K$, is admissible.\nWe may of course also suppose that $y \\le N$, so that \n$\\cH \\subseteq [0,N]$.\nThus, $\\cH$ satisfies each of the hypotheses of \nTheorem \\ref{thm:BFM4.3}.\n\nLet $\\cZ(N^{\\eta})$ be as in Hypothesis \\ref{hyp:EH}, so that \n$\\cZ(N^{\\eta})$ is repulsive (cf.\\ Definition \\ref{def:sparse}), \nand note that since $x \\le f_1(N^{\\eta}) \\le \\log N^{\\eta}$ (cf.\\ \nDefinition \\ref{def:w} (ii)), \n\\[\n \\sums[p \\in \\cZ(N^{\\eta})][p \\ge p']\n  \\frac{1}{p}\n   \\ll\n    \\frac{1}{p'}\n     \\ll\n      \\frac{1}{\\log_2 N^{\\eta}}\n       \\le \n        \\frac{1}{\\log f_1(N^{\\eta})}\n         \\le \n          \\frac{1}{\\log x}.\n\\]\nThus, \\eqref{eq:Zsparse} is satisfied with \n$\\cZ = \\cZ(N^{\\eta})$.\nTherefore, by Corollary \\ref{cor:thm2} there exists a vector of \nresidue classes \n$\n(c_p \\pod{p})_{p \\le f_1(N^{\\eta}), \\, p \\, \\not\\in \\, \\cZ(N^{\\eta})}\n$ \nsuch that \n\\begin{equation}\n \\label{eq:Hsurvives}\n \\cH\n  =\n   \\big(\\ZZ \\cap (x,y]\\big)\n    \\setminus \\, \n     {\\textstyle \\bigcup_{p \\le f_1(N^{\\eta}), \\, p \\, \\not\\in \\, \\cZ(N^{\\eta})}} \n       \\, c_p \\pod{p}.\n\\end{equation} \n\nLet $b \\pod{W}$ be the arithmetic progression modulo \n\\[\n W \\defeq \\prods[p \\le f_1(N^{\\eta})][p \\not\\in \\cZ(N^{\\eta})] p\n\\]\nsuch that $b \\equiv  -c_p \\pod{p}$ for all primes \n$p \\le f_1(N^{\\eta})$ with $p \\not\\in \\cZ(N^{\\eta})$.\nBy \\eqref{eq:Hsurvives} we have $(\\prod_{i=1}^K(b + q_i),W) = 1$.\n\nEach hypothesis of Theorem \\ref{thm:BFM4.3} now accounted for, we \nconclude that there is some $n \\in (N,2N] \\cap b \\pod{W}$, and a \npair of indices \n$i_1,i_2 \\in \\{1,\\ldots,\\lceil (2\/\\theta)  \\rceil + 1\\}$, \n$i_1 < i_2$, such that \n\\[\n \\#(\\bP \\cap n + \\cH_{i_1}) \\ge 1\n  \\quad\n   \\text{and}\n    \\quad \n   \\#(\\bP \\cap n + \\cH_{i_2})\n    \\ge  \n     1.\n\\] \nIf there are more than two such indices, we take $i_2 - i_1$ to \nbe minimal.\n\nThus, if $p$ is the largest prime in $\\bP \\cap n + \\cH_{i_1}$ and \n$p'$ is the smallest prime in $\\bP \\cap n + \\cH_{i_2}$, then $p$ \nand $p'$ are {\\em consecutive}, that is, $p = p_t$ and \n$p' = p_{t+1}$ for some $t$.\nIndeed, by \\eqref{eq:Hsurvives} and the definition of \n$b \\pod{W}$, for any $n \\equiv b \\pod{W}$ with \n$n + x \\ge f_1(N^{\\eta})$ we have\n\\[\n \\bP \\cap (n + x,n + y] = \\bP \\cap n + \\cH. \n\\]\n\nBy \\eqref{eq:rightsize} and \\eqref{eq:betaeta}, and since \n$x\/\\log x = o(f_2(x))$, we have \n\\[\n p_{t+1} - p_t\n  =\n  (\\beta_{i_2} - \\beta_{i_1})f_2(x) + O\\Big(\\frac{x}{\\log x}\\Big)\n   =\n    (\\alpha_{i_2} - \\alpha_{i_1} + o(1))CL_{\\eta}f_2(x).   \n\\]\nSince there are only $O(1\/\\theta^2)$ distinct pairs of indices \nfrom which $i_1$ and $i_2$ may be chosen, we deduce that there \nexists a single pair $i_1 < i_2$ such that, for arbitrarily large  \n$N$, we have    \n\\[\n p_{t+1} - p_t \n  = (\\alpha_{i_2} - \\alpha_{i_1} + o(1))CL_{\\eta}f_2(x),\n\\]\nfor some pair of consecutive primes \n$p_t,p_{t+1} \\in (N,N + y] \\subseteq (N,3N]$.\n\nFinally, using Definition \\ref{def:w} (i), (iii) and (iv) and \nDefinition \\ref{def:2ndkind} (i) and (iii), we find that \n$\n CL_{\\eta} f_2(x)\\sim f_2(f_1(N)) \\sim f_2(f_1(3N)).\n$ \nWe conclude that \n\\[\n \\frac{p_{t+1} - p_t}{f_2(f_1(p_t))}\n  = \n   (1 + o(1))(\\alpha_j - \\alpha_i).\n\\]\nWe deduce \\eqref{eq:genthm2} and \\eqref{eq:genthm3} by using the \nargument of \\cite[Corollary 1.2]{BFM}.\n\\end{proof}\n\nAs in \\cite[Theorem 1.3]{BFM}, we may also consider ``chains'' of \nnormalized, consecutive gaps between primes. \nUsing essentially the same argument as above, but using (the \nunconditional) Theorem 4.3 (ii) of \\cite{BFM} in place of \nTheorem \\ref{thm:BFM4.3}, one may verify the \nfollowing result. \n\n\\begin{theorem}\n \\label{thm:chains}\nFix any integer $a$ with $a \\ge 2$.\nFix functions $f_1 : [T_1,\\infty) \\to [x_1,\\infty)$ and \n$f_2 : [x_1,\\infty) \\to [z_2,\\infty)$ of the first and second \nkinds respectively, and let $f \\defeq f_2 \\circ f_1$.\nLet $d_n \\defeq p_{n+1} - p_n$, where $p_n$ denotes the $n$th \nsmallest prime, and let $\\LP[a,f]$ denote the set of limit \npoints in $[0,\\infty]^a$ of the sequence of ``chains'' \n\\[\n \\textstyle \n  \\Big( \n   \\frac{d_n}{f(p_n)},\\ldots,\\frac{d_{n + a - 1}}{f(p_{n + a -1})}\n  \\Big)\n\\]\nfor $p_n \\ge T_1$.\nGiven \n$\\boldsymbol{\\alpha} = (\\alpha_1,\\ldots,\\alpha_K) \\in \\RR^K$, let \n$S_a(\\boldsymbol{\\alpha})$ be the set \n\\[\n  \\big\\{ \n  \\big( \n   \\alpha_{J(2)} - \\alpha_{J(1)},\n    \\ldots,\n     \\alpha_{J(a+1)} - \\alpha_{J(a)}\n  \\big)\n    :\n     1 \\le J(1) < \\cdots < J(a + 1) \\le K\n  \\big)\n \\big\\}.\n\\]\nFor any $8a^2 + 16a$ nonnegative real numbers \n$\\alpha_1 \\le \\cdots \\le \\alpha_{8a^2 + 16a}$, we have \n\\[\n S_a(\\boldsymbol{\\alpha})  \n  \\cap \n   \\LP[f]^a\n    \\ne \n     \\emptyset.\n\\]\n\\end{theorem}\n\nLet us call a function ``reasonable'' if it is of the form \n$f_2 \\circ f_1$, where $f_1$ is a function of the first kind and \n$f_2$ is a function of the second kind. \nTheorem \\ref{thm:chains} shows that for any $a$ there are \ninfinitely many chains of consecutive prime gaps with \n$d_n,\\ldots,d_{n + a - 1} > f(p_n)$ for any reasonable function \n$f$.\nThere are reasonable functions $f$ for which $f(T)\/(R(T)\\log_3 T)$ \ntends to $0$ arbitrarily slowly (recall that \n$R(T) = \\log T\\log_2 T\\log_4 T\/(\\log_3 T)^2$ is the \nErd{\\H o}s--Rankin function).\nWe believe that in a forthcoming paper \\cite{FMT}, Ford, Maynard \nand Tao show that for any $a$ there are infinitely many chains of \nconsecutive prime gaps with \n$d_n,\\ldots,d_{n + a - 1} \\gg R(p_n)\\log_3 p_n$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTwo knots $K$ and $K'$ are \\emph{equivalent} if there is a homeomorphism of \n${\\mathbb S}^{3}$ sending $K$ to $K'$. Given a knot $K \\subset {\\mathbb S}^{3}$ and an integer \n$p \\geq 2$ one can construct the (total space of the) $p$-fold cyclic cover \n$M_{p}(K)$ of ${\\mathbb S}^{3}$ branched along $K$: it is a fundamental object in knot \ntheory. There are non-prime knots all of whose cyclic branched covers are \nhomeomorphic. This is no longer true for prime knots: S. Kojima \\cite{Ko} \nproved that for each prime knot $K \\subset {\\mathbb S}^{3}$ there is an integer $n_{K} \n\\geq 2$ such that two prime knots $K$ and $K'$ are equivalent if their $p$-fold \ncyclic branched covers are homeomorphic for some $p > \\max (n_{K}, n_{K'})$.\n\n\\bigskip\n\nThere are many examples  of prime knots in ${\\mathbb S}^{3}$ which are not equivalent \nbut share homeomorphic $p$-fold cyclic branched covers due to C. Giller \n\\cite{Gi}, C. Livingston \\cite{Li}, Y. Nakanishi \\cite{Na}, M. Sakuma \n\\cite{Sa1}. Moreover there is no universal bound for $n_{K}$.\n\nThe main goal of this article is to study the relationship between prime knots \nand their cyclic branched covers when the number of sheets is an odd prime \nnumber. \n\n\\smallskip\n\n\\begin{Definition} \nLet $K \\subset {\\mathbb S}^3$ be a prime knot. A knot $K' \\subset {\\mathbb S}^3$ which is not \nequivalent to $K$ and which has the same $p$-fold cyclic branched cover as $K$ \nis called a \\emph{$p$-twin} of $K$.\n\\end{Definition}\n\n\\medskip\n\nThere are examples of prime knots, even hyperbolic knots (e.g. Montesinos \nknots) with an arbitrarily large number of non-equivalent $2$-twins. In \ncontrast, for an odd prime number $p$, the number of $p$-twins is very \nrestricted, according to our main result:\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:twins} \nLet $K\\subset {\\mathbb S}^3$ be a prime knot. Then:\n\n\\item{(i)} There are at most two odd prime numbers $p$ for which $K$ admits a \n$p$-twin.\n\n\\item{(ii)} For a given odd prime number $p$, $K$ admits at most one $p$-twin.\n\n\\item{(iii)} Suppose that a prime knot $K$ admits the same knot $K'$ as a \n$p$-twin and a $q$-twin for two distinct odd prime numbers $p$ and $q$. Then \n$K$ has two commuting rotational symmetries of order $p$ and $q$ with trivial \nquotients.\n\\end{Theorem}\n\n\\medskip\n\nA \\emph{rotational symmetry of order $p$} of a knot $K \\subset {\\mathbb S}^3$ is an \norientation preserving periodic diffeomorphism $\\psi$ of the pair $({\\mathbb S}^3, K)$ \nwith period $p$ and non-empty fixed-point set disjoint from $K$. We say that \nthe rotational symmetry $\\psi$ has \\emph{trivial quotient} if $K\/\\psi$ is the \ntrivial knot.\n\nFor hyperbolic knots Theorem \\ref{thm:twins} is in fact a consequence of B. \nZimmermann's result in \\cite{Zim1} whose proof uses the orbifold theorem and \nthe Sylow theory for finite groups. \n\nThe result in Theorem \\ref{thm:twins} is sharp: for any pair of coprime \nintegers $p> q >2$ B. Zimmermann has constructed examples of prime hyperbolic \nknots with the same $p$-fold and $q$-fold branched coverings \\cite{Zim2}. \n\nThe second named author \\cite{Pao2} has proved that a hyperbolic knot is \ndetermined by three cyclic branched covers of pairwise distinct orders. The \nfollowing, straightforward corollary of Theorem \\ref{thm:twins}, shows that a \nstronger conclusion holds for arbitrary prime knots when we focus on branched \ncoverings with odd prime orders.\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor: three covers} A prime knot is determined by three \ncyclic branched covers of pairwise distinct odd prime orders. More \nspecifically, for every knot $K$ there is at least one integer $p_K \\in \n\\{3, 5, 7 \\}$ such that $K$ is determined by its $p_K$-cyclic branched cover. \n\\end{Corollary}\n\n\\medskip\n\nAnother straightforwards consequence of Theorem \\ref{thm:twins} is:\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor:composite}\nLet $K=K_1\\sharp...\\sharp K_t$ and $K'=K'_1\\sharp...\\sharp K'_t$ be two \ncomposite knots with the same cyclic branched covers of orders $p_j$, \n$j=1,2,3$, for three fixed, pairwise distinct, odd prime numbers. Then, after\na reordering, the (non oriented) knots $K_i$ and $K'_i$ are equivalent for all\n$i=1,...,t$.\n\\end{Corollary}\n\n\\medskip\n\nPart (ii) of Theorem \\ref{thm:twins} states that for a given odd prime number \n$p$ a closed, orientable $3$-manifold can be the $p$-fold cyclic branched cover \nof at most two non-equivalent knots in ${\\mathbb S}^3$. In \\cite{BPZ} it has been shown \nthat an integer homology sphere which is a $n$-fold cyclic branched cover of  \n${\\mathbb S}^3$ for four distinct odd prime numbers $n$ is in fact ${\\mathbb S}^3$. By putting \ntogether these two results we get the following corollary:\n\n\\begin{Corollary}\\label{cor:homologysphere} \nLet $M$ be an irreducible integer homology $3$-sphere. Then: there are at most \nthree distinct knots in ${\\mathbb S}^3$ having $M$ as cyclic branched cover of odd prime \norder.\n\\end{Corollary}\n\nOur main task will be to prove Theorem \\ref{thm:twins} for a satellite knot: \nthat is a knot whose exterior ${\\mathbb S}^3\\setminus{\\mathcal U}(K)$ has a non trivial \nJaco-Shalen-Johannson decomposition \\cite{JS}, \\cite{Jo} (in the sequel we use \n$JSJ$-decomposition for short). Otherwise the knot is called simple: in this \ncase, due to Thurston's hyperbolization theorem \\cite{Th2}, its exterior is \neither hyperbolic, and the proof follows already from the works in \\cite{Pao2} \nand \\cite{Zim1}, or it is a torus knot and a simple combinatorial argument \napplies.\n \nThe proof of Theorem \\ref{thm:twins} for satellite knots relies on the study \nof the \\emph{partial symmetries} of the exterior $E(K)$ of $K$ induced by the \ncovering transformations associated to the twins of $K$ and on the localization \nof their axes of fixed points in the components of the $JSJ$-decomposition of \n$E(K)$. In particular the proof uses the following result about rotational \nsymmetries of prime knots which is of interest in its own right.\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:three rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$ admitting three rotational symmetries  with trivial \nquotients and whose orders are three pairwise distinct numbers $>2$. Then $K$ \nis the trivial knot.\n\\end{Theorem}\n\n\\medskip\n\nSince the trivial knot admits a rotational symmetry with trivial quotient of\norder $p$ for each integer $p \\ge 2$, the above Theorem \\ref{thm:three\nrotations} can be interpreted as a characterisation of the trivial knot, i.e. a\nknot is trivial if and only if it admits three rotational symmetries of\npairwise distinct orders $>2$ and trivial quotients. \n\n\n\\section{Rotational symmetries of knots}\n\nA \\emph{rotational symmetry} of order $p$ of a knot $K \\subset {\\mathbb S}^3$ is an \norientation preserving, periodic diffeomorphism $\\psi$ of the pair $({\\mathbb S}^3, K)$ \nof order $p$ and non-empty fixed-point set disjoint from $K$. We say that the \nrotational symmetry $\\psi$ has \\emph{trivial quotient} if $K\/\\psi$ is the \ntrivial knot.\n\n\\smallskip\n\n\\begin{Remark}\\label{rem:lift} \nLet $K$ be a knot and let $\\psi$ be a rotational symmetry of $K$ of order $p$. \nThe symmetry $\\psi$ lifts to a periodic diffeomorphism $\\tilde\\psi$ of the \n$p$-fold branched cover $M_p(K)$ with order $p$ and non-empty fixed-point set, \nwhich commutes with the covering transformation $h$ of $K$ acting on $M_p(K)$. \nThen the symmetry $\\psi$ has trivial quotient if and only if \n$(M,Fix(\\tilde\\psi))\/<\\tilde\\psi> \\cong ({\\mathbb S}^3,K')$. Moreover in this case $K$ \nand $K'$ have a common quotient link with two trivial components (see \n\\cite{Zim1}).\n\nIn particular a symmetry of a knot $K$ induced by the covering transformation \nassociated to a $p$-twin $K'$ of $K$ is a $p$-rotational symmetry with trivial \nquotient. This follows from the fact that the two commuting deck \ntransformations associated to the two twins induce on $M_p(K)$ a \n${\\mathbb Z}\/p{\\mathbb Z} \\oplus {\\mathbb Z}\/p{\\mathbb Z}$-cover of ${\\mathbb S}^3$ branched over a link with two unknotted \ncomponents.\n\\end{Remark}\n\n\\medskip\n\nThe main result of this section is the following theorem whose assertion (i) is \nTheorem \\ref{thm:three rotations}:\n\n\\smallskip\n\n\\begin{Theorem}\\label{thm:rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$.\n\n\\item{(i)} Assume that $K$ admits three rotational symmetries  with trivial \nquotients and whose orders are three pairwise distinct numbers $>2$. Then $K$ \nis the trivial knot.\n\n\\item{(ii)} Assume that $K$ admits two rotational symmetries $\\psi$ and \n$\\varphi$ with trivial quotients and of distinct orders $>2$. Then the \nfixed-point sets $Fix(\\psi)$ and $Fix(\\varphi)$ sit in the $JSJ$-component of \n$E(K)$ which contains $\\partial E(K)$.\n\n\\end{Theorem}\n\n\\medskip\n\nWe prove first a weaker version of Theorem \\ref{thm:three rotations} that we \nshall use in the remaining of this section (see also \\cite[Scholium]{Pao2}).\n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:commuting rotations} \nLet $K$ be a knot in ${\\mathbb S}^3$ admitting three commuting rotational symmetries of\norders $p>q>r\\ge2$. If the symmetries of order $q$ and $r$ have trivial\nquotients, then $K$ is the trivial knot.\n\\end{Proposition}\n\n {\\bf Proof.} \nDenote by $\\varphi$, $\\psi$ and $\\rho$ the three symmetries. If two of them \n-say $\\varphi$, $\\psi$- have the same axis, then by hypothesis the one with \nsmaller order -say $\\psi$- must have trivial quotient, i.e. $K\/\\psi$ is the \ntrivial knot. Since the three symmetries commute, $\\varphi$ induces a \nrotational symmetry of $K\/\\psi$ which is non trivial for the order of \n$\\varphi$ is larger than that of $\\psi$. The axis ${\\mathcal A}$ of this induced symmetry \nis the image of $Fix(\\psi)$ in the quotient by the action of $\\psi$. In \nparticular $K\/\\psi$ and ${\\mathcal A}$ form a Hopf link and $K$ is the trivial knot: this \nfollows from the equivariant Dehn lemma, see \\cite{Hil}. We can thus assume \nthat the axes are pairwise disjoint. Note that even if $r=2$, since the \nsymmetries commute, the symmetry of order $2$ cannot act as a strong inversion \non the axes of the other two symmetries. In this case we would have that the \naxis of $\\rho$, which is a trivial knot, admits two commuting rotational \nsymmetries, $\\varphi$ and $\\psi$, with distinct axes, which is impossible: this \nfollows, for instance, from the fact (see \\cite[Thm 5.2]{EL}) that one can find \na fibration of the complement of the trivial knot which is equivariant with \nrespect to the two symmetries.\n\\qed\n\n\\bigskip\n\nThe proof of Theorem \\ref{thm:rotations} is based on a series of Lemmata. \n\nThe first result concerns the structure of the $JSJ$-decomposition of the \n$p$-fold cyclic branched cover $M$ of a prime knot $K \\subset {\\mathbb S}^3$. Let $h$ be \nthe covering transformation, then the quotient space $M\/<h>$ has a natural \norbifold structure, denoted by ${\\mathcal O}_p(K)$, with underlying space ${\\mathbb S}^3$ and  \nsingular locus $K$ with local group a cyclic group of order $p$ (cf.\n\\cite[Chap. 2]{BMP}). According to Bonahon-Siebenmann \\cite{BS} and the \norbifold theorem \\cite{BoP}, \\cite{CHK}, such an orbifold admits a \ncharacteristic collection of toric $2$-suborbifolds, which split ${\\mathcal O}_p(K)$ \ninto geometric suborbifolds. Moreover this characteristic collection of toric \n$2$-suborbifolds lifts to the $JSJ$-collection of tori for $M$. It follows that \nfor $p > 2$ the Bonahon-Siebenmann characteristic collection of toric \n$2$-suborbifolds coincides with the $JSJ$-collection of tori for the exterior \n$E(K) = {\\mathbb S}^3\\setminus{\\mathcal U}(K)$ of $K$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:JSJ}\nLet $p >2$ be an integer and let $M$ be the $p$-fold cyclic branched cover of a \nprime knot $K$ in the $3$-sphere. Then:\n\n\\item{(a)} The dual graph associated to the $JSJ$-decomposition of $M$ is a \ntree.\n\n\\item{(b)} The fixed-point set of the group of deck transformations is entirely \ncontained in one geometric piece of the decomposition. \n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf (a)} Note, first of all, that $M$ is irreducible since $K$ is \nprime. Hence the Bonahon-Siebenmann decomposition of the orbifold ${\\mathcal O}_p(K)$ \nlifts to the $JSJ$-collection for $M$ since $p>2$. Moreover, the graph dual to \nthe Bonahon-Siebenmann decomposition of the orbifold ${\\mathcal O}_p(K)$, which lifts to \nthe $JSJ$-decomposition for $M$, is a tree. Cutting along a torus of former \ndecomposition and considering the component $C$ which does not contain \n$K$ one gets the complement of a knot in ${\\mathbb S}^3$. The lemma follows now from the \nfact that each connected component of a cyclic branched cover of $C$ has a\nunique boundary component.\n\n\\noindent{\\bf (b)} Note that the group of deck transformations preserves the \n$JSJ$-collection of tori. If $p>2$, the fixed-point set of this group does not \nmeet any torus of the $JSJ$-decomposition, because each $JSJ$-torus is\nseparating and the fixed point set is connected. Since the fixed point set is \nconnected, it is entirely contained in one geometric piece of the \n$JSJ$-decomposition.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\nNote that the conclusion of the first part of the lemma holds also\nfor covers of order $2$. For covers of prime order this\nproperty follows also from the fact that $M_p(K)$ is a ${\\mathbb Z}\/p{\\mathbb Z}$-homology sphere (see\n\\cite{Go}). \n\\end{Remark}\n\n\\medskip\n\n\\begin{Lemma}\\label{lem:prime} \nIf a knot $K \\subset {\\mathbb S}^3$ has a rotational symmetry with trivial quotient, \nthen $K$ is prime.\n\\end{Lemma}\n\n {\\bf Proof.}  \nM. Sakuma \\cite[Thm 4]{Sa2} showed that the only possible rotational symmetries \nof a composite knot must either permute cyclically its prime summands, or act \nas a symmetry of one prime summand while permuting the remaining ones. In \nparticular the quotient knot cannot be trivial.\n\\qed\n\n\\bigskip\n\nThe following is a key lemma for the proofs of Theorems \\ref{thm:twins} and \n\\ref{thm:rotations}. \n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:companion} \nLet $K$ be a knot admitting a rotational symmetry $\\psi$ of order $p>2$ and \nconsider the $JSJ$-decomposition of its exterior $E(K) = {\\mathbb S}^3 \\setminus{\\mathcal U}(K)$.\n\n\\item{(i)} $T$ is a torus of the decomposition which does not separate\n$\\partial E(K)$ from $Fix(\\psi)$ if and only if the orbit $\\psi T$ has $p$ \nelements.\n\n\\item{(ii)} Under the assumption that $\\psi$ has trivial quotient, each torus \nwhich separates $\\partial E(K)$ from $Fix(\\psi)$ corresponds to a prime \ncompanion of $K$ on which $\\psi$ acts with trivial quotient.\n\\end{Lemma}\n\n {\\bf Proof.}  \nLet $T$ be a torus of the $JSJ$-decomposition of $E(K)$ considered as a torus \ninside $S^3$: $T$ separates the $3$-sphere into a solid torus containing $K$ \nand the exterior of a non trivial knot $K_T$ which is a companion of $K$. Note \nthat, since the order of the symmetry $\\psi$ is $>2$, its axis cannot meet $T$. \nAssume that the axis $Fix(\\psi)$ of the symmetry is contained in the solid \ntorus. \n\nIf the orbit of $T$ under $\\psi$ does not contain $p$ elements, then a \nnon-trivial power of $\\psi$ leaves $T$ invariant, and thus it also leaves the \nsolid torus and the knot exterior invariant. The restriction of this power of \n$\\psi$ to the solid torus acts as a rotation of order $m >1$ around its core \nand leaves invariant each meridian. This non-trivial power of $\\psi$ would then \nbe a rotational symmetry about the non trivial knot $K_T$ which is absurd \nbecause of the proof of the Smith's conjecture (see \\cite{MB}). \n\nFor the reverse implication, it suffices to observe that the geometric pieces \nof the decomposition containing $\\partial E(K)$ and $Fix(\\psi)$ must be \ninvariant by $\\psi$, and so must be the unique geodesic segment joining the \ncorresponding vertices in the tree dual to the decomposition.\n\nFor the second part of the Lemma, note that $K_T\/\\psi$ is a companion of \n$K\/\\psi$, which is trivial by hypothesis. In particular $K_T\/\\psi$ is also \ntrivial and thus, by Lemma \\ref{lem:prime}, must be prime.\n\\qed\n\n\\bigskip\n\nThe following lemma gives a weaker version of assertion (ii) of Theorem \n\\ref{thm:rotations} under a commutativity hypothesis:\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:two rotations}\nLet $K$ be a prime knot admitting two commuting rotational symmetries $\\psi$ \nand $\\varphi$ of orders $p,q>2$. Then:\n\n\\item{(i)} The fixed-point sets of $\\psi$ and $\\varphi$ are contained in the\nsame geometric component of the $JSJ$-decomposition for $E(K)$;\n\n\\item{(ii)} If $\\psi$ has trivial quotient and $p \\not = q$, the fixed-point \nsets of $\\psi$ and $\\varphi$ sit in the component which contains \n$\\partial E(K)$.\n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Part (i)}  Let $v_{\\psi}$ (respectively $v_{\\varphi}$) the\nvertex of the graph $\\Gamma_K$ dual to the $JSJ$-decomposition of $E(K)$ \ncorresponding to the geometric component containing $Fix(\\psi)$ (respectively \n$Fix(\\varphi)$). Since the two rotational symmetries commute, $\\psi$ \n(respectively $\\varphi$) must leave $Fix(\\varphi)$ (respectively $Fix(\\psi)$) \ninvariant, and so the geodesic segment of $\\Gamma_K$ joining $v_{\\psi}$ to \n$v_{\\varphi}$ must be fixed by the induced actions of $\\psi$ and $\\varphi$ on \n$\\Gamma_K$. If this segment contains an edge $e$, the corresponding $JSJ$-torus\n$T$ in $E(K)$ cannot separate both $Fix(\\varphi)$ and $Fix(\\psi)$ from \n$\\partial E(K)$. This would contradict part (i) of Lemma \\ref{lem:companion}.\n\n\\medskip\n\n\\noindent{\\bf Part (ii)} Let $M$ be the $p$-fold cyclic branched cover of $K$ \nand let $h$ be the associated covering transformation. According to Remark \n\\ref{rem:lift} the lift $\\tilde \\psi$ of $\\psi$ to $M$ is the deck \ntransformation of a cyclic cover of ${{\\mathbb S}}^3$ branched along a knot $K'$. Note \nthat both $\\tilde \\psi$ and $\\tilde \\varphi$ (the lift of $\\varphi$ to $M$) \ncommute on $M$ with the covering transformation $h$. In particular \n$\\tilde \\varphi$ and $h$ induce commuting rotational symmetries of $K'$ with \norder $q$ and $p$ respectively. According to part $(i)$, $Fix(\\varphi)$ and \n$Fix(h)$ belong to the same piece of the $JSJ$-decomposition of $M$. Since \n$Fix(h)$ maps to $K$ and $p \\not = q$, $Fix(\\varphi)$ sits in the $JSJ$-piece \nof $E(K)$ which contains $\\partial E(K)$ and the conclusion follows since \n$Fix(\\psi)$ belongs to the same $JSJ$-piece as $Fix(\\varphi)$.\n\\qed\n\n\\bigskip\n\n\\begin{Lemma}\\label{lem:torus} \nLet $K$ be a knot admitting a rotational symmetry $\\psi$ with trivial quotient \nand of order $p>2$. Let $M$ be the $p$-fold cyclic branched cover of $K$ and \ndenote by $\\pi:M \\longrightarrow({\\mathbb S}^3,K)$ the associated branched cover. Let \n$T$ be a torus in the $JSJ$-collection of tori of $E(K)$.\n\n\\item{(i)} The torus $T$ is left invariant by $\\psi$ if and only if \n$\\pi^{-1}(T)$ is connected.\n\n\\item{(ii)} If $\\pi^{-1}(T)$ is connected, then the companion $K_T$ of $K$ \ncorresponding to $T$ is prime and the winding number of $T$ with respect to $K$ \nis prime with $p$, so in particular it is not zero.\n\n\\item{(iii)} The torus $T$ is not left invariant by $\\psi$ if and only if \n$\\pi^{-1}(T)$ has $p$ components.\n\\end{Lemma}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Part (i)}. According to Remark \\ref{rem:lift}, the $p$-fold \ncyclic branched cover $M$ of $K$ admits two commuting diffeomorphisms of order \n$p$, $h$ and $h'=\\tilde\\psi $, such that: $(M,Fix(h))\/<h> \\cong ({\\mathbb S}^3,K)$ on \nwhich $h'$ induces the $p$-rotational symmetry $\\psi$ with trivial quotient, \nand $(M,Fix(h'))\/<h'> \\cong ({\\mathbb S}^3,K')$ on which $h$ induces a $p$-rotational \nsymmetry $\\psi'$ with trivial quotient. The preimage $\\pi^{-1}(T) = \\tilde T$ \nis connected if and only if it corresponds to a torus $\\tilde T$ of the \n$JSJ$-decomposition of $M$ which is left invariant by $h$. If, by \ncontradiction, $\\psi$ does not leave $T$ invariant, then the $h'$-orbit of \n$\\tilde T$ consists of $m>1$ elements. Cutting $M$ along these $m$ separating \ntori, one gets $m+1$ connected components. \n\n\\smallskip\n\n\\begin{Claim}\\label{claim:component} \nBoth $Fix(h)$ and $Fix(h')$ must be contained in the same connected component.\n\\end{Claim}\n\n {\\bf Proof.} \nThe diffeomorphism $h'$ cyclically permutes the $m$ connected components which \ndo not contain $Fix(h')$. Since $h$ and $h'$ commute, $h$ leaves invariant each \nof these $m$ components and it acts in the same way on each of them (that is, \nthe restrictions of $h$ to each component are conjugate). Since the set \n$Fix(h)$ is connected, the claim follows.\n\\qed\n\n\\bigskip\n\nThe $m$ components permuted by $h'$ project to a connected submanifold of the \nexterior $E(K')$ of the knot $K'$ with connected boundary the image $T'$ of $\\tilde T$. This submanifold  is invariant by the action of $\\psi'$ \nbut does not contain $Fix(\\psi')$. This contradicts Lemma \n\\ref{lem:companion}(i). To conclude the proof of Lemma \\ref{lem:torus} (i), it \nsuffices to observe that $h$ and $h'$ play symmetric roles.\n\n\\medskip\n\n\\noindent{\\bf Part (ii)} The first part of assertion (ii) is a straightforward \nconsequence of assertion (i) and of Lemma \\ref{lem:companion}. The second part \nfollows from the fact that for $\\pi^{-1}(T)$ to be connected, the winding \nnumber of $T$ and $p$ must be coprime.\n\n\\medskip\n\n\\noindent{\\bf Part (iii)} is a  consequence of the proof of part (i) of Lemma \n\\ref{lem:companion} and of the fact that $h$ and $h'$ play symmetric roles.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of Theorem \\ref{thm:rotations}.} The proof is achieved in three \nsteps.\n\n\\medskip\n\n\\noindent {\\bf Step 1.}  Theorem \\ref{thm:rotations} is true under the \nassumption that the rotational symmetries commute pairwise.\n\nIn this case, assertion (i) is the statement of Proposition \\ref{prop:commuting \nrotations}. Assertion (ii) follows from Lemma \\ref{lem:two rotations}. \n\n\\bigskip\n\n\\noindent{\\bf Step 2.} Theorem \\ref{thm:rotations} is true under the assumption \nthat every companion of $K$ is prime (i.e. $K$ is \\emph{totally prime}) and has \nnon vanishing winding number (i.e. $K$ is \\emph{pedigreed}).\n\nAssume that we are in the hypotheses of Theorem \\ref{thm:rotations}. Then Lemma \n\\ref{lem:prime} assures that $K$ is a prime knot. If $K$ is also totally prime  \nand pedigreed then M. Sakuma \\cite[Thm4 and Lemma 2.3]{Sa2} proved that, up to \nconjugacy, the rotational symmetries belong either to a finite cyclic subgroup \nor to an $S^1$-action in $Diff^{+,+}(S^3,K)$. Thus after conjugacy, step 1 \napplies. For part (ii) note that the distances of the fixed point set of the \nsymmetries to the vertex containing $\\partial E(K)$ in the $JSJ$-graph \n$\\Gamma_ K$ do not change by conjugacy.\n\n\\bigskip\n\n\\noindent {\\bf Step 3.} Reduction of the proof to step 2.\n\nIf $K$ is not totally prime or pedigreed, then it is non-trivial. We shall\nconstruct a non trivial, totally prime and pedigreed knot verifying the \nhypothesis of Theorem \\ref{thm:rotations}. Assertion (i) then follows by \ncontradiction. For Assertion (ii) we need to verify that the construction does \nnot change the distance of the pieces containing the axes of rotations to the \nroot containing $\\partial E(K)$. Roughly speaking we consider the $JSJ$-tori \nclosest to $\\partial E(K)$ and corresponding either to non-prime or to winding \nnumber zero companions. Then we cut $E(K)$ along these tori and keep the \ncomponent $W$ containing $\\partial E(K)$ and suitably Dehn-fill $W$ along these \ntori to get the exterior of a non-trivial knot $\\hat K$ in ${\\mathbb S}^3$, which \nverifies Sakuma's property. \n\nMore precisely, let $\\Gamma_K$ be the tree dual to the $JSJ$-decomposition of \n$E(K)$ and let $\\Gamma_0$ be its maximal (connected) subtree with the following \nproperties: \n\\begin{itemize}\n\n\\item $\\Gamma_0$ contains the vertex $v_\\partial$ corresponding to the \ngeometric piece whose boundary contains $\\partial E(K)$. Note that the\ngeometric piece of the decomposition corresponding to $v_\\partial$ cannot be a\ncomposing space for $K$ is prime;\n\n\\item no vertex of $\\Gamma_0$ corresponds to a composing space (i.e. a space \nhomeomorphic to a product $S^1 \\times B$ where $B$ is an $n$-punctured disc \nwith $n \\geq 2$);\n\n\\item no edge of $\\Gamma_0$ corresponds to a torus whose meridian has linking \nnumber $0$ with $K$.\n\n\\end{itemize}\nDenote by $X(\\Gamma_0)$ the submanifold of $E(K)$ corresponding to $\\Gamma_0$.\n\nThe following claim describes certain properties of $X(\\Gamma_0)$ with respect \nto a rotational symmetry $\\psi$ of $({\\mathbb S}^3,K)$.\n\n\\begin{Claim}\\label{claim:sym}  \nLet $\\psi$ be a rotational symmetry of $({\\mathbb S}^3,K)$ with order $p >2$ and trivial\nquotient. Then:\n\n\\item{(i)} The fixed-point set of $\\psi$ is contained in $X(\\Gamma_0)$.\n\n\\item{(ii)} The tree $\\Gamma_0$ is invariant by the automorphism of $\\Gamma_K$ \ninduced by $\\psi$ and the submanifold $X(\\Gamma_0)$ is invariant by $\\psi$.\n\\end{Claim}\n\n {\\bf Proof.} \n\n\\noindent{\\bf Assertion {(i)}.} Let $\\gamma$ be the unique geodesic segment in \n$\\Gamma_K$ which joins the vertex $v_\\partial$ to the vertex corresponding to \nthe geometric piece containing $Fix(\\psi)$ (see Lemma \\ref{lem:JSJ}; note that \nhere we use $p>2$). According to assertion (ii) of Lemma \\ref{lem:companion}, \nno vertex along $\\gamma_i$ can be a composing space. Since the linking number \nof $K$ and $Fix(\\psi)$ must be coprime with $p$, no torus corresponding to an \nedge of $\\gamma$ can have winding number $0$ (see Lemma \\ref{lem:torus}). \n\n\\medskip\n\n\\noindent{\\bf Assertion {(ii)}.} This is just a consequence of the maximality \nof $\\Gamma_0$ and the fact that elements of the group $\\langle\\psi \\rangle$ \ngenerated by $\\psi$ must preserve the $JSJ$-decomposition of $E(K)$ and the \nwinding numbers of the $JSJ$-tori, as well as send composing spaces to \ncomposing spaces.\n\\qed\n\n\\bigskip\n\nLet $\\pi :M_{p}(K) \\longrightarrow({\\mathbb S}^3,K)$ be the $p$-fold cyclic \nbranched cover. Let $T$ be a torus of the $JSJ$-collection of tori for $E(K)$. \nDenote by $E_T$ the manifold obtained as follows: cut $E(K)$ along $T$ and \nchoose the connected component whose boundary consists only of $T$. Note that \n$E_T$ is the exterior of the companion $K_T$ of $K$ corresponding to $T$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:meridian-longitude}\nLet $T$ be a torus of $\\partial X(\\Gamma_0)\\setminus\\partial E(K)$. The \npreimage $\\pi^{-1}(T)$ consists of $p$ components, each bounding a copy of \n$E_T$ in $M_{p}(K)$. In particular, there is a well-defined\nmeridian-longitude system $(\\mu_T,\\lambda_T)$ on each boundary component of\n$X(\\Gamma_0)$, different from $\\partial E(K)$, which is preserved by taking the\n$p$-fold cyclic branched covers.\n\\end{Claim}\n {\\bf Proof.} \nAccording to Lemma \\ref{lem:torus}, the preimage of $T$ is either connected or \nconsists of $p$ components. If the preimage of $T$ were connected, the tree \n$\\Gamma_0$ would not be maximal according to Lemma \\ref{lem:torus}(ii). The \nremaining part of the Claim is then easy.\n\\qed\n\n\\bigskip\n\nWe wish now to perform Dehn fillings on the boundary of $X(\\Gamma_0)$ in order \nto obtain a totally prime and pedigreed knot admitting pairwise distinct \nrotational symmetries with trivial quotients. On each component $T$ of\n$\\partial X(\\Gamma_0)\\setminus\\partial E(K)$ we fix the curve $\\alpha_n = \n\\lambda_T+n\\mu_T$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:surgery} \nFor all but finitely many $n \\in {\\mathbb Z}$ the Dehn filling of each component $T$ of \n$\\partial X(\\Gamma_0)\\setminus\\partial E(K)$ along the curve $\\alpha_n$\nproduces the exterior of a non-trivial, prime and pedigreed knot $\\hat K$ in \n${\\mathbb S}^3$.\n\\end{Claim}\n\n {\\bf Proof.} \nNote that by the choice of surgery curves the resulting manifold \n$\\hat X(\\Gamma_0)$ is the exterior of a knot $\\hat K$ in the $3$-sphere, i.e. \n$\\hat X(\\Gamma_0)\\subset {\\mathbb S}^3$, and thus is irreducible. We distinguish two \ncases:\n\n\\medskip\n\n\\noindent{\\bf {(1)}} The $JSJ$-component $X_T$ of $X(\\Gamma_0)$ adjacent to $T$ \nis Seifert fibred. Then, by the choice of $\\Gamma_0$, $X_T$ is a cable space \n(i.e. the exterior of a $(a,b)$-torus knot in the solid torus bounded by $T$ in\n${\\mathbb S}^3$). Moreover the fiber $f$ of the Seifert fibration of $X(\\Gamma_0)$ is \nhomologous to $a\\mu_{T} + b\\lambda_{T}$ on $T$ and the intersection number\n$\\vert \\Delta(f,\\mu_T) \\vert = b > 1$. The intersection number of the filling \ncurve $\\alpha_n$ with the fiber $f$ is then $\\vert \\Delta(f,\\alpha_n) \\vert =\n\\vert na -b \\vert$ and is $> 1$  for all but finitely many $n \\in {\\mathbb Z}$. In this \ncase the resulting manifold $X_T(\\alpha_n)$ is the exterior of a non trivial \ntorus knot which is prime and pedigreed \\cite{CGLS}.\n\n\\medskip\n\n\\noindent {\\bf{( 2)}} The $JSJ$-component $X_T$ of $X(\\Gamma_0)$ adjacent to \n$T$ is hyperbolic. By Thurston's hyperbolic Dehn filling theorem \n\\cite[Chap. 5]{Th1} (see also \\cite[Appendix B]{BoP}) for all but finitely many \n$n \\in {\\mathbb Z}$ the Dehn filling of each component $T \\subset \\partial X_T \\cap \n(\\partial X(\\Gamma_0)\\setminus\\partial E(K))$ along the curve $\\alpha_n$ \nproduces a hyperbolic manifold $X_T(\\alpha_n)$ with finite volume.\n\n\\medskip\n\nTherefore for all but finitely many $n$'s $\\in {\\mathbb Z}$ the Dehn filling of each \ncomponent $T \\subset \\partial X(\\Gamma_0)\\setminus\\partial E(K)$ along the \ncurve $\\alpha_n$  produces a $\\partial$-irreducible $3$-manifold $\\hat \nX(\\Gamma_0) \\subset {\\mathbb S}^3$ such that each Seifert piece of its \n$JSJ$-decomposition is either a Seifert piece of $X(\\Gamma_0)$ or a non-trivial \ntorus knot exterior. Hence it corresponds to the exterior of a non-trivial knot \n$\\hat{K} \\subset {\\mathbb S}^3$ which is totally prime. It is also pedigreed by the \nchoice of $\\Gamma_0$.\n\\qed\n\n\\bigskip\n\nLet $\\psi$ a rotational symmetry of $({\\mathbb S}^3,K)$ with order $p >2$. Then the \nrestriction ${\\psi}_{\\vert_{X(\\Gamma_0)}}$, given by Claim \\ref{claim:sym} \nextends to $\\hat X(\\Gamma_0)$, giving a $p$-rotational symmetry $\\hat\\psi$ of \nthe non-trivial, totally prime and pedigreed knot $({\\mathbb S}^3,\\hat K)$. In order to \nbe able to apply  step 2 to the knot $\\hat K$ and the induced rotational \nsymmetries, we still need to check that the rotational symmetry $\\hat \\psi$ has \ntrivial quotient when $\\psi$ has trivial quotient. This is the aim of the \nfollowing:\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:quotient} \nIf the knot $K\/\\psi$ is trivial, then the knot $\\hat K\/\\hat\\psi$ is trivial.\n\\end{Claim}\n\n {\\bf Proof.}  \nLet $\\pi:M_{p}(K) \\longrightarrow({\\mathbb S}^3,K)$ be the $p$-fold cyclic \nbranched cover. Let $h$ be the deck transformation of this cover and $h'$ \nthe lift of $\\psi$. According to Remark \\ref{rem:lift}, $h'$ is the deck \ntransformation for the $p$-fold cyclic cover of the $3$-sphere branched along \na knot $K'$. Note that, by Claim \\ref{claim:meridian-longitude},\n$M_{p}(K)\\setminus\\pi^{-1}(X(\\Gamma_0) \\cup {\\mathcal U}(K))$ is a disjoint union of $p$ \ncopies of $E(K) \\setminus X(\\Gamma_0)$. It follows that the $p$-fold cyclic \nbranched cover $M_{p}(\\hat K)$ of $\\hat K$ is the manifold obtained by a \n$(\\lambda_T+n\\mu_T)$-Dehn filling on all the boundary components of \n$\\pi^{-1}(X(\\Gamma_0) \\cup {\\mathcal U}(K))$. The choice of the surgery shows that both \n$h$ and $h'$ extend to diffeomorphisms $\\hat h$ and $\\hat h'$ of order \n$p$ of $M_{p}(\\hat K)$. By construction we have that $M_{p}(\\hat K)\/<\\hat\nh> \\cong {\\mathbb S}^3$. In the same way $M_{p}(\\hat K)\/<\\hat h'>$ is obtained\nfrom $M_{p}(K)\/<h'> \\cong{\\mathbb S}^3$ by cutting off a copy of $E(K) \\setminus\nX(\\Gamma_0)$ and Dehn filling along $\\partial X(\\Gamma_0)$. The choice of\nthe surgery curve assures that the resulting manifold is again ${\\mathbb S}^3$ and the \nconclusion follows from Remark \\ref{rem:lift}.\n\\qed\n\n\\bigskip\n\nFrom the non-trivial prime knot $K$, we have thus constructed a non-trivial, \ntotally prime and pedigreed knot $\\hat K$ which has the property that every \nrotational symmetry $\\psi$ of $K$ with trivial quotient and order $>2$  \ninduces a rotational symmetry $\\hat \\psi$ of $\\hat K$ with trivial quotient and \nthe same order. Moreover by the choice of the Dehn filling curve in the \nconstruction of $\\hat K$, the vertex containing $Fix(\\hat \\psi)$ remains at the \nsame distance from the vertex containing $\\partial E(\\hat K)$ in the \n$JSJ$-tree $\\Gamma_ {\\hat K}$ as the vertex containing $Fix(\\psi)$ from the \nvertex containing $\\partial E(K)$ in the $JSJ$-tree $\\Gamma_ K$. Then the \nconclusion is a consequence of step 2.\\qed \n\n\n\\section{Twins of a prime knot}\n\nIn this section we prove Theorem \\ref{thm:twins}. If $K$ is trivial, the\ntheorem is a consequence of the proof of Smith's conjecture (see \n\\cite{MB}). We shall thus assume in the remaining of this section that $K$ is \nnon trivial and $p$ is an odd prime number.\n\nLet $M$ be the common $p$-fold cyclic branched cover of two prime knots $K$ and \n$K'$ in ${\\mathbb S}^3$. Let $h$ and $h'$ be the deck transformations for the coverings \nof $K$ and $K'$ respectively. By the orbifold theorem \\cite{BoP}, see also \n\\cite{CHK} one can assume that $h$ and $h'$ act \\emph{geometrically} on the \ngeometric pieces of the $JSJ$-decomposition of $M$, i.e. by isometries on the \nhyperbolic pieces and respecting the fibration on the Seifert fibred ones.\n\nThe following lemma describes the Seifert fibred pieces of the \n$JSJ$-decomposition of the $p$-fold branched cyclic cover $M$ (see also\n\\cite{Ja} and \\cite[Lemma 2]{Ko}).\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:seifert}\nLet $p$ be an odd prime integer and let $M$ be the $p$-fold cyclic branched \ncover of ${\\mathbb S}^3$ branched along a prime, satellite knot $K$. If $V$ is a Seifert \npiece in the $JSJ$-decomposition for $M$. Then the base $B$ of $V$ can be:\n\n\\begin{enumerate}\n\n\\item A disc with $2$, $p$ or $p+1$ singular fibres;\n\n\\item A disc with $1$ hole, i.e. an annulus, with $1$ or $p$ singular fibres;\n\n\\item A disc with $p-1$ holes and $1$ singular fibre;\n\n\\item A disc with $p$ holes and $1$ singular fibre;\n\n\\item A disc with $n$ holes, $n\\ge2$.\n\n\\end{enumerate}\n\\end{Lemma}\n\n {\\bf Proof.} \nIt suffices to observe that $V$ projects to a Seifert fibred piece $V'$ of the \nBonahon-Siebenmann decomposition for the orbifold ${\\mathcal O}_p(K)$. There are four \npossible cases:\n\n\\noindent{\\bf (a)} $V'$ contains $K$: $V'$ is topologically a non trivially \nfibred solid torus and $K$ is a regular fibre of the fibration, i.e. a torus \nknot $K(a,b)$, since it cannot be the core of the fibred solid torus. The knot\n$K$ lifts to a singular fibre of order $p$ if $p$ does not divide $ab$ and to \na regular fibre otherwise. The core of the solid torus is a singular fibre of \norder -say- $a$. It lifts to a regular fibre if $a=p$, a singular fibre of \norder $a\/(a,p)$ if $p$ does not divide $b$, or to $p$ singular fibres of order \n$a$ if $p$ divides $b$. Thus $V$ has $p$ boundary components if $p$ divides $a$ \nand $1$ otherwise. An Euler characteristic calculation shows that $B$ is either \na disc with $2$ or $p$ singular fibres, or a disc with $p-1$ holes and with at \nmost $1$ singular fibre.\n\n\\noindent{\\bf (b)} $V'$ is the complement of a torus knot $K(a,b)$ in ${\\mathbb S}^3$. \nIn this case, $V$ is either a copy of $V'$, and $B$ is a disc with $2$ singular\nfibres or $V$ is a true $p$-fold cover of $V'$. In this case $V$ has exactly\none boundary component. Reasoning as in case (a), we see that the two singular\nfibres of $V'$ must lift to either $2$ singular fibres, or $1$ regular fibre\nand $p$ singular fibres or $1$ singular fibre and $p$ singular fibres. In\nparticular $B$ is a disc with $2$, $p$ or $p+1$ singular fibres.\n\n\\noindent{\\bf (c)}  $V'$ is the complement of a torus knot $K(a,b)$ in a solid \ntorus, i.e. a cable space, and its base is an annulus with $1$ singular fibre. \nReasoning as in (b) we find that $B$ can be a disc with $1$ hole and $1$ or $p$ \nsingular fibres or a disc with $p$ holes and at most $1$ singular fibre.\n\n\\noindent{\\bf (d)}  $V'$ is a composing space with at least $3$ boundary\ncomponents and thus so is $V$. More precisely, note that either $V'$ lifts to\n$p$ disjoint copies of itself, or $V$ and $V'$ are homeomorphic and $V'$ is \nobtained by quotienting $V$ via the $p$-translation along the ${\\mathbb S}^1$ fibre. In \nthis case $B$ is a disc with at least $2$ holes.\n\nThis analysis ends the proof of Lemma \\ref{lem:seifert}.\n\\qed\n\n\\bigskip\n\n\\begin{Proposition}\\label{prop:subtree}\nLet $M$ be the common $p$-fold cyclic branched cover of two prime knots $K$ and \n$K'$ in ${\\mathbb S}^3$, $p$ an odd prime number, and let $h$ be the deck transformation \nfor the covering of $K$. Let $\\Gamma$ be the tree dual to the\n$JSJ$-decomposition of $M$. The deck transformation $h'$ for the covering of\n$K'$ can be chosen (up to conjugacy) in such a way that:\n\n\\item{(i)} There exists a subtree $\\Gamma_f$ of $\\Gamma$ on which the actions\ninduced by $h$ and $h'$ are trivial;\n\n\\item{(ii)} The vertices of $\\Gamma$ corresponding to the geometric pieces of\nthe decomposition which contain $Fix(h)$ and $Fix(h')$ belong to $\\Gamma_f$;\n\n\\item{(iii)} Let $M_f$ the submanifold of $M$ corresponding to $\\Gamma_f$. The\nrestrictions of $h$ and $h'$ to $M_f$ commute.\n\\end{Proposition}\n\n {\\bf Proof.}  \nThe proof relies on the study of the actions of the two covering \ntransformations $h$ and $h'$ on the $JSJ$-decomposition of the common $p$-fold  \ncyclic branched covering $M$. Since $\\Gamma$ is finite, the group generated by \nthe tree automorphisms induced by $h$ and $h'$ is finite as well. Standard \ntheory of group actions on trees assures that a finite group acting on a tree \nwithout inversion must have a global fixed point and that its fixed-point set \nis connected. Thus part (i) of the proposition follows, using the fact that $h$ \nand $h'$ have odd orders.\n\n\\medskip\n\nChoose now $h'$, up to conjugacy in $Diff^+(M)$, in such a way that\n$\\Gamma_f$ is maximal. We want to show that, in this case, $M_f$ contains \n$Fix(h)$ and $Fix(h')$. Assume by contradiction that the vertex $v_h$ of \n$\\Gamma$ corresponding to the geometric piece containing $Fix(h)$, whose\nexistence is ensured by Lemma \\ref{lem:JSJ}, does not belong to $\\Gamma_f$. Let \n$\\gamma_h$ the unique geodesic path in $\\Gamma$ connecting $v_h$ to $\\Gamma_f$. \nLet $e_h$ the edge in $\\gamma_h$ adjacent to $\\Gamma_f$ and denote by $T$ the \ncorresponding torus of the $JSJ$-collection of tori for $M$. Let $U$ be the \nconnected component of $M\\setminus T$ which contains $Fix(h)$. Consider the \n$\\langle h,h'\\rangle$-orbit of $U$. This orbit is the disjoint union of $h$ \n(and $h'$) orbits of $U$. Remark that the $h$-orbit of $U$ is $\\{U\\}$.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:orbit}\nThe orbit $\\langle h,h'\\rangle U$ must contain an $h$-orbit, different from \n$\\{U\\}$ and containing a unique element.\n\\end{Claim}\n\n {\\bf Proof.}  \nOtherwise all the $h$-orbits in $\\langle h,h'\\rangle U$ different from $\\{U\\}$ \nwould have $p$ elements, since $p$ is prime. In particular, the cardinality of \n$\\langle h,h'\\rangle U$ would be of the form $kp+1$. This implies that at least \none of the $h'$-orbits in $\\langle h,h'\\rangle U$ must contain one single \nelement $U'$. Up to conjugacy with an element of $\\langle h,h'\\rangle$ (whose \ninduced action on $\\Gamma_f$ is trivial), we can assume that $U=U'$, \ncontradicting the hypothesis that $h'$ was chosen up to conjugacy in such a \nway that $\\Gamma_f$ is maximal.\n\\qed\n\n\\bigskip\n\nLet $U'\\neq U$ the element of $\\langle h,h'\\rangle U$ such that $h(U')=U'$. \nNote that $U$ and $U'$ are homeomorphic since they belong to the same \n$\\langle h,h'\\rangle$-orbit.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:knot complement}\n$U$ is homeomorphic to the exterior $E({{\\mathcal K}})$ of a knot ${\\mathcal K} \\subset {\\mathbb S}^3$\nadmitting a free symmetry of order $p$.\n\\end{Claim}\n\n {\\bf Proof.}  \nThe first part of the Claim follows from the fact that, by maximality of \n$\\Gamma_f$, $h'$ cannot leave $U$ invariant, so must freely permute $p$ copies\nof $U$ belonging to $\\langle h,h'\\rangle U$. Thus $U$ must appear as a union of\ngeometric pieces of the $JSJ$-splitting of $E(K')$. The second part follows \nfrom the fact that $h$ must act freely on $U'$ which is homeomorphic to $U$.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:free quotient} \nNote that the quotient of $U$ by the action of its free symmetry of order $p$ \nis also a knot exterior because $h$ acts freely on $U'$ and $U'$ must project \nto a union of geometric pieces of the $JSJ$-splitting of $E(K)$.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Claim}\\label{claim:non-free quotient}\n$U$ admits a rotational symmetry of order $p$ whose quotient $U\/\\langle \nh\\rangle$ is topologically a solid torus.\n\\end{Claim}\n\n {\\bf Proof.} \nThe quotient $U\/\\langle h\\rangle$ is obtained by cutting ${\\mathbb S}^3$ along an \nessential torus in $E(K)$. Since $K \\subset U\/\\langle h\\rangle$, it must be a \nsolid torus. \n\\qed\n\n\\bigskip\n\nIt follows from Claim \\ref{claim:non-free quotient} and Lemma \\ref{lem:prime} \nthat the knot ${\\mathcal K}$ is prime. Moreover, according to Claims \\ref{claim:knot \ncomplement} and \\ref{claim:non-free quotient}, ${\\mathcal K}$ admits a rotational \nsymmetry and a free symmetry, both of order $p$. This is however impossible  \nbecause M. Sakuma \\cite[Thm. 3]{Sa2} showed that a prime knot can only have one \nsymmetry of odd order up to conjugacy. This contradiction proves part (ii) of \nProposition \\ref{prop:subtree}.\n\n\\medskip\n\nTo prove part (iii) we shall consider two cases, according to the structure\nof $\\Gamma_f$.\n\n\\medskip\n\n\\noindent {\\bf Case (a)}: {\\it $\\Gamma_f$ contains an edge.}\nChoose an edge in $\\Gamma_f$ and let $T$ be the corresponding torus in the\n$JSJ$-collection of tori for $M$. Let $V$ be a geometric piece of the \n$JSJ$-decomposition of $M$ adjacent to $T$. Then Lemma \\ref{lem:commutation} \nbelow together with a simple induction argument show that $h'$ can be chosen \n(up to conjugacy) in such a way that its restriction to $M_f$ commutes with the \nrestriction of $h$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:commutation}\nIf the covering transformations $h$ and $h'$ preserve a $JSJ$-torus $T$ of $M$ \nthen, up to conjugacy in $Diff^{+}(M)$, $h$ and $h'$ commute on the union of \nthe geometric components of the $JSJ$-decomposition adjacent to $T$.\n\\end{Lemma}\n\n {\\bf Proof.} \nFirst we show that $h$ and $h'$ commute on each geometric component adjacent to \n$T$. Since $h$ and $h'$ preserve the orientation of $M$, we deduce that \n$h(V)=V$ and $h'(V)=V$, and that $h$ and $h'$ act geometrically on the \ngeometric piece $V$. A product structure on $T$ can always be induced by the \ngeometric structure on $V$: either by considering the induced Seifert fibration \non $T$ if $V$ is Seifert fibred, or by identifying $T$ with a section of a cusp \nin the complete hyperbolic manifold $V$. Since $h$ and $h'$ are isometries of \norder $p$, for such a product structure on $T$ they act as (rational) \ntranslations, i.e. their action on $T={{\\mathbb S}}^1\\times{{\\mathbb S}}^1$ is of the form \n$(\\zeta_1,\\zeta_2) \\mapsto (e^{2i\\pi r_1\/p}\\zeta_1,e^{2i\\pi r_2\/p}\\zeta_2)$, \nwhere $p$ and at least one between $r_1$ and $r_2$ are coprime. Thus $h$ and \n$h'$ commute on $T$.\n\n\\medskip\n\nIf $V$ is hyperbolic, we have just seen that $h$ and $h'$ are two isometries of \n$V$ which commute on the cusp corresponding to $T$. Thus they must commute on \n$V$.\n\n\\medskip\n\nIf $V$ is Seifert fibred, then the Seifert fibration is unique up to isotopy, \nand $h$ and $h'$ preserve this fibration. \n\n\\smallskip\n\n\\begin{Remark}\nNote that the quotient of $V$ by a fiber-preserving diffeomorphism of\nfinite order $h$ only depends on the combinatorial behaviour of $h$, i.e. its\ntranslation action along the fibre and the induced permutation on cone points \nand boundary components of the base. In particular, the conjugacy class of $h$ \nonly depends on these combinatorial data. Note moreover that two geometric\nsymmetries having the same combinatorial data are conjugate via a\ndiffeomorphism isotopic to the identity.\n\\end{Remark}\n\n\\medskip\n\nSince the translation along the fibres commutes with every fiber-preserving \ndiffeomorphism of $V$, it suffices to see whether $h$ and $h'$ commute, up to a \nconjugation of $h'$, on the base $B$ of $V$. It is enough then to consider the \npossible actions of order $p$ on the possible bases. According to Lemma \n\\ref{lem:seifert} the possible actions of $h$ and $h'$ are described below:\n\n\\begin{enumerate}\n\n\\item If $B$ is a disc with $2$ singular fibres, or an annulus with $1$ \nsingular fibre, or a disc with $n$ holes, $n\\neq p$, or a disc with $p-1$ holes\nand $1$ singular fibre, then the action on $B$ is necessarily trivial and there \nis nothing to prove. Note that, according to the proof of Lemma \n\\ref{lem:seifert}, if $B$ is a disc with $p-1$ holes with one singular fibre, \nno boundary torus is left invariant, so this possibility in fact does not \noccur.\n\n\\item If $B$ is a disc with $p$ holes and $1$ singular fibre or a disc with \n$p+1$ singular fibres, then the only possible action is a rotation about a \nsingular fibre cyclically permuting the holes or the remaining singular fibres. \n\n\\item If $B$ is a disc with $p$ singular fibres then the action must be a\nrotation about a regular fibre which cyclically exchanges the singular fibres.\n\n\\item If $B$ is an annulus with $p$ singular fibres the action must be a free\nrotation cyclically exchanging the singular fibres. Note that in the three\nlatter cases the action can never be trivial on the base.\n\n\\item If $B$ is a disc with $n$ holes then two situations can arise: either the\naction is trivial on the base (case (d) in the proof of Lemma \n\\ref{lem:seifert}; note that in case (a), when $n=p-1$, all boundary components \nmust be cyclically permuted), or $n=p$ and the action is a rotation about a \nregular fibre which cyclically permutes the $p$ holes (see part (c) of Lemma \n\\ref{lem:seifert}).\n\n\\end{enumerate}\n\nWe shall now show that, if both $h$ and $h'$ induce non trivial actions on the\nbase of $V$, then, up to conjugacy, $h$ and $h'$ can be chosen so that their \nactions on $B$ coincide. Note that for $h$ and $h'$ to commute it suffices that \nthe action of $h'$ on $B$ coincides with the action of some power of $h$, \nhowever this stronger version will be needed in the proof of Corollary\n\\ref{cor:extension}.\n\nFirst of all remark that, if $B$ is a disc with $p+1$ singular fibres (case 2) \nand $h$ and $h'$ leave invariant distinct singular fibres, then all the\nsingular fibres must have the same order (in fact, must have the same\ninvariants). This means that, after conjugating $h'$ by a homeomorphism of $V$\nwhich is either an isotopy exchanging two regular fibres or a Dehn twist along\nan incompressible torus exchanging two singular fibres, one can assume that, in\ncases 2 and 3, $h$ and $h'$ leave set-wise invariant the same fibre. Note that\nthis homeomorphism is isotopic to the identity on $\\partial V$ and thus extends\nto $M$. In fact, using Lemma \\ref{lem:seifert} one can show that the fibres\ncannot all have the same order.\n\nSince the actions of $h$ and $h'$ consist in permuting exactly $p$ holes or \nsingular fibres, it suffices to conjugate $h'$ via a homeomorphism of $V$ \n(which is a composition of Dehn twists along incompressible tori) in such a way \nas to exchange the order of the holes or singular fibres so that $h'$ and $h$\ncyclically permute them in the same order. Note that in the case of singular \nfibres this product of Dehn twists is isotopic to the identity on $\\partial V$ \nand thus extends to $M$. In the case of holes, the product of Dehn twists \nextends to $M$ since it induces the identity on the fundamental groups of the \ntori of $\\partial V$ and the connected components of $M\\setminus V$ adjacent to \nboundary tori different from $T$ are necessarily homeomorphic. \n\nOnce the two diffeomorphisms $h$ and $h'$ commute on the two geometric pieces \nadjacent to $T$, the commutation can be extended on a product neighborhood of \n$T$, since the two finite abelian groups generated by the restrictions of $h$ \nand $h'$ on each side of $T$ have the same action on $T$. Indeed, the slope of \nthe translation induced by $h'$ on $T$ has been left unchanged by the \nconjugation.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:same action}\nNote that in case 1 of the proof of the above Lemma, the actions of $h$ and\n$h'$ must coincide after taking a power, i.e. $h$ and $h'$ generate the same\ncyclic group. This is not necessarily true in the remaining cases, even if $h$ \nand $h'$ induce the same action on $B$. Indeed, they can induce different\ntranslations along the fibres. Nevertheless, in both cases, to assure that the\nactions of $h$ and $h'$ coincide on $V$, it suffices to check that they \ncoincide on $T$.\n\\end{Remark}\n\n\\medskip\n\n\\noindent {\\bf Case (b)}: {\\it $\\Gamma_f$ is a single vertex.}\nLet $V=M_f$ be the geometric piece corresponding to the unique vertex of \n$\\Gamma_f$. If $V=M$, then the result is already known. We can thus assume that\n$V\\neq M$. According to part (ii) of Proposition \\ref{prop:subtree}, we \ncan assume that the fixed-point sets of $h$ and $h'$ are contained in $V$.\nIf $V$ is Seifert fibred then, case (a) of the proof of Lemma \\ref{lem:seifert} \nshows that the base $B$ of $V$ is either a disc with $2$ or $p+1$ singular \nfibres, or a disc with $p-1$ holes and with $1$ or $2$ singular fibres. In the \nfirst case the boundary torus of $V$ is preserved by $h$ and $h'$ and the \nassertion follows from Lemma \\ref{lem:commutation}. In the second case the \naction on the base is necessarily a rotation fixing two points (either the \nunique singular fibre and a regular one, or the two singular fibres) and \ncyclically permuting the $p$ boundary components. Then conjugating $h'$ by a \nproduct of Dehn twists along incompressible tori, which extends to $M$ as in \nthe proof of Lemma \\ref{lem:commutation}, leads to the desired conclusion.\n\n\\medskip\n\nThe case where $V$ is hyperbolic is due to B. Zimmermann \\cite{Zim1}. We give \nthe argument for completeness. Since $V$ is hyperbolic, we consider the group \n${\\mathcal I}_V$ of isometries of $V$ induced by diffeomorphisms of $M$ which leave $V$ \ninvariant. Let ${\\mathcal S}$ be the $p$-Sylow subgroup of ${\\mathcal I}_V$. Up to conjugacy, we \ncan assume that both $h=h_{\\vert_V}$ and $h'=h'_{\\vert_V}$ belong to ${\\mathcal S}$. If \nthe groups $\\langle h\\rangle$ and $\\langle h'\\rangle$ generated by $h$ and $h'$ \nare conjugate, we can assume that $h=h'$ and we are done. So we assume that \n$\\langle h\\rangle$ and $\\langle h'\\rangle$ are not conjugate. Then it suffices \nto prove that $h'$ normalises $\\langle h\\rangle$ because each element \nnormalising $\\langle h\\rangle$ must leave invariant $Fix(h)$ and the subgroup \nof ${\\mathcal I}_V$ which leaves invariant a simple closed geodesic, like $Fix(h)$, must \nbe a finite subgroup of ${\\mathbb Z}\/2{\\mathbb Z}\\ltimes({\\mathbb Q}\/{\\mathbb Z}\\oplus{\\mathbb Q}\/{\\mathbb Z})$. In particular, \nelements of odd order must commute. Assuming that $\\langle h\\rangle$ and \n$\\langle h'\\rangle$ are not conjugate, we have that $\\langle h\\rangle \n\\subsetneq {\\mathcal S}$ and, by \\cite[Ch 2, 1.5]{Su}, either $\\langle h\\rangle$ is \nnormal in ${\\mathcal S}$ and we have reached the desired conclusion, or there exist an \nelement $\\hat{h}=ghg^{-1}$, conjugate to $h$ in ${\\mathcal S}$, which normalises \n$\\langle h\\rangle$ and such that \n$\\langle h\\rangle \\cap\\langle{\\hat{h}}\\rangle=\\{1\\}$.\n\nWe want to show that $h'$ normalises $\\langle h\\rangle$. Assume, by \ncontradiction that $h'$ is not contained in $\\langle h,\\hat{h}\\rangle =\n{\\mathbb Z}\/p{\\mathbb Z}\\oplus{\\mathbb Z}\/p{\\mathbb Z}$. Then this group is smaller than ${\\mathcal S}$ and again we are \nable to find a new cyclic group $H$ of order $p$ whose intersection with \n$\\langle h,\\hat{h}\\rangle$ is reduced to the identity and which normalises \n$\\langle h,\\hat{h}\\rangle$. Since the order of $H$ is an odd prime number and \nsince $\\langle h\\rangle$ and $\\langle\\hat{h}\\rangle$ are the only subgroups of \n$\\langle h,\\hat{h}\\rangle$ which fix point-wise a geodesic by \\cite[Proposition \n4]{MZ}, $H$ would commute with $\\langle h,\\hat{h}\\rangle$ which is a \ncontradiction to the structure of a group leaving a geodesic invariant. This \nfinal contradiction shows that, up to conjugacy, the subgroups $\\langle \nh\\rangle$ and $\\langle h'\\rangle$ either commute or coincide on $V$. This \nfinishes the proof of Proposition \\ref{prop:subtree}.\n\\qed\n\n\\bigskip\n\nThe following proposition shows that a prime knot $K$ having a $p$-twin either \nadmits a rotational symmetry of order $p$, or a well-specified submanifold \n$E_p(K)$ built up of geometric pieces of the $JSJ$-decomposition of $E(K)$ \nadmits a symmetry of order $p$ with non-empty fixed-point set.\n\n\\smallskip\n\n\\begin{Definition} \nLet $K$ be a prime knot in ${\\mathbb S}^3$. For each odd prime number $p$ we define \n$E_p(K)$ to be the connected submanifold of $E(K)$ containing $\\partial E(K)$ \nand such that $\\partial E_p(K) \\setminus \\partial E(K)$ is the union of the \n$JSJ$-tori of $E(K)$ with winding number $p$ which are closest to $\\partial \nE(K)$.\n\\end{Definition}\n\n\\medskip\n\n\\begin{Proposition}\\label{prop:orbifold}\nLet $K$ be a prime knot and let $p$ be an odd prime number. Then for any \n$p$-twin $K'$, the deck transformation of the branched cover\n$M\\longrightarrow({{\\mathbb S}}^3,K')$ induces on $E_p(K)$ a symmetry of order $p$, with\nnon-empty fixed-point set and which extends to ${\\mathcal U}(K)$.\n\\end{Proposition}\n\n {\\bf Proof.} \nFirst we show that the deck transformation of the branched cover\n$M\\longrightarrow({{\\mathbb S}}^3,K')$ associated to a $p$-twin of $K$ induces on \n$E_p(K)$ a symmetry of order $p$.\n  \nLet $K'$ be a $p$-twin of $K$. Let $h$ and $h'$ be the deck transformations on \n$M$ for the $p$-fold cyclic branched covers of $K$ and $K'$. We shall start by \nunderstanding the behaviour of $h$ and $h'$ on $M$. We have seen in Proposition\n\\ref{prop:subtree} that $h$ and $h'$ can be chosen to commute on the \nsubmanifold $M_f$ of $M$ corresponding to the maximal subtree of $\\Gamma$ on \nwhich both $h$ and $h'$ induce a trivial action. Let $\\Gamma_c$ the maximal \n$\\langle h,h'\\rangle$-invariant subtree of $\\Gamma$ containing $\\Gamma_f$, such \nthat, up to conjugacy, $h$ and $h'$ can be chosen to commute on the \ncorresponding submanifold $M_c$ of $M$.\n\nIf $M_c = M$ then after conjugation $h'$ commutes with $h$ on $M$, but is\ndistinct from $h$ because the knots $K$ and $K'$ are not equivalent. Hence it \ninduces a rotational symmetry of order $p$ of the pair $(S^3,K)$ and we are \ndone.\n\nSo we consider now the case where $\\partial M_c$ is not empty. It is sufficient \nto show that $E_p(K) \\subset M_c\/<h>$: then the symmetry of order $p$ induced \nby $h'$ on $M_c\/<h>$ must preserve $E_p(K)$ since each $JSJ$-torus of $E(K)$ \ncan only be mapped to another torus of the family with the same winding number \nand the same distance from $\\partial E(K)$. First we show:\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:permutation}\nLet $T$ be a connected component of $\\partial M_c$. The $h$-orbit of $T$ \nconsists of $p$ elements which are permuted in the same way by $h$ and $h'$.\n\\end{Claim}\n\n {\\bf Proof.}  \nLet $T$ be a torus in $\\partial M_c$ and let $U$ be the connected component of \n$M\\setminus M_c$ adjacent to $T$. Because of Lemma \\ref{lem:commutation}, $T$ \ncannot be preserved by both $h$ and $h'$ for else $M_c$ would not be maximal.\nWithout loss of generality, we can assume that either:\n\\medskip\n\n\\noindent{\\bf (a)} $h(T) \\neq T$ and $h'(T) \\neq T$;\n\n\\medskip\n\n\\noindent or\n\n\\medskip\n\n\\noindent{\\bf (b)} $h(T)=T$ but $h'(T)\\neq T$; in this case since $h$ and\n$h'$ commute on $M_c$, we have that $h(h{'}^{\\alpha}(U)) = h{'}^{\\alpha}(U)$. \nThen part (ii) of Proposition \\ref{prop:subtree} implies that $h$ acts freely \non $h{'}^{\\alpha}(U)$ for each $\\alpha=0,...,p-1$.\n\n\\medskip\n\nIn case (a), the orbit of $T$ by the action of the group $\\langle h,h'\\rangle$ \nconsists of $p$ or $p^2$ elements which bound on one side $M_c$ and on the \nother side a manifold homeomorphic to $U$. If the orbit consist of $p$ \nelements, since $h$ and $h'$ commute on $M_c$, up to choosing a different\ngenerator in $\\langle h'\\rangle$ we can assume that $h$ and $h'$ permute the\nelements of the orbit in the same way. Indeed, we have \n$h'h(T)=hh'(T)=h(h^{\\alpha}(T))=h^{\\alpha}(h(T))$.\n\nIf the orbit consist of $p^2$ elements, $U$ is a is a knot exterior and there \nis a well-defined longitude-meridian system on each component of the $\\langle \nh,h'\\rangle$-orbit of $T$. In particular, there is a unique way to glue a copy \nof $U$ along the projection of $T$ in $M_c\/\\langle h,h'\\rangle$. This implies \nthat $h$ and $h'$ commute up to conjugacy on $M_c\\cup\\langle h,h'\\rangle U$, \ncontradicting the maximality of $M_c$. Note also that in this latter case the \nstabiliser of each component of $\\langle h,h'\\rangle U$ is reduced to the \nidentity which clearly extends to $\\langle h,h'\\rangle U$.\n\n\\medskip\n\nAssume we are in case (b). Consider the restriction of $h$ and\n$h_\\alpha=h{'}^{-\\alpha}hh{'}^\\alpha$ to $U$. Since $h$ and $h'$ commute on\n$M_c$, $h$ and $h_\\alpha$ coincide on $T$. Let $V$ be the geometric piece of \nthe $JSJ$-decomposition for $M$ adjacent to $T$ and contained in $U$. Using \nLemma \\ref{lem:commutation}, we see that $h$ and $h_\\alpha$ commute on $V$ and \nthus coincide on it, because they coincide on $T$. Thus $h$ and $h'$ commute on\n$M_c\\cup_{\\alpha=0}^{p-1}h{'}^\\alpha(V)$, and again we reach a contradiction to\nthe maximality of $M_c$.\n\\qed\n\n\\bigskip\n\nWe can thus assume to be in case (a) and that the $\\langle h,h'\\rangle$-orbit\nof $T$ has $p$ elements.\n\n\\smallskip\n\n\\begin{Claim}\\label{claim:winding number}\nEach torus in the boundary of $M_c\/<h>$ has winding number $p$ with respect\nto $K$.\n\\end{Claim}\n\n {\\bf Proof.} \nSince a boundary component $T$ of $M_c\/<h>$ lifts to $p$ boundary components of \n$M_c$, the winding number of $T$ with respect to $K$ must be a multiple of $p$. \nWe shall now reason by induction on  the number $n$ of boundary components of \n$M_c\/<h>$. If $n=0$ there is nothing to prove.\n\nIf $n = 1$ the quotient spaces $M_c\/<h>$ and $M_c\/<h'>$ are solid tori, i.e. \nthe exterior of a trivial knot which can be identified with a meridian of each \nsolid torus. Note that the winding number of $T$ is precisely the linking \nnumber of $K$ with such a meridian. Note, moreover, that the spaces $M_c\/<h>$ \nand $M_c\/<h'>$ have a common quotient ${\\mathcal O}$ which is obtained by quotienting \n$M_c\/<h>$ (respectively $M_c\/<h'>$) via the the symmetry $\\psi$ (respectively \n$\\psi'$) of order $p$ and with non-empty fixed-point set, induced by $h'$ \n(respectively $h$). Since $\\psi'$ preserves $\\partial{(M_c\/<h'>)}$ and has \nnon-empty fixed-point set, $Fix(\\psi')$ and the meridian of \n$\\partial{(M_c\/<h'>)}$ must form a Hopf link, in particular, their linking \nnumber is $1$. The image of $Fix(\\psi')$ and of the meridian of \n$\\partial{(M_c\/<h'>)}$ form again a Hopf link in ${\\mathcal O} =(M_c\/<h'>)\/\\psi$. By \nlifting them up to $M_c\/<h>$ we see that the meridian lifts to a meridian and \nthe image of $Fix(\\psi')$ lifts to $K$ which thus have linking number $p$. \nHence the property is proved in this case.\n\nIf $n>1$, we shall perform trivial Dehn surgery on $n-1$ boundary components of\n$M_c\/<h>$. Note that such a surgery does not change the winding number of the \nremaining boundary components (for the boundary components are unlinked), that \nthe symmetry of order $p$ of $M_c\/<h>$ extends to the resulting solid torus, \nand that the surgery can be lifted on $M_c$ in such a way that the quotient of \nthe resulting manifold by the action of the diffeomorphism induced by $h'$ is \nagain a solid torus. This last property follows from the fact that each \nconnected component of $(E(K)\\setminus(M_c\/\\langle h\\rangle)$ is the exterior \nof a knot which lifts in $M$ to $p$ diffeomorphic copies. These $p$ copies of \nthe knot exterior are permuted by $h'$ and a copy appears in the \n$JSJ$-decomposition of $E(K')$. This means that on each boundary component \nthere is a well-defined meridian-longitude system which is preserved by $h$ and \n$h'$ and by passing to the quotient. The claim follows now from case $n=1$.\n\\qed\n\n\\bigskip\n\nNow Claims \\ref{claim:permutation} and \\ref{claim:winding number} imply that \n$E_p(K)$ is a submanifold of $M_c\/\\langle h\\rangle \\cap E(K)$. \n\nNote, moreover, that the fixed-point set of the induced symmetry is contained \nin $M_f\/\\langle h\\rangle\\subset M_c\/\\langle h\\rangle$. In particular, each \ntorus of the $JSJ$-family separating such fixed-point set from $K$ lifts to a \nsingle torus of the $JSJ$-family for $M$ and its winding number cannot be a \nmultiple of $p$. We can thus conclude that the fixed-point set of the symmetry \ninduced by $h'$ is contained in $E_p(K)$. This finishes the proof of\nProposition \\ref{prop:orbifold}.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:orbifold} \nNote that $M_c\/h \\cap E(K)$ can be larger than $E_p(K)$ for there might be tori \nof the $JSJ$-collection for $M$ which have an $\\langle h,h'\\rangle$-orbit \ncontaining $p^2$ elements and which project to tori with winding number $p$. \nNote also that $E_p(K)$ coincides with $E(K)$ if there are no \n$JSJ$-tori in $E(K)$ with winding number $p$.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Remark}\\label{rem:commutation} \nThe deck transformations $h$ and $h'$ cannot commute on the submanifolds $U$ of \n$M$ corresponding to branches of $\\Gamma$ whose $h$- and $h'$-orbits coincide \nand consist of $p$ elements, if $h$ and $h'$ are different; that is, the \nstabiliser $h'h^{-1}$ is a finite order diffeomorphism of $U$ if and only if it \nis trivial. To see this, assume that there is a unique orbit of this type and \nassume by contradiction that $h$ and $h'$ commute on $M$ and are distinct. The \ndiffeomorphism $h'$ would induce a non trivial symmetry of $E(K)$ of order $p$ \nand non-empty fixed-point set which fixes set-wise the projection of $U$ and \nacts freely on it. This contradicts the first part of Lemma \n\\ref{lem:companion}. If there are $n>1$ such orbits an equivariant Dehn surgery \nargument on $n-1$ components leads again to a contradiction\n\\end{Remark}\n\n\\medskip\n\nHere is a straightforward corollary of Proposition \\ref{prop:orbifold} which \ngeneralises a result proved by B. Zimmermann \\cite{Zim1} for hyperbolic knots.\n\n\\smallskip\n\n\\begin{Corollary}\\label{cor:p-symmetry}\nLet $K$ be a prime knot and let $p$ be an odd prime number. If $K$ has no\ncompanion of winding number $p$ and has a $p$-twin, then $K$ admits a \nrotational symmetry of order $p$ with trivial quotient.\n\\qed\n\\end{Corollary}\n\n\\bigskip\n\nSo far we have proved that if a prime knot $K$ has a $p$-twin either $E(K)$ \nadmits a $p$-rotational symmetry or a well-specified submanifold $E_p(K)$ of \n$E(K)$ admits a symmetry of order $p$ with non-empty fixed-point set. We shall \nsay that the $p$-twin induces a \\emph{symmetry}, respectively a \\emph{partial \nsymmetry}, of $K$.\n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:twin}\nLet $K$ be a prime knot. Assume that $K$ has a $p$-twin and a $q$-twin for two \ndistinct odd prime numbers.\n\n\\item{(i)} At least one twin, say the $q$-twin, induces a $q$-rotational  \nsymmetry $\\psi_q$ of $K$. Moreover:\n\n\\item{(ii)} If the $p$-twin induces a partial $p$-symmetry of $K$, then \n$\\partial E_p(K) \\setminus \\partial E(K)$ is a $JSJ$-torus which separates \nthe fixed point set  $Fix(\\psi_q)$ from $\\partial E(K)$.\n\\end{Proposition}\n\n\\medskip\n\nFirst we study some properties of partial symmetries induced by $p$-twins for \nan odd prime number $p$.\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:partial companion}\nLet $K$ be a prime knot and let $\\psi$ be the partial symmetry of order $p$ \ninduced on $E_p(K)$ by a $p$-twin. Let $T$ be a torus of the $JSJ$-collection \nof $E_p(K)$ which is not in the boundary. Then $T$ does not separate $\\partial \nE(K)$ from $Fix(\\psi)$ if and only if its $\\psi$-orbit has $p$ elements. \nMoreover, this is the case if and only if the lift of $T$ to the $p$-fold \ncyclic branched cover of $K$ has $p$ elements.\n\\end{Lemma}\n\n {\\bf Proof.} \nIt suffices to perform $\\psi$-equivariant Dehn fillings on the boundary\ncomponents $\\partial E_p(K) \\setminus \\partial E(K)$ of $E_p(K)$ in such a way \nthat the resulting manifold is a knot exterior $E(\\hat K)$ and that the graph \ndual to the $JSJ$-decomposition of $E(\\hat K)$ remains unchanged after filling \n(see the proof of Theorem \\ref{thm:rotations}). Part (i) of Lemma \n\\ref{lem:companion} then applies to the resulting knot $\\hat K$ and the induced \nrotational symmetry. To apply Lemma \\ref{lem:torus} it suffices to note that, \nas in the proof of Claim \\ref{claim:winding number}, the fillings can be chosen \nin such a way that the induced fillings on the quotient $E_p(K)\/\\langle \\psi \n\\rangle$ give also a solid torus (see Remark \\ref{rem:lift}).\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:case b} \nIn particular, case (b) of the proof of Claim \\ref{claim:permutation} cannot\nhappen for a torus $T$ in the situation of Lemma \\ref{lem:partial companion}.\n\\end{Remark}\n\n\\medskip\n\n\\begin{Lemma}\\label{lem:vertex}\nLet $K$ be a prime knot and let $\\psi$ be the partial symmetry of order $p$ \ninduced on $E_p(K)$ by a $p$-twin. Let $T \\subset \\partial E_p(K) \\setminus \n\\partial E(K)$ be a torus which is $\\psi$-invariant. Let $e_{T}$ be the \ncorresponding edge in the tree dual to the $JSJ$-decomposition of $E_p(K)$. Let \n$v_K$ and $v_\\psi$ be the vertices corresponding to the geometric pieces \ncontaining $\\partial E(K)$ and $Fix(\\psi)$ respectively. Then $v_\\psi$ belongs \nto the unique geodesic joining $v_K$ to $e_{T}$ in this $JSJ$-tree.\n\\end{Lemma}\n\n {\\bf Proof.}  \nIf we cut ${\\mathbb S}^3$ along a torus of the $JSJ$-collection of $E_p(K)$, the \nconnected component which does not contain $K$ is a knot exterior and is thus \ncontained in a ball in ${\\mathbb S}^3$. If the conclusion of the Lemma were false, then \nwe could find two tori of the $JSJ$-decomposition of $E_p(K)$ contained in two \ndisjoint balls, one torus separating $Fix(\\psi)$ from $K$ and the other \ncoinciding with $T$ or separating it from $K$. In particular the linking number \nof $Fix(\\psi)$ and a meridian of the solid torus bounded by $T$ (i.e. the \nwinding number of $T$ with respect to $Fix(\\psi)$) would be zero. This is \nimpossible since $\\psi$ leaves set-wise invariant $T$.\n\\qed\n\n\\bigskip\n\n\\begin{Remark}\\label{rem:adjacency}\nLemma \\ref{lem:vertex} has two interesting consequences. Since $h$ and $h'$\nplay symmetric roles, we deduce that $Fix(\\psi)$ and $\\partial E(K)$ must \nbelong to the same geometric piece of the $JSJ$-decomposition of $E_p(K)$. This \nfollows from the fact that, in $E_p(K') \\cup {\\mathcal U}(K')$, $Fix(\\psi)$ maps to \n$K'$, $K$ maps to $Fix(\\psi')$, and $T$ maps to a $\\psi'$-invariant torus. \nMoreover, each invariant boundary torus $T$ is adjacent to the geometric \ncomponent containing $Fix(\\psi)$ and $K$, else, we would get a contradiction to \nLemma \\ref{lem:partial companion}.\n\\end{Remark}\n\n\\medskip\n\n{\\bf Proof of Proposition \\ref{prop:twin}(i).}\nWe argue by contradiction, assuming that there are a $p$-twin and a $q$-twin of \n$K$ which induce only partial symmetries of $E(K)$ for two distinct odd \nprime numbers $p$ and $q$. Then $\\partial E_p(K)$ and $\\partial E_q(K)$ are not \nempty. Moreover, we must have $E(K)\\setminus E_p(K) \\subset E_q(K)$ since the \nwinding number along nested tori is multiplicative and thus the winding number \nof any $JSJ$-torus  contained in $E(K)\\setminus E_p(K)$ must be of the form \n$kp$ and cannot be $q$. In particular $\\partial E_p(K) \\setminus \\partial E(K) \n\\subset \\text{int}(E_q(K))$.\n\nLet $T \\in \\partial E_p(K) \\setminus \\partial E(K)$ be a torus and let $\\psi$ \nbe the $q$-symmetry with non-empty fixed-point set induced on $E_q(K)$ by the \n$q$-twin. Since the winding number of $T$ is $p$, its lift to the $q$-fold \ncyclic branched cover of $K$ is connected. According to part (i) of Lemma \n\\ref{lem:companion} and to Lemmata \\ref{lem:torus} and \\ref{lem:partial \ncompanion}, $T$ must separate $\\partial E(K)$ from $Fix(\\psi)$. Since \n$Fix(\\psi)$ is connected, we see that so must be $\\partial E_p(K) \\setminus \n\\partial E(K) = T$. The final contradiction is then reached by applying Remark \n\\ref{rem:adjacency}.\n\\qed \n\n\\bigskip\n\n{\\bf Proof of Proposition \\ref{prop:twin}(ii).} \nThis is a consequence of the proof of part (i): note that in the proof $\\psi$ \nmay be a global or partial symmetry.\n\\qed\n\n\\bigskip\n\nWe are now in a position to prove Theorem \\ref{thm:twins}.\n\n\\medskip\n\n{\\bf Proof of part (i) of Theorem \\ref{thm:twins}.}\nWe argue by contradiction, assuming that $K$ admits twins for three distinct, \nodd prime numbers $p, q, r$. Under this assumption, it follows that $K$ is a \nnon-trivial knot. \n\nIf the three twins induce rotational symmetries of the knot $K$, then part (i) \nof Theorem \\ref{thm:rotations} gives a contradiction.\n\nTherefore part (i) of Proposition \\ref{prop:twin} implies that twins of orders, \nsay $q$ and $r$, induce rotational symmetries $\\psi_q$ and $\\psi_r$ of $K$ \nhaving order $q$ and $r$ respectively, while a $p$-twin induces only a partial \nrotational symmetry of $E(K)$ of order $p$. \n\nThen part (ii) of Proposition \\ref{prop:twin} shows that $\\partial E_p(K) \n\\setminus \\partial E(K)$ is a $JSJ$-torus in $E(K)$ which separates $\\partial \nE(K)$ from both $Fix(\\psi_q)$ and $Fix(\\psi_r)$. This contradicts part (ii) of \nTheorem \\ref{thm:rotations} which states that $Fix(\\psi_q)$ and \n$Fix(\\psi_r)$ must sit in the $JSJ$-component containing $\\partial E(K)$.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of part (ii) of Theorem \\ref{thm:twins}.}\nLet $K$ be a prime knot and let $p$ be an odd prime number. We assume that $K$ \nhas at least two non-equivalent $p$-twins $K_1$ and $K_2$ and look for a \ncontradiction. \n\n\\medskip\n\nIf both  $\\psi_{1}$ and $\\psi_{2}$ are rotational symmetries of order $p$ of\n$K$, then by M. Sakuma \\cite[Thm. 3]{Sa2} they are conjugate since $K$ is \nprime. This would contradict the hypothesis that the knots $K_1$ and $K_2$ \nare not equivalent.\n\n\\medskip\n\nAssume now that at least one symmetry, say $\\psi_{1}$ is partial. Then \n$\\psi_{1}$ and $\\psi_{2}$ are rotational symmetries of order $p$ of the \nsubmanifold $E_p(K) \\subset E(K)$. Let $X_0$ be the geometric piece of \nthe $JSJ$-decomposition of $E(K)$ containing $\\partial E(K)$. Then $\\psi_1$ \n(respectively $\\psi_2$) generates a finite cyclic subgroup $G_1$ (respectively\n$G_2$) of the group $Diff^{+,+}(X_0, \\partial E(K))$ of diffeomorphisms of the \npair $(X_0, \\partial E(K))$ which preserve the orientations of $X_0$ and of \n$\\partial E(K)$. Moreover, one can assume that $G_1$ and $G_2$ act\ngeometrically on $X_0$.\n\nIf $X_0$ admits a hyperbolic structure, it is a consequence of the proof of the \nSmith conjecture (see for example \\cite[Lemma 2.2]{Sa2}) that the subgroup of \n$Diff^{+,+}(X_0, \\partial E(K))$ consisting of restrictions of isometries of \n$X_0$ is finite cyclic. Hence $G_1 = G_2$ and up to taking a power \n$\\psi_1 = \\psi_2$ on $X_0$.\n\nIf $X_0$ is Seifert fibred, then it must be a cable space, since $K$ is prime. \nThe uniqueness of the Seifert fibration and the fact that the basis of the \nSeifert fibration has no symmetry of finite order imply that the cyclic groups \n$G_1$ and $G_2$ belong to the circle action $S^1 \\subset Diff^{+,+}(X_0, \n\\partial E(K))$ inducing the Seifert fibration of $X_0$, see \n\\cite[Lemma 2.3]{Sa2}. Since $G_1$ and $G_2$ have the same prime order, up to \ntaking a power $\\psi_1 = \\psi_2$ on $X_0$.\n\nLet $h_1$ and $h_2$ be the deck transformations on $M$ associated to the\n$p$-fold cyclic coverings branched along $K_1$ and $K_2$, and which induce \n$\\psi_1$ and $\\psi_2$. Then by taking a suitable powers, $h_1$ and $h_2$ \ncoincide up to conjugacy on the geometric piece $\\widetilde X_0$ of the \n$JSJ$-decomposition of $M$ containing the preimage of $K$. The following lemma \nshows that they will coincide on $M$, contradicting our hypothesis.\n\\qed\n\n\\bigskip\n\n\\begin{Lemma}\\label{cor:extension}\nIf the covering transformations $h$ and $h'$ preserve a $JSJ$-piece or a\n$JSJ$-torus of $M$ and coincide on it, then they can be chosen, up to\nconjugacy, to coincide everywhere.\n\\end{Lemma}\n\n {\\bf Proof.}  \nThis is  a consequence of the proofs of Propositions \\ref{prop:subtree} and\n\\ref{prop:orbifold}. We shall start by showing that we can always assume that \nthere is a piece $V$ of the $JSJ$-decomposition on which $h$ and $h'$ coincide. \nTo this purpose, assume that $h$ and $h'$ coincide only on a $JSJ$-torus $T$. \nAccording to Lemma \\ref{lem:commutation} and Remark \\ref{rem:same action}, $h$ \nand $h'$ coincide on the geometric pieces of the decomposition adjacent to $T$, \nwhich are also invariant. Consider now the maximal subtree $\\Gamma_1$ of \n$\\Gamma$ such that the restrictions of $h$ and $h'$ to the corresponding \nsubmanifold $M_1$ of $M$ coincide, up to conjugacy, and such that \n$V\\subset M_1$. Let $S$ be a $JSJ$-torus for $M$ in the boundary of $M_1$. \nSince $h$ and $h'$ coincide on $M_1$, the $h$-orbit and the $h'$-orbit of $S$ \ncoincide as well and consist of either one single element $\\{S\\}$ or $p$ \nelements $\\{S,h(S)=h'(S),...,h^{p-1}(S)={h'}^{p-1}(S)\\}$. In the former case, \naccording to Lemma \\ref{lem:commutation}, $\\Gamma_1$ would not be maximal. In \nthe latter case, we are precisely in the situation described in part (a) of \nClaim \\ref{claim:permutation}. Once more, $\\Gamma_1$ is not maximal because one \ncan impose that $h$ and $h'$ act in the same way on the $p$ connected \ncomponents with connected boundary obtained by cutting $M$ along the $\\langle \nh,h'\\rangle$-orbit of $S$ (see Remark \\ref{rem:commutation}). This \ncontradiction shows that $M=M_1$ and the lemma is proved.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of part (iii) of Theorem \\ref{thm:twins}.}\nFirst we analyse the case of a knot admitting two twins, one of which induces a \npartial symmetry. \n\n\\smallskip\n\n\\begin{Proposition}\\label{prop:partial}\nLet $K$ be a prime knot admitting a $p$-twin $K'$ and a $q$-twin $K''$ for two distinct\nodd prime numbers $p$ and $q$. If $K'$ induces a partial symmetry of $K$ then\n$K'$ and $K''$ are not equivalent.\n\\end{Proposition}\n\n {\\bf Proof.} \nBy part (ii) of Proposition \\ref{prop:twin}, $E_p(K)$ has a unique boundary \ncomponent which separates $\\partial E(K)$ from the fixed-point set of the\n$q$-rotational symmetry $\\psi$ induced by $K''$. By cutting ${\\mathbb S}^3$ along \n$T = \\partial E_p(K)$ we obtain a solid torus $V=E_p(K)\\cup {\\mathcal U}(K)$ containing \n$K$, and a knot exterior $E_T$. $K$ admits a $q$-rotational symmetry $\\psi$ \ninduced by $K''$ which preserves this decomposition and induces a \n$q$-rotational symmetry with trivial quotient (see Lemma \\ref{lem:companion}) \non $E_T$ and a free $q$-symmetry $\\tilde\\psi$ on $V$. The covering \ntransformation for the knot $K'$ induces a $p$-symmetry $\\varphi$ of $V$ with \nnon-empty fixed-point set.\n\nAssume now by contradiction that $K'=K''$. Since $K'$ induces a partial \nsymmetry of $K$ and vice versa, $S^3$ admits a decomposition into two \npieces: $V'=E_p(K')\\cup {\\mathcal U}(K')$ and $E_T $. On the other hand, since $K''$ \ninduces a genuine $q$-rotational symmetry of $K$, $K''$ admits a \n$q$-rotational symmetry $\\psi''$ induced by $K$ which preserves the \naforementioned decomposition and induces a $q$-rotational symmetry with trivial \nquotient on $E_T$. Using the fact that $E_T$ is the exterior of a prime knot \n(see Lemma \\ref{lem:prime}) and M. Sakuma's result \\cite[Thm. 3]{Sa2}, we see \nthat the two $q$-rotational symmetries with trivial quotient induced by $\\psi$ \nand $\\psi''$ on $E_T$ act in the same way. Let now $E_0$ be the smallest knot \nexterior of the $JSJ$-decomposition of $E_T$ on which $\\psi=\\psi''$ induces a \n$q$-rotational symmetry with trivial quotient (this is obtained by cutting \n$E_T$ along the torus of the $JSJ$-decomposition closest to $Fix(\\psi)$ \n-respectively $Fix(\\psi'')$- and separating it from $T$. Consider now the lift,\ndenoted by $(X,{\\mathcal K})$, to $(S^3,K'')$ of $(E_0, Fix(\\psi))\/\\psi$. We claim that \n$(X,{\\mathcal K})=(V',K')$. Indeed, $X$ contains $K''=K'$ by construction, and its \nboundary is the unique torus of the $JSJ$-decomposition which is left invariant \nby the $q$-rotational symmetry of $K''$ -by construction again- and which is \nclosest to $K''$ (compare Remark \\ref{rem:adjacency}). Since \n$E_0\/\\psi=E_0\/\\psi''$, and a solid torus has a unique $q$-fold cyclic cover, we \ndeduce that $(V',K')=(X,{\\mathcal K})=(V,K)$. In particular, the deck transformations \nfor $K$ and $K'$ on their common $p$-fold cyclic branched cover can be chosen \nto coincide on the lift of $V=V'$. Lemma \\ref{cor:extension} implies that \n$K=K'$ contradicting the fact that $K'$ is a $p$-twin.\n\\qed\n\n\\bigskip\n\nLet $K'$ be a $p$-twin and a $q$-twin of $K$ for two distinct odd prime numbers \n$p$ and $q$. Proposition \\ref{prop:partial} implies that $K'$ induces two \nrotational symmetries $\\psi_p$ and $\\psi_q$ of $K$ with trivial quotients and \norders $p$ and $q$. Part (ii) of Theorem \\ref{thm:rotations} shows that the \nfixed-point sets $Fix(\\psi_p)$ and $Fix(\\psi_q)$ lie in the $JSJ$-component of \n$E(K)$ which contains $\\partial E(K)$. Then the proof of part (iii) of Theorem \n\\ref{thm:twins} follows from the following:\n\n\\smallskip\n\n\\begin{Lemma}\\label{lem:commuting symmetries}\nLet $K$ be a prime knot admitting two rotational symmetries $\\psi$ and\n$\\varphi$ of odd prime orders $p > q$. If the fixed-point sets of $\\psi$ and \n$\\varphi$ lie in the component which contains $\\partial E(K)$, then the two \nsymmetries commute up to conjugacy.\n\\end{Lemma}\n\n {\\bf Proof.} \nReasoning as in the proof of part (ii) of Theorem \\ref{thm:twins}, one can show\nthat $\\psi$ and $\\varphi$ commute on the component which contains $\\partial\nE(K)$. Since all other components are freely permuted according to part (i) of\nLemma \\ref{lem:companion}, the conclusion follows as in the proof of part (a)\nof Claim \\ref{claim:permutation}.\n\\qed\n\n\\bigskip\n\n{\\bf Proof of Corollary \\ref{cor:composite}.}\nFirst of all note that, because of the uniqueness of the Milnor-Kneser \ndecomposition of the covers of $K$ and $K'$, the number of prime summands of $K$ \nand $K'$ is the same. After ditching components of $K$ and $K'$ that appear in \nboth decompositions in equal number, we can assume that $K_i$ is not equivalent \nto $K'_\\ell$, for all $i,\\ell=1,...,t$. If $K$ and $K'$ have three common \ncyclic branched covers of odd prime orders, we deduce that for each \n$i=1,...,t$, $K_i$ is not determined by its $p_j$-fold cyclic branched cover, \n$j=1,2,3$, for it is also the $p_j$-fold cyclic branched cover of some \n$K'_{i_j}$ not equivalent to $K_i$. Hence $K_i$ would have twins for three\ndistinct odd prime orders which is impossible by Theorem \\ref{thm:twins}. \n\\qed\n\n\n\\section{Examples}\n\nExamples of prime knots admitting a $p$-twin which induces a global rotational\nsymmetry of order $p$ were first constructed by Y. Nakanishi \\cite{Na} and M.\nSakuma \\cite{Sa1}. They considered a prime link with two trivial components \nwhose linking number is $1$. By taking the $p$-fold cyclic cover of ${\\mathbb S}^3$ \nbranched along the first (respectively the second) component of the link one\ngets again ${\\mathbb S}^3$ and the second (respectively first) component lifts to a\nprime knot. The two knots thus constructed have the same $p$-fold cyclic\nbranched cover by construction (see also Remark \\ref{rem:lift}), moreover, by \ncomputing their Alexander polynomial they were shown to be distinct.\n\nIn \\cite[Thm 3 and Cor. 1]{Zim1} B. Zimmerman showed that if a hyperbolic knot \nhas a $p$-twin, for $p\\ge 3$, then the $p$-twin induces a global symmetry and \nthe two knots are thus obtained by Y. Nakanishi and M. Sakuma's construction \nwhere the quotient link is hyperbolic and admits no symmetry which exchanges \nits two components.\n\nAs a matter of fact, the links considered by Y. Nakanishi and M. Sakuma are in \nfact hyperbolic and so are the resulting twins if $p$ is at least $3$, \naccording to the orbifold theorem \\cite{BoP}, see also \\cite{CHK}. Note that, \nwhen $p=2$, the situation, even in the case of hyperbolic knots, is much more \ncomplex and there are several ways to construct $2$-twins of a given knot. In \nthis section we shall see how one can construct, for each given odd prime $p$, \ntwo prime, non simple, knots which are $p$-twins, and such that the symmetries \nthey induce are not global.\n\nThe first construction shows that the number $\\nu$ of components of $\\partial \nE_p(K)\\setminus\\partial E(K)$ can be arbitrarily large. This means that the \nsituation encountered in Proposition \\ref{prop:twin}(ii) is extremely special. \nThe second construction shows that our result is indeed best possible even for\nprime knots with $p$-twins inducing partial symmetries: we shall construct \nprime knots admitting a $p$-twin inducing a partial symmetry and a $q$-twin \ninducing a global rotational symmetry. \n\n\\bigskip\n\n\\subsection{Knots admitting a $p$-twin inducing only a partial symmetry} \n\n\\medskip\n\nAssume we are given a hyperbolic link $L=L_1\\cup...\\cup L_{\\nu+2}$, with $\\nu+2\n\\ge 3$ components, satisfying the following requirements:\n\n\\medskip\n\n{\\bf Property $*$}\n\n\\begin{enumerate}\n\n\\item The sublink $L_3\\cup...\\cup L_{\\nu+2}$ is the trivial link;\n\n\\item For each $i=1,2$ and $j=3,...,\\nu+2$, the sublink $L_i\\cup L_j$ is a Hopf\nlink;\n\n\\item ${\\rm lk}(L_1,L_2)$ is prime with $p$;\n\n\\item No symmetry of $L$ exchanges $L_1$ and $L_2$.\n\n\\end{enumerate}\n\n\\medskip\nWe shall consider the orbifold ${\\mathcal O}=({\\mathbb S}^3,(L_1\\cup L_2)_p)\\setminus\n{\\mathcal U}(L_3\\cup...\\cup L_{\\nu+2})$ which is the $3$-sphere with singular set of\norder $p$ the (sub)link $L_1\\cup L_2$ and an open tubular neighbourhood of the\n(sub)link $L_3\\cup...\\cup L_{\\nu+2}$ removed. ${\\mathcal O}$ is hyperbolic if $p\\ge3$, \nand will represent the quotient of ${\\mathcal O}_p=E_p(K)\\cup{\\mathcal U}(K)$ and \n${\\mathcal O}_p'=E_p(K')\\cup{\\mathcal U}(K')$ via the action of the partial $p$-symmetries. \nIndeed, to obtain ${\\mathcal O}_p$ (respectively ${\\mathcal O}_p'$) take the $p$-fold cyclic \norbifold cover of $({\\mathbb S}^3,(L_1\\cup L_2)_p)\\setminus\n{\\mathcal U}(L_3\\cup...\\cup L_{\\nu+2})$ which desingularises $L_2$ (respectively $L_1$).\nObserve that one can fix a longitude-meridian system on each boundary \ncomponent of ${\\mathcal O}$, induced by those of $L_i$, $i=3,\\dots,\\nu+2$. Note that, \nbecause of condition 4 of Property $*$, the two orbifolds ${\\mathcal O}_p$ and ${\\mathcal O}_p'$ \nwith the fixed peripheral systems are distinct. \n\nRemark that ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ can be obtained by the orbifold covers,\nanalogous to those described above, of $({\\mathbb S}^3,(L_1\\cup L_2)_p)$ (which are \ntopologically ${\\mathbb S}^3$) by removing open regular neighbourhoods of the lifts of \nthe components $L_3\\cup...\\cup L_{\\nu+2}$. Note that these components \nlift to trivial components whose linking number with the lift of $L_i$, \n$i=1,2$, is precisely $p$, because of condition 2, and which form again a \ntrivial link.\n\nFor each $j=3,...,\\nu+2$, choose a knot exterior $E({\\mathcal K}_j)$ to be glued along \nthe $j$-th boundary component of ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ in such a way that a \nfixed longitude-meridian system on $E({\\mathcal K}_j)$ is identified with the lift of \nthe longitude-meridian system on the $j$-th boundary component of ${\\mathcal O}$. The underlying spaces of the\norbifolds ${\\mathcal O}_p\\cup_{j=3}^{\\nu+2} E({\\mathcal K}_j)$ and ${\\mathcal O}'_p\\cup_{j=3}^{\\nu+2} \nE({\\mathcal K}_j)$ are topologically ${\\mathbb S}^3$ and it is easy to see that their singular \nsets are connected (see condition 3). The resulting knots have the same \n$p$-fold cyclic branched cover, however, since ${\\mathcal O}_p$ and ${\\mathcal O}'_p$ are \ndistinct, they are not equivalent.\n\n\\bigskip\n\n\\begin{Remark}\nObserve that we have just shown that the number of connected components of\n$\\partial E_p(K)\\setminus\\partial E(K)$, which is precisely $\\nu$, can be \narbitrarily large. Note also that if $\\nu\\ge2$, according to Proposition\n\\ref{prop:twin}, the knot $K$ has no $q$-twins for $q\\neq p$ odd prime.\n\\end{Remark}\n\n\\bigskip\n\nWe shall now prove that links with Property $*$ exist. Notice that for $\\nu=1$\nlinks satisfying all the requirements where constructed by Zimmermann in\n\\cite{Zim2}, see also \\cite{Pao1}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{partial.eps}\n\\end{center}\n\\caption{The link $L$ and its Bonahon-Siebenmann decomposition.}\n\\label{fig:one}\n\\end{figure}\n\nConsider the link given in Figure \\ref{fig:one} for $\\nu=3$ (the generalization \nfor arbitrary $\\nu\\ge1$ is obvious). Most conditions are readily checked just \nby looking at the figure, and we only need to show that $L$ is hyperbolic and \nhas no symmetries which exchange $L_1$ and $L_2$. To this purpose, we shall \ndescribe the Bonahon-Siebenmann decomposition of the orbifold $({\\mathbb S}^3,(L)_2)$, \nwhere all components have ${\\mathbb Z}\/2{\\mathbb Z}$ as local group. The decomposition consists \nof one single hyperbolic piece (see Figure \\ref{fig:one}) and $\\nu+1$ \n(respectively $1$) Seifert fibred pieces if $\\nu\\ge2$ (respectively $\\nu=1$). \nSince the Seifert fibred pieces contain no incompressible torus, the \nhyperbolicity of $L$ follows.\n\nNote now that every symmetry of $L$ must leave invariant the unique hyperbolic\npiece of the decomposition. This piece is obtained by quotienting the\nhyperbolic knot $10_{155}$ via its full symmetry group ${\\mathbb Z}\/2{\\mathbb Z}\\oplus{\\mathbb Z}\/2{\\mathbb Z}$ and \nthus has no symmetries (for more details see \\cite{Pao1}), so we conclude that \nthe components $L_1$ and $L_2$ are non exchangeable.\n\n\\bigskip\n\n\\subsection{Knots admitting a $p$-twin inducing a partial symmetry and a\n$q$-twin inducing a global symmetry}\n\n\\medskip\n\nLet ${\\mathcal K}$ be a hyperbolic knot admitting a $p$-twin and a $q$-twin; the twins \nof ${\\mathcal K}$ induce global symmetries, so that ${\\mathcal K}$ admits a $p$- and a\n$q$-rotational symmetry with trivial quotient (see \\cite{Zim2}, where a method\nto construct hyperbolic knots with two twins is described). Remove a tubular\nneighbourhood of the axis of the symmetry of order $q$ (note that the two\nsymmetries have disjoint axes), and use the resulting solid torus $V$ to \nperform Dehn surgery on the exterior $E$ of the $(2,q)$-torus knot. Denote by \n$K$ the image of ${\\mathcal K}$ after surgery. We require that:\n\n\\begin{enumerate}\n\n\\item The resulting manifold is ${\\mathbb S}^3$;\n\n\\item The $q$-rotational symmetry of $E$ and the restriction of the\n$q$-rotational symmetry of ${\\mathcal K}$ to $V$ give a global $q$-rotational symmetry \nof $K$;\n\n\\item The $q$-rotational symmetry of $K$ has trivial quotient.\n\n\\end{enumerate}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=10cm]{trivial.eps}\n\\end{center}\n\\caption{Satellising so that the induced rotation has trivial quotient.}\n\\label{fig:two}\n\\end{figure}\n\nNote that the last requirement can be met by choosing appropriately the\nlongitude when satellising, as illustrated in Figure \\ref{fig:two}. We claim \nthat $K$ admits a $q$-twin, $K''$, and a $p$-twin, $K'$. $K''$ is obtained by \nthe standard method described in Remark \\ref{rem:lift}. Note that $K\\neq K''$, \nfor the roots of the $JSJ$-decompositions of the exteriors of $K$ and $K''$ are \nhyperbolic and Seifert fibred respectively. To construct $K'$, consider the \n$p$-twin ${\\mathcal K}'$ of ${\\mathcal K}$ and let $V'$ be the solid torus obtained by removing \nthe axis of the $q$-rotational symmetry of ${\\mathcal K}'$. Note that $V$ and $V'$ have \na common quotient obtained by taking the space of orbits of the $p$-rotational \nsymmetries, however $V$ and $V'$ are different orbifolds by construction. Fix a \nlongitude-meridian system on $V$ (the one used for the surgery): by first \nquotienting and then lifting it, get a longitude-meridian system on $V'$ that \nmust be used to perform surgery along a copy of $E$. The image of ${\\mathcal K}'$ after \nthe surgery will be $K'$. Note that, when taking the $p$-fold cyclic branched \ncovers of $K$ and $K'$, the hyperbolic orbifolds $V$ and $V'$ lift to the same \nmanifold by construction, while the Seifert fibred part lifts, in both cases, \nto $p$ copies of $E$. Again by construction, the gluings are compatible and the \ntwo covers coincide. It is also evident that $K'$ can only induce a partial \nsymmetry of $K$, and the claim is proved.\n\n\\bigskip\n\n\\begin{Remark}\nNote that according to Proposition \\ref{prop:partial} the $p$-twins and \n$q$-twins obtained in this construction cannot be equivalent.\n\\end{Remark}\n\n\n\\section{Homology spheres as cyclic branched covers}\n\nBy the proof of the Smith conjecture Corollary \\ref{cor:homologysphere} is true \nfor the $3$-sphere $S^3$. So from now on we assume that the integral homology \nsphere $M$ is not homeomorphic to $S^3$. Then by \\cite[Thm1]{BPZ}, $M$ can be a \n$p_i$-fold cyclic branched cover of ${\\mathbb S}^3$ for at most three pairwise distinct \nodd prime numbers $p_i$. Moreover if $M$ is irreducible and is the $p_i$-fold \ncyclic branched cover of ${\\mathbb S}^3$ for three pairwise distinct odd prime numbers \n$p_i$, then the proof of \\cite[Corollary 1.(i)]{BPZ} shows that for each prime \n$p_i$, $M$ is the $p_i$-fold cyclic branched cover of precisely one knot. Since \na knot admits at most one $p$-twin for an odd prime integer $p$, we need only \nto consider the case when the irreducible integral homology sphere $M$ is the \nbranched cover of ${\\mathbb S}^3$ for precisely two distinct odd primes, say $p$ and \n$q$. Moreover \\cite[Corollary 1.(ii)]{BPZ} shows that $M$ has a non trivial \n$JSJ$-decomposition. \n\nLooking for a contradiction, we can assume that, for each prime, $M$ is the \nbranched covering of two distinct knots with covering transformations $\\psi$, \n$\\psi'$ of order $p$ and $\\varphi$, $\\varphi'$ of order $q$. \n\nIf each rotation of order $p$ commutes with each rotation of order $q$ up to \nconjugacy, then the contradiction follows from the following claim which is an \neasy consequence of Sakuma's result \\cite[Thm. 3]{Sa2} (see \\cite[Claim \n8]{BPZ}). \n\n\\begin{Claim}\\label{claim:unique symmetry} \nLet $n\\ge3$ be a fixed odd integer. Let $\\rho$ be a rotation with trivial \nquotient of an irreducible manifold $M$. All the rotations of $M$ of order $n$ \nwhich commute with $\\rho$ are conjugate in $Diff(M)$ into the same cyclic group \nof order $n$.  \n\\qed\n\\end{Claim}\n\nOtherwise, consider the subgroup $G=\\langle \\psi, \\psi', \\varphi, \\varphi' \n\\rangle$ of diffeomorphisms of $M$. According to the proof of \\cite[Proposition \n4]{BPZ}, each rotation of order $p$ commutes with each rotation of order $q$ up \nto conjugacy, unless the induced action of $G$ on the dual tree of the \n$JSJ$-decomposition for $M$ fixes precisely one vertex corresponding to a \nhyperbolic piece $V$ of the decomposition and $\\{p,q\\}=\\{3,5\\}$. In this case, \none deduces as in the proof of \\cite[Corollary 1.(ii)]{BPZ} that the \nrestrictions of $\\psi$ and $\\psi'$ (respectively $\\varphi$ and $\\varphi'$) \ncoincide up to conjugacy on $V$. Then the desired contradiction follows from \nLemma \\ref{cor:extension} which implies that $\\psi$ and $\\psi'$ (respectively \n$\\varphi$ and $\\varphi'$) coincide up to conjugacy on $M$.\n\\qed\n\n\n\\begin{footnotesize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\\large \\textbf{Supplemental Material}\\\\Stochastic Thermodynamics of\n  Learning}\n\n\\begin{center}\n  Sebastian Goldt and Udo Seifert\\\\\n  \\emph{II. Institut f\u00fcr Theoretische Physik, Universit\u00e4t Stuttgart, 70550\n    Stuttgart, Germany}\\\\\n  (Dated: \\today)\n\\end{center}\n\n  In this supplemental material, we discuss the stochastic thermodynamics of\n  neural networks in detail in section~\\ref{sec:stochastic-thermodyanmics} and\n  derive our main result, eq.~\\eqref{eq:inequality} of the main text, in\n  section~\\ref{sec:inequalities}. Furthermore, we complement our discussion Hebbian\n  learning in the thermodynamic limit with additional analytical calculations in\n  section~\\ref{sec:hebbian-calculations}.\n\n\\section{Stochastic thermodynamics of neural networks}\n\\label{sec:stochastic-thermodyanmics}\n\nWe now give a detailed account of the stochastic thermodynamics of neural\nnetworks. For simplicity, here we will focus on batch learning; the\ngeneralisation to online learning is straightforward. For a network with $N$\nweights $\\wn\\in\\mathbb{R}^N$ learning $P$ samples $\\vsample[\\mu]\\in\\{\\pm1\\}^N$\nwith their labels $\\tlab[\\mu]=\\pm1$, $\\mu=1,2,\\dots,P$, we have $N$ Langevin\nequations~\\cite{s_vankampen1992}\n\\begin{equation}\n  \\label{eq:langevin}\n  \\dot{\\w}_n(t) = - \\wn(t) + f(\\wn(t), \\{\\sample[\\mu]_n, \\tlab[\\mu]\\}, t) + \\zeta_n(t).\n\\end{equation}\nThe Gaussian noise $\\zeta_n(t)$ has correlations\n$\\avg{\\zeta_n(t)\\zeta_m(t')}=2T\\delta_{nm}\\delta(t-t')$ for $n,m=1,\\dots,N$\nwhere $T$ is the temperature of the surrounding medium and we have set\nBoltzmann's constant to unity to render entropy dimensionless. The\nweights $\\vw$ determine the transition rates of the $P$ independent two-state\nprocesses for the predicted labels $\\lab[\\mu]$ via\n\\begin{equation}\n  k_\\mu^+\/k_\\mu^- = \\exp \\left(\\act^\\mu\/T\\right)\n\\end{equation}\nwhere $\\act^\\mu$ is the input-dependent activation\n\\begin{equation}\n  \\act^\\mu\\equiv\\frac{1}{\\sqrt{N}}\\vw\\cdot\\vsample[\\mu]\n\\end{equation}\nFor the remainder of this supplemental material, we set $T=1$, rendering energy\ndimensionless. We assume that the thermal noise in each subsystem, like $\\wn$ or\n$\\lab[\\mu]$, is independent of all the others. This multipartite\nassumption~\\cite{s_horowitz2015} allows us to write the master equation for the\ndistribution $p(\\tlabs,\\vw,\\labs, t)$ with\n$\\tlabs\\equiv(\\tlab[1],\\dots,\\tlab[P])$ and $\\labs\\equiv(\\lab[1],\\dots,\\lab[P])$\nas\n\\begin{equation}\n  \\label{eq:master}\n  \\partial_t p(\\tlabs,\\vw,\\labs, t)=-\\sum_{n=1}^N \\partial_n j_n(t) + \\sum_{\\mu=1}^Pj_\\mu(t),\n\\end{equation}\nwhere $\\partial_t\\equiv\\partial\/\\partial t$,\n$\\partial_n\\equiv \\partial\/\\partial\\, \\wn$ and the probability currents for the\n$n$-th weight $\\wn$ and the $\\mu$-th predicted label $\\lab[\\mu]$ are given by\n\\begin{subequations}\n  \\label{eq:currents}\n  \\begin{align}\n    j_n(t)=& \\left[-\\wn+f(\\wn, \\vsample[\\mu(t)], \\tlab[\\mu(t)], t) - \\partial_n\\right]\n             p(\\tlabs,\\vw,\\labs, t), \\\\\n      j_\\mu(t) =& k^+ p(\\tlabs,\\vw,\\lab[1],\\dots,-\\lab[\\mu],\\dots,\\lab[P], t)\n      - k^- p(\\tlabs, \\vw, \\labs, t).\n  \\end{align}\n\\end{subequations}\nWe choose symmetric rates $k^\\pm_\\mu=\\gamma\\exp(\\pm\\act^\\mu\/2)$ with\n$\\gamma\\gg1$. Initially, the true labels $\\tlabs$, weights $\\vw$ and predicted\nlabels are all uncorrelated with\n\\begin{align}\n  p_0(\\tlab[\\mu])=&1\/2, \\\\\n  p_0(\\lab[\\mu])=&1\/2, \\quad \\text{and} \\\\\n  p_0(\\vw) =& \\frac{1}{(2\\pi)^{N\/2}} \\exp(-\\vw\\cdot\\vw\/2).\n\\end{align}\nSince the following discussion applies to the time-dependent\ndynamics~\\eqref{eq:master}, we understand that all quantities that will be\nintroduced in the remainder of this section have an implicit\ntime-dependence via the distribution $p(\\tlabs, \\vw, \\labs, t)$ or the\ncurrents~\\eqref{eq:currents}.\n\nOur starting point for the stochastic thermodynamics of this system is the\nwell-known total entropy production $\\dot{S}^\\tot$ of the network which obeys\nthe following second-law like inequality~\\cite{s_Seifert2012}\n\\begin{equation}\n  \\dot{S}^\\tot = \\partial_t S(\\tlabs, \\vw, \\labs) + \\dot{S}^\\m \\ge 0\n\\end{equation}\nwith equality in equilibrium only. Here, we have the Shannon\nentropy~\\cite{s_cover2006} of the system,\n\\begin{equation}\n  \\label{eq:shannon_all}\n  S(\\tlabs, \\vw, \\labs) = - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  p(\\tlabs, \\vw, \\labs) \\ln p(\\tlabs, \\vw, \\labs).\n\\end{equation}\nHere, we include the variables $\\tlabs$, $\\vw$ and $\\labs$ as arguments of the\nfunction $S$ in a slight abuse of notation to emphasise that we consider the\nShannon entropy of the full distribution $p(\\tlabs, \\vw, \\labs)$.  $\\dot{S}^\\m$\ngives the rate of entropy production in the medium. For a system at constant\ntemperature $T=1$, $\\dot{S}^\\m\\equiv\\dot{Q}$, the rate of heat dissipation into\nthe medium~\\cite{s_Seifert2012}. Let us first focus on the change in Shannon\nentropy by differentiating~\\eqref{eq:shannon_all} with respect to time,\n\\begin{equation}\n  \\partial_t S(\\tlabs, \\vw, \\labs) = - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  \\dot{p}(\\tlabs, \\vw, \\labs) \\ln p(\\tlabs, \\vw, \\labs),\n\\end{equation}\nwhere we have used that $p(\\tlabs, \\vw, \\labs)$ is, of course, normalised. Using\nthe master equation~\\eqref{eq:master}, we find that\n\\begin{equation}\n  \\partial_t S(\\tlabs, \\vw, \\labs) = \\sum_{n=1}^N \\dot{S}_n + \\sum_{\\mu=1}^P \\dot{S}_\\mu\n\\end{equation}\nwhere \n\\begin{align}\n  \\dot{S}_n \\equiv&  \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  \\partial_n j_n(t) \\ln p(\\tlabs, \\vw, \\labs),\\\\\n  \\dot{S}_\\mu \\equiv& - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  j_\\mu \\ln p(\\tlabs, \\vw, \\labs),\n\\end{align}\nare the rate of change of the Shannon entropy $S(\\tlabs, \\vw, \\labs)$ due to the\ndynamics of $\\w_n$ and $\\lab[\\mu]$, respectively. The key point here is that\nmultipartite dynamics, a consequence of the uncorrelated noise across\nsubsystems, lead to a linear splitting of the probability currents and hence to\na linear splitting of all quantities which are functions of the total\nprobability current. Similarly, for the rate of heat dissipation $\\dot{Q}$,\nwe can write\n\\begin{equation}\n  \\dot{Q} = \\sum_{n=1}^N \\dot{Q}_n + \\sum_{\\mu=1}^P\\dot{Q}_\\mu\n\\end{equation}\nwhere\n\\begin{equation}\n  \\dot{Q}_n =  \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  j_n(t) F_n(\\tlabs, \\vw, \\labs)\n\\end{equation}\nwith the total force on the $n$-th weight\n$F_n=- \\wn(t) + f(\\wn(t), \\{\\sample[\\mu]_n, \\tlab[\\mu]\\}, t)$, while\n\\begin{equation}\n\\dot{Q}_\\mu = \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n  j_\\mu(t) \\lab[\\mu] \\vw\\cdot\\vsample[\\mu]\/2.\n\\end{equation} \nFinally, total entropy production $\\dot{S}^\\tot$ can also be split,\n\\begin{equation}\n  \\dot{S}^\\tot = \\sum_{n=1}^N \\dot{S}^\\tot_n + \\sum_{\\mu=1}^P \\dot{S}^\\tot_\\mu.\n\\end{equation}\nIt can easily be shown that each of these total entropy productions of a\nsubsystem obeys a separate second-law like inequality, \\emph{e.g.}\n\\begin{equation}\n  \\label{eq:2nd-law-short-n}\n  \\dot{S}^\\tot_n =\\dot{S}_n(\\tlabs, \\vw, \\labs) + \\dot{Q}_n \\geq 0\n\\end{equation}\nfor the $n$-th weight. \n\nWriting\n\\begin{equation}\np(\\tlabs, \\vw, \\labs) = p(\\wn)p(\\tlabs, \\vwOthers, \\labs |\\wn)\n\\end{equation}\nwith $\\vwOthers\\equiv(\\cdots, \\w_{n-1}, \\w_{n+1},\\cdots)$, we can split\n$\\dot{S}_n(\\tlabs,\\vw,\\labs)$ into two parts: first, the change of Shannon\nentropy of the marginalized distribution $p(\\wn)$,\n\\begin{equation}\n  \\dot{S}_n(\\wn) = \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\;\n   \\partial_n j_n(t) \\ln p(\\wn) = \\partial_t S(\\wn),\n\\end{equation}\nwhere the last equality follows from the fact that an entropy change of the\nmarginalized distribution $p(\\wn)$ can only come from the dynamics of $\\wn$. The\nsecond part is called the learning rate \\cite{s_hartich2014}\n\\begin{equation}\n  \\label{eq:lw}\n  l_n(\\w_n; \\tlabs, \\labs, \\vwOthers) = \n    - \\sum_{\\tlabs,\\labs} \\int_{-\\infty}^\\infty \\diff{\\vw} \\; \\partial_n j_n(t) \\ln p(\\tlabs,\\labs, \\vwOthers |\\w_n)\n\\end{equation}\nor information flow \\cite{s_allahverdyan2009,s_horowitz2014}. We emphasise that this\nlearning rate $l_n$ is thermodynamic and has nothing to do with the learning\nrate $\\lr$ that goes into the definition of the learning algorithms, see for\nexample eq.~\\eqref{eq:hebbian-force} of the main text. To avoid confusion, we\nwill refer to $l_n$ as the thermodynamic learning rate for the remainder of this\nsupplemental material. The second law \\eqref{eq:2nd-law-short-n} for the $n$-th\nweight hence becomes\n\\begin{equation}\n  \\label{eq:2nd-law-long-n}\n  \\dot{S}^\\tot_n = \\partial_t S(\\wn) + \\dot{Q}_n - l_n(\\w_n; \\tlabs, \\labs, \\vwOthers)\\geq 0\n\\end{equation}\nThe thermodynamic learning rate is a thermodynamically consistent measure of how\nmuch the dynamics of $\\wn$ change the mutual information\n$\\mutual{\\wn}{\\tlabs, \\vwOthers, \\labs}$, in particular for a system that\ncontinuously rewrites a single memory~\\cite{s_horowitz2014a}.\n\nWe can further refine the second law~\\eqref{eq:2nd-law-long-n} by exploiting the\ncausal structure of the dynamics, as was recently suggested by\nHorowitz~\\cite{s_horowitz2015}. The subsystem $\\wn$ directly interacts only with\nthose degrees of freedom that appear in its probability current $j_n(t)$\n\\eqref{eq:currents}. From inspection of the current $j_n(t)$, we see that $\\wn$\nis directly influenced only by itself and the given labels $\\tlabs$. Keeping\nthis in mind, we use the chain rule for mutual information~\\cite{s_cover2006} to\nwrite\n\\begin{equation}\n  \\mutual{\\wn}{\\tlabs, \\vwOthers, \\labs}= \\mutual{\\wn}{\\tlabs} +\n  \\mutualGiven{\\wn}{\\vwOthers, \\labs}{\\tlabs},\n\\end{equation}\nwhere we use the conditional mutual information\n\\begin{align}\n  \\label{eq:mutual-info-conditional}\n  \\mutualGiven{\\wn}{\\vwOthers, \\labs}{\\tlabs} =& S(\\wn|\\tlabs) -\n                                                 S(\\wn|\\vwOthers, \\labs, \\tlabs) \\\\\n =&-\n  \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty \\diff{\\vw} \\; p(\\tlabs, \\vw, \\labs) \\ln\n  \\frac{p(\\tlabs, \\vw, \\labs)p(\\tlabs)}{p(\\wn, \\tlabs)p(\\vwOthers, \\labs, \\tlabs)}.\n\\end{align}\nAccordingly, we split the thermodynamic learning rate~\\eqref{eq:lw} into a\nthermodynamic learning rate of the $n$-th weight with the degrees of freedom\nthat it directly interacts with, \\emph{i.e.} the true labels $\\tlabs$,\n\\begin{equation}\n  \\label{eq:lw-refined}\n  l_n(\\wn; \\tlabs) = - \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty \\diff{\\vw}\n  \\; \\partial_n j_n(t)\\ln p(\\tlabs|\\wn),\n\\end{equation}\nand a thermodynamic learning rate with the other subsystems given the true labels,\n\\begin{equation}\n  \\label{eq:lw-conditional}\n  l_n(\\wn; \\vwOthers, \\labs|\\tlabs) = - \\sum_{\\labs,\\tlabs}\\int_{-\\infty}^\\infty\n    \\diff{\\vw} \\; \\partial_n j_n(t)\\ln \\left(\\frac{p(\\wn, \\vwOthers, \\labs\n        |\\tlabs)}{p(\\wn|\\tlabs)p(\\vwOthers, \\labs|\\tlabs)}\\right).\n\\end{equation}\nHorowitz proved \\cite{s_horowitz2015} the following second-law like inequality\nincluding the refined thermodynamic learning rate~\\eqref{eq:lw-refined},\n\\begin{equation}\n  \\label{eq:2nd-law-refined}\n  \\partial_t S(\\wn) + \\dot{Q}_n - l_n(\\wn; \\tlabs) \\geq 0.\n\\end{equation}\nwhich is the basis for our proof of the main inequality,\nequation~\\eqref{eq:inequality} of the main text.\n\n\\section{Derivation of inequality~$\\eqref{eq:inequality}$ of the main text}\n\\label{sec:inequalities}\n\nThe stochastic thermodynamics of neural networks yields $N$ inequalities of the\nform~\\eqref{eq:2nd-law-refined}. Integrating over time and summing over all the\nweights, we find\n\\begin{equation}\n  \\sum_{n=1}^N \\left[ \\Delta S(\\wn) + \\Delta Q_n\\right] \\ge \\sum_{n=1}^N \\int_0^\\infty \\diff{t} \\; l_n(\\wn; \\tlabs) = \\sum_{n=1}^N \\Delta\n  \\mutual{\\wn}{\\tlabs}\n\\end{equation}\nThe precise definition of all the terms are discussed in the main text and in\nsection \\ref{sec:stochastic-thermodyanmics} of this supplemental material. The\ncrucial point for the last equality is that the labels $\\tlabs$ are static, so\nthat the mutual information $\\mutual{\\wn}{\\tlabs}$ changes only due to the\ndynamics of $\\wn$ and hence $\\partial_t \\mutual{\\wn}{\\tlabs}=l_n(\\wn;\\tlabs)$\n\\cite{Note6}.  To make progress towards our main result,\ninequality~\\eqref{eq:inequality} of the main text, we need to show that\n\\begin{equation}\n  \\label{eq:1}\n  \\sum_{n=1}^N \\Delta \\mutual{\\wn}{\\tlabs} \\ge \\sum_{\\mu=1}^P \\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]}.\n\\end{equation}\n\nFirst, we note that from the chain rule of mutual information \\cite{s_cover2006},\nwe have\n\\begin{equation}\n  \\Delta \\mutual{\\vw}{\\tlabs}=\\Delta \\mutual{\\w_1, \\dots, \\w_n}{\\tlabs}=\\sum_{n=1}^N\\Delta \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1}\n\\end{equation}\nwith the conditional mutual information \\cite{s_cover2006}\n\\begin{equation}\n  \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1} \\equiv S(\\wn|\\w_{n-1},\\dots,\\w_1) -\n  S(\\wn|\\tlabs, \\w_{n-1},\\dots,\\w_1).\n\\end{equation}\nDue to the form of the Langevin equation for the single weight,\neq.~\\eqref{eq:langevin}, individual weights are uncorrelated, and hence the\nconditional mutual information simplifies to\n\\begin{align}\n  \\Delta \\mutualGiven{\\wn}{\\tlabs}{\\w_{n-1},\\dots,\\w_1} &=\n                                                    \\Delta S(\\wn|\\w_{n-1},\\dots,\\w_1)-\\Delta S(\\wn|\\tlabs,\\w_{n-1},\\dots,\\w_1)\\\\\n  &=  \\Delta S(\\wn) - \\Delta S(\\wn|\\tlabs)\\\\\n  &= \\Delta \\mutual{\\wn}{\\tlabs}\n\\end{align}\nsuch that\n\\begin{equation}\n  \\sum_{n=1}^N\\Delta \\mutual{\\wn}{\\tlabs} = \\Delta \\mutual{\\vw}{\\tlabs}.\n\\end{equation}\n\nNext, we show that\n\\begin{equation}\n  \\label{eq:first-inequality}\n  \\Delta \\mutual{\\vw}{\\tlabs} = \\sum_{\\mu=1}^P\n  \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\stackrel{!}{\\ge} \\sum_{\\mu=1}^P\\Delta \\mutual{\\vw}{\\tlab[\\mu]}.\n\\end{equation}\nusing the independence of the given labels $\\tlabs$.  We first note that\n\\begin{align}\n  \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}= & \\Delta S(\\tlab[\\mu]|\\tlab[\\mu-1],\\dots,\\tlab[1])\n                                                               - \\Delta S(\\tlab[\\mu]|\\vw, \\tlab[\\mu-1],\\dots,\\tlab[1]) \\\\\n  =& \\Delta S(\\tlab[\\mu]) - \\Delta S(\\tlab[\\mu]|\\vw, \\tlab[\\mu-1],\\dots,\\tlab[1]) \n\\end{align}\nwhile\n\\begin{equation}\n  \\Delta \\mutual{\\vw}{\\tlab[\\mu]}=\\Delta S(\\tlab[\\mu]) - \\Delta S(\\tlab[\\mu]|\\vw)\n\\end{equation}\nHence for\n$\\Delta\n\\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\stackrel{!}{\\ge}\\Delta\n\\mutual{\\vw}{\\tlab[\\mu]}$, we need\n\\begin{align}\n  \\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}-\n           \\Delta \\mutual{\\vw}{\\tlab[\\mu]} & \\\\ \n  = \\quad & \\Delta S(\\tlab[\\mu]|\\vw) - \\Delta S(\\tlab[\\mu]|\\vw,\n            \\tlab[\\mu-1],\\dots,\\tlab[1]) \\label{eq:first-step} \\\\\n  = \\quad & \\Delta \\mutualGiven{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}{\\vw} \\\\\n  \\ge \\quad &  0\n\\end{align}\nwhere we first used that the $\\tlab[\\mu]$ are independent and identically\ndistributed. The last inequality follows since any mutual information,\nconditional or not, is always greater than or equal to zero \\cite{s_cover2006}. We\nhave thus shown that\n$\\Delta \\mutualGiven{\\vw}{\\tlab[\\mu]}{\\tlab[\\mu-1],\\dots,\\tlab[1]}\\ge\\Delta\n\\mutual{\\vw}{\\tlab[\\mu]}$ and hence~\\eqref{eq:first-inequality} is true.\n\nFinally, to prove that\n$\\Delta \\mutual{\\vw}{\\tlab[\\mu]}>\\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]}$, we\nconsider the full probability distribution $p(\\tlabs,\\vw,\\labs)$. From the\nmaster equation, eq.~\\eqref{eq:master}, we can write this distribution as\n\\begin{equation}\n  p(\\tlabs, \\vw, \\labs) = p(\\tlabs)p(\\vw|\\tlabs)\\left[p^{(0)}(\\labs|\\vw) +\n    \\frac{1}{\\gamma}p^{(1)}(\\labs|\\vw) + \\mathcal{O}(1\/\\gamma^2)\\right]\n\\end{equation}\nwith $\\gamma\\gg1$ for physiological reasons as described in the text -- it takes\nthe neuron longer to learn than to generate an action potential. Hence to first\norder, $\\tlabs\\rightarrow\\vw\\rightarrow\\labs$ is by definition a Markov chain~\\cite{s_cover2006}. Integrating out all the labels, true and predicted, except\nfor the $\\mu$-th one, we have the Markov chain\n$\\tlab[\\mu]\\rightarrow\\vw\\rightarrow\\lab[\\mu]$. For such a Markov chain, it is\neasy to show the following data processing inequality \\cite{s_cover2006},\n\\begin{equation}\n  \\Delta \\mutual{\\tlab[\\mu]}{\\vw} \\ge \\Delta \\mutual{\\tlab[\\mu]}{\\lab[\\mu]},\n\\end{equation}\nwhich completes our derivation.\n\n\\section{Hebbian learning in the thermodynamic limit}\n\\label{sec:hebbian-calculations}\n\nIn this section, we provide additional analytical calculations for Hebbian\nlearning in the thermodynamic limit for long times $t\\rightarrow\\infty$.\n\n\n\n\\subsection{Direct integration of the full distribution $p(\\tlabs, \\vw, \\labs)$ }\n\nTo compute the mutual information between the true and predicted label of a\ngiven sample, $\\mutual{\\tlab[\\mu]}{\\lab[\\mu]}$, we need the distribution\n$p(\\tlab[\\mu], \\lab[\\mu])$ or, since both $\\tlab[\\mu]$ and $\\lab[\\mu]$ are\nsymmetric binary random variables, the probability that\n$\\tlab[\\mu] = \\lab[\\mu]$. Our aim in this section is to obtain this probability\nfor Hebbian learning in the thermodynamic limit with $t\\rightarrow\\infty$ by\ndirect integration of the full distribution over the true labels, weights and\npredicted labels for a given set of samples $\\{\\vsample[\\mu]\\}$, which will also\ngive additional motivation for introducing the stability $\\Delta^\\mu$ of a\nsample.\n\nWe start with the full probability distribution\n\\begin{equation}\n  p(\\tlabs, \\vw, \\labs) = \\left(\\frac{1}{2}\\right)^P \\left(\\prod_{n=1}^N\n    \\frac{e^{-(\\wn-\\lr \\mathcal{F}_n)^2\/2}}{\\sqrt{2\\pi}}\\right)\n  \\left(\\prod_{\\mu=1}^P \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\\right),\n\\end{equation}\nwhere $\\lr$ is the learning rate and $\\mathcal{F}_n$ is a suitably scaled\naverage over the samples and labels,\n\\begin{equation}\n  \\mathcal{F}_n=\\frac{1}{\\sqrt{N}}\\sum_{\\rho=1}^P \\tlab[\\rho]\\sample[\\rho]_n\n\\end{equation}\nWhile the sum over the predicted labels $\\lab[\\rho\\neq\\mu]=\\pm1$ is trivial, we\ncan integrate over the true labels by noting that we can rewrite the exponent as\n\\begin{equation}\n  p(\\tlabs, \\vw, \\lab[\\mu]) = \\left(\\frac{1}{2}\\right)^P \\left(\\prod_{n=1}^N\n    \\frac{e^{-(\\wn-\\lr \\tlab[\\mu] \\sample[\\mu]_n\/\\sqrt{N}- \\lr \\mathcal{F}^{\\overline{\\mu}}_n)^2\/2}}{\\sqrt{2\\pi}}\\right)\n   \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nwhere the only dependence of the weight distribution on the true labels\n$\\tlab[\\rho\\neq\\mu]$ is now confined to the sum\n\\begin{equation}\n  \\mathcal{F}^{\\overline{\\mu}}_n\\equiv\\frac{1}{\\sqrt{N}}\\sum_{\\rho\\neq\\mu}^P \\tlab[\\rho]\\sample[\\rho]_n.\n\\end{equation}\nIn the thermodynamic limit, this allows us to replace the sum over all\n$\\tlab[\\mu\\neq\\rho]$ by an integral over the stochastic variable\n$\\mathcal{F}^{\\overline{\\mu}}_n$, which is normally distributed by the central\nlimit theorem and has mean 0 and variance $\\alpha$. Carrying out the integral,\nwe find\n\\begin{equation}\n  \\label{eq:3}\n  p(\\tlab[\\mu], \\vw, \\lab[\\mu]) =  \\left(\\prod_{n=1}^N\n    \\frac{e^{-(\\wn-\\lr \\tlab[\\mu] \\sample[\\mu]_n\/\\sqrt{N})^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\\right)\n  \\frac{e^{\\lab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nSince both $\\tlab[\\mu]$ and $\\lab[\\mu]$ are binary random variables and\n$\\tlab[\\mu]=\\pm1$ with equal probabilities, the mutual information between the\ntrue and predicted label can be written as\n\\begin{equation}\n  \\mutual{\\tlab[\\mu]}{\\lab[\\mu]} = \\ln 2 - S[p(\\tlab[\\mu]=\\lab[\\mu])]\n\\end{equation}\nwith the shorthand for the binary entropy\n$S[p]=-p \\ln p - (1-p)\\ln(1-p)$~\\cite{s_cover2006}. With $\\lab[\\mu]=\\tlab[\\mu]$ in\nthe exponential term of eq.~\\eqref{eq:3} and noting that\n$(\\tlab[\\mu]\\sample[\\mu]_n)^2=1$ for all $\\tlab[\\mu]$, $\\sample[\\mu]_n$, we then\nhave\n\\begin{equation}\n  \\label{eq:4}\n  p(\\tlab[\\mu] =\\lab[\\mu], \\vw) =  \\left(\\prod_{n=1}^N\n    \\frac{e^{-(\\wn \\tlab[\\mu] \\sample[\\mu]_n -\\lr\/\\sqrt{N})^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\\right)\n  \\frac{e^{\\tlab[\\mu]\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}{e^{-\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}+e^{\\vw\\cdot\\vsample[\\mu]\/2\\sqrt{N}}}\n\\end{equation}\nIt thus becomes clear that $\\vw\\cdot\\vsample[\\mu]\\tlab[\\mu]$ is the sum of $N$\nrandom variables with mean $\\lr\/\\sqrt{N}$ and variance $1+\\alpha\\lr^2$. We are\nthen motivated to introduce the stability of a sample,\n\\begin{equation}\n  \\Delta^\\mu \\equiv \\frac{1}{\\sqrt{N}} \\vw\\cdot\\vsample[\\mu]\\tlab[\\mu] = \\act^\\mu \\tlab[\\mu].\n\\end{equation}\nwhich, from eq.~\\eqref{eq:4}, is normally distributed with mean $\\lr$ and\nvariance $1+\\alpha\\lr^2$. Introducing the stability allows us to replace the\nintegral over all the weights by an integral over the stability,\n\\begin{equation}\n  p(\\tlab[\\mu] = \\lab[\\mu]) = \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\n  \\frac{e^{-(\\Delta^\\mu -\\lr)^2\/2(1+\\alpha\\lr^2)}}{\\sqrt{2\\pi(1+\\alpha\\lr^2)}}\n  \\frac{e^{\\Delta^\\mu}}{1+e^{\\Delta^\\mu}} = \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\n  p(\\Delta^\\mu) \\frac{e^{\\Delta^\\mu}}{1+e^{\\Delta^\\mu}}\n\\end{equation}\nwhich is the distribution obtained as eq.~\\eqref{eq:pR} of the main text.\n\n\\subsection{Direct derivation of the distribution of stabilities}\n\n\\label{sec:stabilities}\n\nLet us quickly show how the distribution of stabilities\n\\begin{equation}\n  \\label{eq:supp_stability}\n  \\Delta^\\mu \\equiv \\frac{1}{\\sqrt{N}} \\vw\\cdot\\vsample[\\mu]\\tlab[\\mu],\n\\end{equation}\n$\\mu=1,\\dots,P$, is obtained directly from its definition. The weights are given\nby\n\\begin{equation}\n  \\label{eq:supp_mean-weight}\n  \\vw = \\frac{1}{\\sqrt{N}}\\lr \\sum_{\\rho=1}^P \\vsample[\\rho]\\tlab[\\rho] + \\mathbf{y}\n\\end{equation}\nwith $\\mathbf{y}=(y_1,y_2,\\dots,y_N)$ where $y_n$ are normally distributed\nrandom variables with mean 0 and variance 1 arising from the thermal\nfluctuations in equilibrium. Substituting eq. \\eqref{eq:supp_mean-weight} into\n\\eqref{eq:supp_stability}, we have\n\\begin{align}\n  \\Delta^\\mu = & \\frac{1}{N} \\lr \\sum_{\\rho=1}^P \\tlab[\\rho]\\tlab[\\mu]\n                 \\vsample[\\rho]\\cdot\\vsample[\\mu] +\n                 \\frac{1}{\\sqrt{N}}\\tlab[\\mu]\\vsample[\\mu]\\cdot\\mathbf{y} \\\\\n  = & \\lr + \\frac{1}{N}\\lr\\sum_{\\rho\\neq\\mu}^P\\tlab[\\rho]\\tlab[\\mu]\n      \\vsample[\\rho]\\cdot\\vsample[\\mu] + \\frac{1}{\\sqrt{N}}\\tlab[\\mu]\\vsample[\\mu]\\cdot\\mathbf{y}\n\\end{align}\nwhere going to the last line we have used the fact that\n$\\vsample[\\mu]\\cdot\\vsample[\\mu]=N$. By inspection, we see that the second term\nis the sum of $N(P-1)\\approx NP$ random numbers $\\pm \\lr \/ N$ and the last term is\nthe sum of $N$ random numbers $y_n\/\\sqrt{N}$. By the central limit theorem,\n$\\Delta^\\mu$ is hence normally distributed with mean $\\overline{\\avg{\\Delta^\\mu}}=\\lr$ and\nvariance\n\\begin{equation}\n  \\overline{\\avg{(\\Delta^\\mu)^2}}-\\overline{\\avg{\\Delta^\\mu}}^2 = \\lr^2 + NP \\frac{\\lr^2}{N^2} +\n  N\\frac{1}{N}-\\lr^2 = 1+\\alpha\\lr^2.\n\\end{equation}\n\n\\subsection{Analytical approximation for $\\mutual{\\tlab}{\\lab}$}\n\n\\label{sec:approximation}\n\nWe quantify the success of learning using the mutual information per sample,\n\\begin{equation}\n  \\mutual{\\tlab[\\mu]}{\\lab[\\mu]} = \\ln 2 - S(p^\\mu_\\textrm{C})\n\\end{equation}\nwhere $S(p)=-[p \\ln p + (1-p)\\ln(1-p)]$ is the binary Shannon entropy and\n$p^\\mu_\\textrm{C}$ is defined as\n\\begin{equation}\n  \\label{eq:pC}\n  p^\\mu_\\textrm{C}\\equiv p(\\lab[\\mu]=\\tlab[\\mu]) = \\int_{-\\infty}^\\infty \\dd \\! \\Delta^\\mu \\; p(\\Delta^\\mu)\\frac{e^{\\Delta^\\mu}}{e^{\\Delta^\\mu}+1}\n\\end{equation}\nThe stabilities $\\Delta^\\mu$ are normally distributed with mean $\\lr$ and variance\n$1+\\alpha \\lr^2$ (see section \\ref{sec:stabilities}). This integral does not\nhave a closed-form analytical solution, but here we will demonstrate a very good\nanalytical approximation.\n\nTo that end, we first rewrite the sigmoid function in the integrand in terms of\nthe hyperbolic tangent and exploit the similarity of the latter to the error\nfunction:\n\\begin{align}\n  \\pC^\\mu = & \\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n          \\frac{e^{\\Delta^\\mu\/2}}{e^{\\Delta^\\mu\/2}+e^{-\\Delta^\\mu\/2}} \\\\\n      = & \\frac{1}{2} + \\frac{1}{2}\\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n          \\tanh(\\Delta^\\mu \/ 2) \\\\\n    \\simeq & \\frac{1}{2} + \\frac{1}{2}\\int_{-\\infty}^\\infty \\dd \\Delta^\\mu\\; p(\\Delta^\\mu)\n          \\erf(\\gamma \\Delta^\\mu \/ 2)\n\\end{align}\nwhere we choose $\\gamma=4\/5$ by inspection of the graphs of the two\nfunctions. Now the convolution of a normal distribution and an error function\nhas an exact solution,\n\\begin{equation}\n  \\frac{1}{\\sqrt{2\\pi d^2}}\\int_{-\\infty}^\\infty \\dd x \\erf(ax+b)\n  \\exp\\left(-\\frac{(x-c)^2}{2d^2}\\right)\n  = \\erf\\left(\\frac{b+ac}{\\sqrt{1+2a^2d^2}}\\right).\n\\end{equation}\nSetting $a=\\gamma\/2$, $b=0$, $c=\\lr$ and $d^2=1+\\alpha\\lr^2$, we find that\n\\begin{align}\n  \\pC^\\mu(\\alpha, \\lr) \\simeq & \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\gamma\\lr\/2}{\\sqrt{1+\\gamma^2(1+\\alpha\n                                \\lr^2)\/2}}\\\\\n  = & \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\lr\/2}{\\sqrt{25\/16+1\/2+\\alpha\n      \\lr^2\/2}}\\\\   \n  \\simeq &  \\frac{1}{2} + \\frac{1}{2}\\erf\\frac{\\lr\/2}{\\sqrt{2(1+\\alpha\n           \\lr^2\/4)}}\\\\\n  = & p(\\Delta^\\mu > 0 | \\alpha, \\lr\/2) \\label{eq:approximation}\n\\end{align}\nwhere in the last line we recognise by inspection that our result is nothing but\nthe integral over the distribution of stabilities $p(\\Delta^\\mu|\\alpha, \\lr\/2)$ from\n0 to $\\infty$. The probability that the neuron predicts the correct label is\nhence given by the probability that the neuron learned the label correctly,\n$\\Delta^\\mu>0$, with \\emph{half the learning rate}.\n\n\\bibliographystyle{aip}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Right Tail}\n\nLet us first describe the essentially inviscid \ninstantons producing the right tails of the PDFs \nfor gradients and differences \\cite{Pol95,95GM,GK96}. \nAt $t=0$, the \nfield $p$ is localized near the origin. At moving backwards in time the \nviscosity will spread the field $p$. Nevertheless, \na positive velocity slope\n``compresses'' the field $p$ so that one can expect that the width of $p$ remains\nmuch smaller than $L$.  Then, it is possible to formulate the closed \nsystem of equations for the quantities $a(t)$ and\n$c(t)=-i \\int dx\\,x\\,p(t,x)$ since for narrow $p$ and small $x$ we can put \n$\\int dx'\\chi(x-x')p(t,x')\\to-i\\partial_x\\chi(x)c(t)\\approx2i\\omega^3xc(t)$:\n\\begin{equation}\n\\partial_tc=2ac, \\quad\n\\partial_ta=-a^2+2\\omega^3c.\n\\label{vca} \\end{equation}\nThe instanton is a separatrix \nsolution of (\\ref{vca}). \nThe initial condition \n$a(0)c(0)=n$ by virtue of the energy conservation \ngives $a(0)=\\omega^3c^2(0)\/n=\\omega n^{1\/3}$. \nFor differences, $w=2a(0)\\rho$.\nOne can check that \n$ {\\cal I}_{\\rm extr}=i\\int dt\\, c\\partial_t a\\sim a(0)c(0)=n $ which \nis negligible in comparison with $n\\ln[a(0)]$ so that\n$\\langle(u')^n\\rangle\\sim[a(0)]^n \\sim \\omega^n n^{n\/3}$ \nwhich gives the right cubic tails of the PDFs \n$\\ln{\\cal P}(u')\\sim-(u'\/\\omega)^3$ \\cite{GK96} and\n$\\ln{\\cal P}(w)\\sim-[w\/(\\rho\\omega)]^3$ \n\\cite{Pol95,95GM}.  \nOne can show that the width of $p$ is much less than $L$ through the \ntime of evolution $T\\sim n^{-1\/3}\\omega^{-1}$ \ngiving the main contribution into the action \\cite{95GM}. \nThe right tails of ${\\cal P}(u')$ and ${\\cal P}(w)$ are thus\nuniversal i.e. independent of the large-scale properties of the pumping. \nAbove consideration does not imply that the instanton is completely inviscid,\nit may well have viscous shock at $x\\sim L$, this has no influence\non the instanton answer (since $p$ is narrow) while may influence the fluctuation\ncontribution i.e. predexponent in the PDF.\n\n\nThe main subject of this paper is the analysis of the instantons that give\nthe tails of ${\\cal P}(u)$ and the left tails of ${\\cal P}(u')$ \nand ${\\cal P}(w)$ corresponding to negative $a$, $w$. \nEven though the field $p$ is narrow at $t=0$, we cannot use the simple \nsystem (\\ref{vca}) to describe those instantons. \nThe reason is that sweeping by a negative velocity slope provides for \nstretching (rather than compression) of the field $p$ at moving backwards \nin time. As a result, the support of $p(x)$ stretches up to $L$ so that\none has to account for the given form of the pumping correlation function\n$\\chi(x)$ at $x\\simeq L$. This leads to a nonuniversality of ${\\cal P}(u)$ and of \nthe left tails of ${\\cal P}(u')$ and ${\\cal P}(w)$ which depend\non the large-scale properties of the pumping. As we shall see, the form of\nthe tails is universal, nonuniversality is related to a single constant in PDF.\nAdditional complication in analytical description is due to\nthe shock forming from negative slope near the\norigin. The shock cannot be described in terms of the inviscid \nequations so that we should use the complete system\n(\\ref{va2},\\ref{vam}) to describe what can be called\nviscous instantons.\n\nApart from a narrow front near $x=0$, the velocity field \nhas $L$ as the only characteristic scale of change. The life time $T$ of\nthe instanton is then determined by the moment when the position of $p$ \nmaximum reaches $L$ due to sweeping by the velocity \n$u_0$: $T\\sim L\/u_0$. Such a velocity $u_0$ itself has been created during \nthe time $T$ by the forcing so that $u_0\\sim|c|_{max}TL\\omega^3$. \nTo estimate the maximal value of $|c(t)|$, let us consider the backward \nevolution from $t=0$. We first notice that the width of $p$ (which was zero \nat $t=0$) is getting larger than the width of the velocity front $\\simeq u_0\/a$\nalready after the short time $\\simeq a^{-1}$. After that time, the \nvalues of $c$ and $a$ are of order of their values at $t=0$. \nThen, one may consider that $p(t,x)$ propagates (backwards in time) \nin the almost homogeneous velocity field $u_0$ so that \n$$\\partial_t c=-i\\int_{-\\infty}^{\\infty} dx\\, xup_x\\approx 2iu_0\\int_0^\\infty dx\\, p\n\\ .$$ The (approximate) integral\nof motion $i\\int dx\\, p$ can be estimated by it's value at $t=0$ which \nis $n\/2u_0$. Therefore, we get $c_{max}\\simeq nT$ so that\n$T\\simeq n^{-1\/3}\\omega^{-1}$ and $u_0\\simeq L\\omega n^{1\/3}$. \nAt the viscosity-balanced shock, the velocity $u_0$ and the gradient $a$ \nare related by $u_0^2\\simeq\\nu a$ so that $a(0)\\simeq \\omega{\\rm Re}\\,n^{2\/3}$.\n\nLet us briefly describe now the consistent analytic procedure of the derivation of\nthe function $c(t)$ that confirms above estimates. We use the\nCole-Hopf substitution \\cite{Burg} for the velocity $\\partial_x\\Psi=-{u}\\Psi\/{2\\nu}$\nand introduce $P=2\\nu\\partial_xp\/\\Psi$.\nThe saddle-point equations for $\\Psi$ and $P$ \n\\begin{eqnarray} &&\n\\partial_t\\Psi-\\nu\\partial_x^2\\Psi+\\nu F\\Psi=0,\n\\label{ha7} \\\\ &&\n\\partial_t P+\\nu\\partial_x^2P-\\nu FP\n-{2\\nu}\\lambda'(x)\\delta(t)\\Psi^{-1}=0\n\\label{ha3} \\end{eqnarray}\ncontain $F$ determined by $\\partial_xF(t,x)=-{i}\n\\int dx'\\chi(x-x')p(t,x')\/{2\\nu^2}$ and fixed by the condition $F(t,0)=0$. \nWe introduce the evolution operator $\\hat U(t)$\nwhich satisfies the equation $\\partial_t\\hat U=\\hat H\\hat U$ with\n$\\hat H(t)=\\nu(\\partial_x^2-F)$. It is remarkable that one \ncan develop the closed description\nin terms of two operators $\\hat A=\\hat U^{-1} x\\hat U$ and $\\hat B=\n\\hat U^{-1}\\partial_x\\hat U$:\n$$\\partial_t\\hat A=-2\\nu\\hat B\\,,\\quad \\partial_t\\hat B=-\\nu F_x(t,\\hat A)\\ .$$\nSince we are \ninteresting in the time interval when $p(t,x)$ is narrow, it is enough for our\npurpose to consider $x\\ll L$ where $F(t,x)=c(t)x^2\\omega^3\/2\\nu^2$. Further\nsimplification can be achieved in this case and the closed ODE for $c(t)$ can\nbe derived after some manipulations:\n$$(\\partial_t c)^2=4\\omega^3c^3+16\\xi_2^2+4\\omega^3\\xi_1^3\\ ,$$\nwhere $\\xi_1\\!=i\\int\\! dx\\lambda(x) x$ and $4\\xi_2\\!=-{i}\\int\\! dx\\lambda(x)\n\\partial_x[xu(0,x)]$. Integrating we get\n\\begin{eqnarray}&&\nt=\\frac{1}{2}\n\\int_{c(0)}^{c}\\frac{dx}{\\sqrt{\\omega^3x^3+4\\xi_2^2+\\xi_1^3}}\\ ,\n\\label{b7}\\end{eqnarray}\nwhich describes $c(t)$ in an implicit form. Further analysis depends on the\ncase considered. For the gradients, we substitute $\\xi_1=n\/a_0$ and $\\xi_2=-n\/2$\nand see that, as time goes backwards, negative $c(t)$ initially decreases by \nthe law $c(t)=c(0)+2nt$ until $T= \\omega^{-1}(n\/2)^{-1\/3}$ then it grows\nand the approximation looses validity when $c(t)$ approaches zero and the account\nof the pumping form $\\chi(x)$ at $x\\simeq L$ is necessary. Requiring the width\nof $p(x)$ at this time to be of order $L$ we \nget the estimate $a(0)\\simeq \\omega{\\rm Re}\\,n^{2\/3}$ and thus confirm the above\npicture.\nThe main contribution to the saddle-point value (\\ref{vaa})\nis again provided by the term $[\\partial_xu(0,0)]^n$ \nand we find $\\langle(u')^n\\rangle\\simeq[a(0)]^n\\simeq\n(\\omega{\\rm Re})^n n^{2n\/3}$, which corresponds to the following left \ntail of PDF at $u'\\gg \\omega{\\rm Re}$\n\\begin{equation} \n{\\cal P}(u')\\propto \n\\exp[-C(-u'\/\\omega{\\rm Re})^{3\/2}]\\ .\n\\label{an2} \\end{equation}\nFor higher derivatives $u^{(k)}$, by using (\\ref{b7}) we get initial growth \n$c(t)=c(0)+n(k+1)t$ which gives\n$u^{(k)}(0,0)\\sim N^{k+1}L^{1-k}\\omega {\\rm Re}^{k}$ leading to\n$\\langle[u^{(k)}]^n\\rangle\\sim\\omega {\\rm Re}^{k} \nL^{1-k} n^{(k+1)\/3}$ which can be rewritten in terms of PDF: \n\\begin{eqnarray} &&\n{\\cal P}\\left(|u^{(k)}|\\right)\\!\\propto\\!\n\\exp\\left[-C_k\\left({|u^{(k)}|L^{k-1}}\/\n{\\omega{\\rm Re}^k}\\right)^{3\/(k+1)}\\right].\n\\label{hd4}\\end{eqnarray}\nNote that the non-Gaussianity increases with increasing $k$. On the other hand,\nthe higher $k$ the more distant is the validity region of (\\ref{hd4}): \n$u^{(k)}\\gg u^{(k)}_{\\rm rms}\\sim L^{1-k}\\omega{\\rm Re}^k$.\n\nFor the differences, $\\xi_1={2n\\rho_0}\/{w}$ and\n$4\\xi_2=-{n}[1+{2\\rho_0u_x(0,\\rho_0)}\/{w}]$ and we get \n$\\langle w^n\\rangle\\simeq (L\\omega)^nn^{n\/3}$\nwhich corresponds to the cubic left tail \n\\begin{equation}  {\\cal P}(w)\\propto \\exp\\{-B[w\/(L\\omega)]^3\\}\n\\label{an3} \\end{equation}\nvalid at $w\\gg L\\omega$. In the intermediate region $L\\omega\\gg w\\gg\\rho\\omega$,\nthere should be a power asymptotics which is the subject of current debate\n\\cite{Pol95,GK96,KS96}.\nIt is natural that $\\rho$-dependence of ${\\cal P}(w)$ cannot be found in a \nsaddle-point approximation; as a predexponent, it can be obtained only at the\nnext step by calculating the contribution of fluctuations around the instanton\nsolution. This is consistent with the known fact that the scaling exponent\nis $n$-independent for $n>1$: $\\langle w^n(\\rho)\\rangle\\propto\\rho$.\n\nFor the velocity, $\\lambda(x)=-{in}\\delta(x)\/u(0,0)$ is an even function\nso that  $F$ is a linear (rather than quadratic)\nfunction of $x$ for narrow $p$: $F(x)={\\chi(0)bx}\/{2\\nu^2}$ with\n$b=-{i}\\int dx p(x)$. Direct calculation shows that energy and momentum\nconservation makes $b$ time independent: $b=n\/u(0,0)$.\nIt is easy then to get the $n$-dependence of $u(0,0)$: \nVelocity stretches the field $p$ so that the width of $p$\nreaches $L$ at $T\\simeq L\/u(0,0)$ while the velocity itself is produced by\nthe pumping during the same time: $u(0,0)\\simeq \\chi(0)bT=\\chi(0)nT\/2u(0,0)\n\\simeq n\\chi(0)L\/u(0,0)$. That gives $u(0,0)\\simeq L\\omega n^{1\/3}$ and\n$$ {\\cal P}(u)\\propto \\exp\\{-D[u\/(L\\omega)]^3\\}\\ .$$ \nThe product $L\\omega$ plays the role of the root-mean square velocity\n$u_{\\rm rms}$. The numerical factors $C$, $B$ and $D$ \nare determined by the evolution at $t\\simeq T$ i.e. by the behavior of pumping\ncorrelation function $\\chi(x)$ at $x\\simeq L$.\n\n\nWe thus found the main exponential factors in the PDF tails. \nComplete description of the tails requires the analysis\nof the fluctuations around the instanton which will be the subject of\nfuture detailed publications. Here, we briefly outline some important\nsteps of this analysis. The account of the fluctuations in the Gaussian\napproximation is straightforward and leads to the shift of ${\\cal I}_{\\rm extr}$\ninsignificant at $n\\gg1$. However, the terms of the perturbation theory with \nrespect to the interaction of fluctuations are infrared divergent (proportional\nto the total observation time). That means that there is a soft mode which\nis to be taken into account exactly. Such an approach has been already developed\nin \\cite{95FKLM} for the simpler problem of the PDF tails for a passive scalar\nadvected by a large-scale velocity where the comparison with the exactly\nsolvable limits was possible. \nA soft mode usually corresponds to a global symmetry\nwith a continuous group: if one allows the slow spatio-temporal\nvariations of the parameters of the transformation then small variations of\nthe action appears. \nOur instantons break Galilean invariance so that the respective\nGoldstone mode has to be taken into account. Namely, under the transformation\n\\begin{equation}x\\rightarrow x-r,\\ u(x)\\rightarrow u(x-r)+v,\\ \nr=\\int_t^0v(\\tau)d\\tau\\ ,\\label{sym}\\end{equation}\nthe action is transformed as ${\\cal I}\\rightarrow{\\cal I}-i\\int dxdtp\\partial_tv$.\nThe source term $\\int dxdt\\lambda u$ is invariant with respect to (\\ref{sym}) for\nantisymmetric $\\lambda(x)$. To integrate exactly along the direction specified by\n(\\ref{sym}) in the functional space we use Faddeev-Popov trick \ninserting  \nthe additional factor \n\\begin{equation}\n1=\\int{\\cal D}v(t)\\delta\\left[u\\biggl(t,\\int_t^0v(\\tau)d\\tau\\biggr)-\nv(t)\\right] {\\cal J}\\ .\n\\label{unity}\\end{equation}\ninto the integrand in (\\ref{si1},\\ref{sio}).\nJacobian ${\\cal J}$ is determined by a regularization of (\\ref{sym})\naccording to our choice of the retarded regularization for the initial\nintegral: at \ndiscretizing time we put $\\partial_tu+u\\partial_xu\n\\rightarrow ({u_n-u_{n-1}})\/{\\epsilon}+u_{n-1} u'_{n-1}$ (otherwise, \nsome additional $u$-dependent term appears \\cite{DP78}). \nThe discrete version of (\\ref{sym}),\n$p_n(x)\\rightarrow p_n(x-\\epsilon\\sum_{j=n}^{N-1}v_j)$,\n$u_n(x)\\rightarrow u_n(x-\\epsilon\\sum_{j=n}^{N-1}v_j)-v_n$, $u_N(x)\\rightarrow\nu_N(x)-v_N$ gives \n$${\\cal J}=\\exp\\left[\\int_{-T}^0dtu'\\biggl(t,\\int_t^0 v(\\tau)d\\tau\\biggr)\\right]\\ .$$\nSubstituting (\\ref{unity}) into (\\ref{si1},\\ref{sio})\nand making (\\ref{sym}) we calculate $\\int{\\cal D}v$ as a Fourier integral \n(the saddle-point method is evidently\ninapplicable to such an integration) and conclude that after the integration\nover the mode (\\ref{sym}) the measure ${\\cal D}u{\\cal D}pe^{i{\\cal I}}$\nacquires the additional factor\n$$\\prod\\limits_t\\!\\delta\\biggl[\\int\\partial_t^2 p(t,x)dx\\biggr]\n\\delta[u(t,0)]\\exp\\left[\n\\int_{-T}^0\\!u'(t,0)dt\\right].$$\nThe last (jacobian) term here exactly corresponds\nto the term $u'{\\cal P}(u')$  in the equation for ${\\cal P}(u')$ \nderived in \\cite{GK93,GK96}. This term \nmakes the perturbation theory for the fluctuations around the instanton to be\nfree from infrared \ndivergences, the details will be published elsewhere. \n\n\nLet us summarize.\nAt smooth almost inviscid ramps, velocity differences and gradients are \npositive and linearly related $w(\\rho)\\approx \n2\\rho u'$ so that the right tails of PDFs have the same cubic form \n\\cite{Pol95,95GM,GK96}.\nThose tails are universal i.e. they are determined by a single characteristics\nof the pumping correlation function $\\chi(r)$, namely, by it's second\nderivative at zero $\\omega=[-(1\/2)\\chi''(0)]^{1\/3}$. \nContrary, the left tails found here contain nonuniversal constant which \ndepends on a large-scale behavior of the pumping. The left tails\ncome from shock fronts where $w^2\\simeq -\\nu u'$ so that cubic tail \nfor velocity differences (\\ref{an3}) corresponds to semi-cubic tail for gradients\n(\\ref{an2}).\nThe formula (\\ref{an3}) is valid for $w\\gg u_{\\rm rms}\\simeq L\\omega$\nwhere ${\\cal P}(w)$ should coincide with \na single-point ${\\cal P}(u)$ since the  \nprobability is small for both $u(\\rho)$ and $u(-\\rho)$ being large \nsimultaneously. Indeed, we saw that the tails of\n$\\ln{\\cal P}(u)$ at $u\\gg u_{\\rm rms}$ are cubic as well.\nNote that \n(\\ref{hd4}) is the same as obtained for decaying turbulence with white (in space)\ninitial conditions\nby a similar method employing the saddle-point approximation in the \npath integral with time as large parameter \\cite{Avel}. \nThat, probably, means that white-in-time forcing\ncorresponds to white-in-space initial conditions. Note that\nif the pumping has a finite correlation time $\\tau$ then our\nresults, strictly speaking, are valid for \n$u,w\\ll L\/\\tau$ and $u'\\ll1\/\\tau$.\n\nWe are grateful to M. Chertkov, V. Gurarie, D. Khmelnitskii, R. Kraichnan \nand A. Polyakov for useful discussions. \nThis work was supported \nby the Minerva Center for Nonlinear Physics (I. K. and V. L.), \nby the Minerva Einstein Center (V. L.), by the Israel Science Foundation\nfounded by the Israel Academy (E.B.) and by \nthe Cemach and Anna Oiserman Research Fund (G.F.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}