diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkjar" "b/data_all_eng_slimpj/shuffled/split2/finalzzkjar"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkjar"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction and results}\n\n\\subsection{Introduction.} \nWe study the basic ordinary differential equation\n\\begin{align*}\n-\\left(a\\left(\\frac{x}{\\varepsilon}\\right) u_{\\varepsilon}'(x)\\right)' &= f(x) \\quad \\mbox{for} ~x \\in (0,1)\\\\\nu_{\\varepsilon}(0) = 0& = u_{\\varepsilon}(1)\n\\end{align*}\nwhere $a:\\mathbb{R} \\rightarrow \\mathbb{R}_{+}$ is a strictly positive function with period 1, the Dirichlet\nboundary conditions $u_{\\varepsilon}(0) = 0 = u_{\\varepsilon}(1)$ are prescribed, and $f$ is assumed to be smooth.\nIt is classical \\cite{avi, ben, jikov, papa, pap} that $u_{\\varepsilon}$ converges to a limiting function $u$ as $\\varepsilon \\rightarrow 0^+$, where\n$u$ solves the homogenized equation\n\\begin{align*}\n-\\left(a u_{}'(x)\\right)' &= f(x) \\quad \\mbox{for} ~x \\in (0,1)\\\\\nu_{}(0) = 0& = u_{}(1) \n\\end{align*}\nand the constant $a > 0$ is given by\n$$ a = \\left( \\int_{0}^{1}{\\frac{1}{a(x)} dx}\\right)^{-1}.$$\nMoreover, convergence occurs with a linear rate as $\\varepsilon \\rightarrow 0^+$ \n$$ \\| u_{\\varepsilon}(x) - u(x)\\|_{L^{\\infty}} \\lesssim_{a(x), f} \\varepsilon$$\nand it is not difficult to see that this estimate is sharp.  It is often the case that local averages are smoother objects and we show that a similar phenomenon occurs here: local averages can improve convergence by an order of magnitude. This can also be motivated by a formal multiple scale expansion: one would expect the second term to have mean value 0 and to oscillate at scale $\\sim \\varepsilon$ suggesting that such a result may hold.\n\n\\subsection{The Result} Our result shows that local averages converge an entire order faster whenever the coefficient $a(\\cdot)$ satisfies a suitable condition. We show that, among other things, $a(x) = a(1-x)$ implies that\n$$ \\left| \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ u_{\\varepsilon}(y) dy} - u(x) \\right| \\lesssim_{a, f}  \\varepsilon^2.$$\nThe result is not generally true and there is an explicitly given linear function $\\ell_{\\varepsilon}$\nthat has to be corrected for first to reach the same order of convergence.\n\n\n\\begin{thm}  Let $u_{\\varepsilon}$ and $u$ be defined as above. There exists an affine corrector $\\ell_{\\varepsilon}(x)$ given by\n \\begin{align*}\n\\ell_{\\varepsilon}(x) = \\varepsilon  \\left(\\int_{0}^{1}{f(x) dx}\\right) \\left( \\int_{0}^{1}{ a(y)^{-1}(y-\\frac12)dy}\\right) x +  \\varepsilon\\left(\\int_{0}^{1}{ \\int_{0}^{y}{f(z)dz} dy}\\right)  \\int_{-\\frac12}^{\\frac12}{ \\int_{0}^{y}{a\\left(z\\right)^{-1}  dz} dy}\n\\end{align*}\nsuch that for $\\varepsilon^{-1} \\in \\mathbb{N}$ and $x \\in (\\varepsilon, 1-\\varepsilon)$\n$$ \\left| \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ u_{\\varepsilon}(y) dy} + \\ell_{\\varepsilon}(x) - u(x) \\right| \\lesssim_{a, f}  \\varepsilon^2.$$\n\\end{thm}\n\nWe note that the algebraic structure of $\\ell_{\\varepsilon}$ implies that $\\ell \\equiv 0$ whenver $a(x) = a(1-x)$, however, the actual condition is actually quite a bit more general than\nthat. We also note that $\\ell$ vanishes for certain types of right-hand sides $f$ (a space of codimension 2 that contains, for example, the function $f(x) = x^2 - x +1\/6$). This seems like an interesting phenomenon.\\\\\n\nWe are not aware of any result at that level of precision. However, there is a great deal of related work at a greater level of generality \\cite{arm, arm2, ch, glorm1, glor0, glorr, glor1, glor2, koz, jikov, sm}.\nThe result is closely related to Allaire's two-scale approach \\cite{all1, all2} and could be derived from a two-scale expansion. It remains to be seen whether our result has analogues in higher dimensions\nor whether it is a one-dimensional 'miracle'. \nOur result deals with integrating an indicator function $\\chi_{[-\\varepsilon\/2, \\varepsilon\/2]}$ against $u_{\\varepsilon}$,\none could ask what happens if one were to take other averaging operators such as $\\chi_{[-\\varepsilon\/2, \\varepsilon\/2]} * \\chi_{[-\\varepsilon\/2, \\varepsilon\/2]}$ and whether there are higher-order analogues of our result in one dimension.\n\n\n\n\\subsection{Another proof.}\nWe conclude the paper with a description of an approach to homogenization that we came across by coincidence.\n A short summary is as follows:\n\\begin{enumerate}\n\\item Extend the elliptic equation in time by turning the elliptic operator $Lu = f$ into a parabolic operator $(\\partial_t - L)u=f$; the solution becomes stationary in time.\n\\item Use the Feynman-Kac formula to produce reproducing identities for $u(x)$ and $u_{\\varepsilon}(x)$ by writing them as weighted integral averages over\na neighborhood around $x$.\n\\item Algebraic manipulation leads to an inequality of the form\n$$ |u_{\\varepsilon}(x) - u(x)| \\leq \\mbox{error} + \\left( \\mbox{solution of heat equation starting from}~|u_{\\varepsilon} - u|\\right).$$\nThis type of bootstrapping can then be exploited to obtain $L^{\\infty}-$estimates for $u_{\\varepsilon} - u$.\n\\end{enumerate}\n\nGenerally, this seems to reduce the problem of obtaining quantitative estimates to controlling the diffusion induced by the infinitesimal operators. That underlying idea is\nnot new, we refer to work of Gloria \\& Otto \\cite{glor1, glor2} in the stochastic setting. \nWe illustrate this technique for the most basic case discussed above in \\S 3 and hope that it might have more general applications.\n\n\n\n\\section{Proof of the Theorem }\n\\begin{proof} We make use of the explicit solution formula\n$$ u_{\\varepsilon}(x) = \\int_{0}^{x}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1} \\left( c_{\\varepsilon} - \\int_{0}^{y}{f(z) dz} \\right) dy},$$\nwhere\n$$ c_{\\varepsilon} = \\left( \\int_{0}^{1}{ a\\left(x\\right)^{-1} dx}\\right)^{-1} \\int_{0}^{1}{ \\left(a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x}{f(y)dy}\\right)dx}.$$\nWe will now use this formula to analyze\n$$ \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ u_{\\varepsilon}(y) dy} =  \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{\n\\int_{0}^{y}{ a\\left(\\frac{z}{\\varepsilon}\\right)^{-1} \\left( c_{\\varepsilon} - \\int_{0}^{z}{f(w) dw} \\right) dy}\n}.$$\nThe analysis decouples into three parts: analyzing the leading oscillation term, suitably approximating the constant $c_{\\varepsilon}$ and analyzing the term involving the function $f$\nand we will carry out the argument in that order.\\\\\n\n\\textit{The leading oscillation term.} The first term is the easiest since it grows linearly. \nDifferentiation in $x$ leads to\n$$ \\frac{\\partial}{\\partial x}   \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2} \\int_{0}^{y}{a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}  dz} dy = \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{a_{}\\left(\\frac{y}{\\varepsilon}\\right)^{-1} dy} = \\int_{0}^{1}{a(y)^{-1}dy}$$\nwhich implies that the function is linear and\n$$  \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2} \\int_{0}^{y}{a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}  dz} dy = \\left(\\int_{0}^{1}{a(y)^{-1}dy}\\right) x + \\frac{1}{\\varepsilon}\n \\int_{-\\varepsilon\/2}^{\\varepsilon\/2} \\int_{0}^{y}{a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}  dz} dy.$$\n\n\n\\textit{The constant $c_{\\varepsilon}$.} We now compute the constant $c_{\\varepsilon}$. This computation makes explicit use of $\\varepsilon^{-1} \\in \\mathbb{N}$. We split the integral into the basic intervals\n\\begin{align*}\n\\int_{0}^{1}{ \\left(a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x}{f(y)dy}\\right)dx} = \\sum_{k=0}^{\\varepsilon^{-1}-1}{ \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{ \\left(a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x}{f(y)dy}\\right)dx}}\n\\end{align*}\nand use \n\\begin{align*}\n \\int_{k \\varepsilon}^{(k+1)\\varepsilon}  a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x} f(y)dy dx \n&= \\int_{k \\varepsilon}^{(k+1)\\varepsilon} a\\left(\\frac{x}{\\varepsilon}\\right)^{-1}\\int_{0}^{(k+1\/2)\\varepsilon}f(y)dy dx\\\\\n&+ \\int_{k \\varepsilon}^{(k+1)\\varepsilon}   a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{(k+1\/2)\\varepsilon}^{x}f(y)dydx.\n\\end{align*}\nA Taylor expansion up to first order, using $f \\in C^1$, shows that\n\\begin{align*}\n  \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{  a\\left(\\frac{x}{\\varepsilon}\\right)^{-1}   \\int_{(k+\\frac12)\\varepsilon}^{x}{f(y)dy}dx} &= \\varepsilon^2 f((k+1\/2)\\varepsilon)\\int_{0}^{1}{a(x)^{-1}(x-\\frac12) dx} \\\\\n&+ \\mathcal{O}(\\|f'\\|_{L^{\\infty}} \\varepsilon^3)\n\\end{align*}\nAltogether, this shows that\n\\begin{align*}\n \\int_{0}^{1}  a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x} f(y)dy &= \\sum_{k=0}^{\\varepsilon^{-1}-1}  \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{   a\\left(\\frac{x}{\\varepsilon}\\right)^{-1}  \\int_{0}^{(k+\\frac12)\\varepsilon}{f(y)dy} dx} \\\\\n&+ \\varepsilon^2 \\left(\\int_{0}^{1}{a(x)^{-1}(x-\\frac12) dx}  \\right)  \\sum_{k=0}^{\\varepsilon^{-1}-1}  f((k+\\frac12)\\varepsilon) \\\\\n&+ \\mathcal{O}(\\|f'\\|_{L^{\\infty}} \\varepsilon^2).\n\\end{align*}\nThe first term simplifies due to the periodicity of $a(\\cdot)$ to\n$$  \\sum_{k=0}^{\\varepsilon^{-1}-1}  \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{   a\\left(\\frac{x}{\\varepsilon}\\right)^{-1}  \\int_{0}^{(k+1\/2)\\varepsilon}{f(y)dy} dx} =\n \\varepsilon \\left(\\int_{0}^{1}{a(z)^{-1} dz} \\right) \\sum_{k=0}^{\\varepsilon^{-1}-1}   \\int_{0}^{(k+1\/2)\\varepsilon}{f(y)dy} $$\nThere are two sums that are left to be evaluated: one is essentially the midpoint rule, a Taylor expansion shows that\n$$\n  \\int_{0}^{(k+1\/2)\\varepsilon}{f(y)dy} = \\frac{1}{\\varepsilon} \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{ \\int_{0}^{y}{f(z)dz} dy} + \\mathcal{O}(\\|f'\\|_{L^{\\infty}} \\varepsilon^2)\n$$\nIn combination, we obtain for the first sum that\n\\begin{align*}\n \\sum_{k=0}^{\\varepsilon^{-1}-1}   \\int_{0}^{(k+1\/2)\\varepsilon}{f(y)dy} &=  \\sum_{k=0}^{\\varepsilon^{-1}-1} \\left(  \\frac{1}{\\varepsilon} \\int_{k \\varepsilon}^{(k+1)\\varepsilon}{ \\int_{0}^{y}{f(z)dz} dy} + \\mathcal{O}(\\|f'\\|_{L^{\\infty}} \\varepsilon^2) \\right) \\\\\n&=  \\frac{1}{\\varepsilon} \\int_{0}^{1}{ \\int_{0}^{y}{f(z)dz} dy} + \\mathcal{O}(\\|f'\\|_{L^{\\infty}} \\varepsilon).\n\\end{align*}\nThe second sum follows from another application of the midpoint rule in the form\n$$ \\varepsilon f((k+1\/2)\\varepsilon) =   \\int_{k\\varepsilon}^{(k+1)\\varepsilon}{ f(x) dx} + \\mathcal{O}(\\|f''\\|_{L^{\\infty}} \\varepsilon^3)$$\nto simplify\n\\begin{align*}\n \\varepsilon^2 \\left(\\int_{0}^{1}{a(x)^{-1}(x-1\/2) dx}  \\right)  \\sum_{k=0}^{\\varepsilon^{-1}-1}  f((k+1\/2)\\varepsilon) &= \\varepsilon  \\left(\\int_{0}^{1}{a(x)^{-1}(x-1\/2) dx}  \\right) \\int_{0}^{1}{f(x) dx} \\\\\n&+ \\mathcal{O}( \\|f''\\|_{L^{\\infty}} \\varepsilon^2)\n\\end{align*}\nCollecting all these computations, we see that\n\\begin{align*} c_{\\varepsilon} = \\left(\\int_{0}^{1}{ \\int_{0}^{y}{f(z)dz}} dy\\right)  &+ \\left(\\int_{0}^{1}{f(x) dx}\\right) \\left( \\int_{0}^{1}{ a(x)^{-1}(x-1\/2)dx}\\right)    \\\\\n&+ \\mathcal{O}(\\varepsilon^2 \\|f'\\|_{L^{\\infty}})  + \\mathcal{O}(\\varepsilon^2 \\|f''\\|_{L^{\\infty}}).\n\\end{align*}\n\n\n\\textit{The remaining term.} We now analyze the remaining term \n$-J$ given by\n$$ J = \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{\\int_{0}^{y}{ a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}  \\int_{0}^{z}{f(w) dw}  dy}}.$$\nAs before, we gain some insight into the expression by differentiating it first in $x$ and using yet another Taylor expansion (omitting higher order terms)\n\\begin{align*}\n \\frac{\\partial}{\\partial x} J &= \\frac{\\partial}{\\partial x} \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{\\int_{0}^{y}{ a\\left(\\frac{z}{\\varepsilon}\\right)^{-1} \\int_{0}^{z}{f(w) dw}  dz dy}} \\\\\n&=\\frac{1}{\\varepsilon} {\\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1} \\int_{0}^{y}{f(z) dz} dy}} \\\\\n&= \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1} \\left( \\int_{0}^{x}{f(z) dz} + (y-x)f(x) + \\frac{(y-x)^2}{2} f'(x)  \\right)dy}.\n\\end{align*}\nThe quadratic term is so small that it has no big effect\n\\begin{align*}\n\\left| \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1}  \\frac{(y-x)^2}{2} f'(x) dy} \\right| \\lesssim_{a(\\cdot)} \\varepsilon^2 \\|f'\\|_{L^{\\infty}}\n\\end{align*}\n and we obtain\n\\begin{align*}\n \\frac{\\partial}{\\partial x} J  &= \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1} \\left( \\int_{0}^{x}{f(z) dz} + (y-x)f(x)  \\right)dy} + \\mathcal{O}( \\varepsilon^2 \\|f'\\|_{L^{\\infty}}) \\\\\n&=  \\left(  \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1} dy} \\right)\\left(\\int_{0}^{x}{f(z) dz}\\right) + \\frac{f(x) }{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1}(y-x) dy} +  \\mathcal{O}( \\varepsilon^2 \\|f'\\|_{L^{\\infty}}) \\\\\n&= \\left(  \\int_{0}^{1}{ a\\left(y\\right)^{-1} dy} \\right)\\left(\\int_{0}^{x}{f(z) dz}\\right) + \\frac{f(x) }{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1}(y-x) dy} +  \\mathcal{O}( \\varepsilon^2 \\|f'\\|_{L^{\\infty}}).\n\\end{align*}\n\nWe study the error term by remarking that the function\n$$ h(x) = \\frac{1}{\\varepsilon}\\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ a\\left(\\frac{y}{\\varepsilon}\\right)^{-1}(y-x) dy}$$\nsatisfies $\\| h\\|_{L^{\\infty}} \\lesssim \\varepsilon$ (which is obvious) and\n\\begin{align*}\n \\frac{1}{\\varepsilon} \\int_{x}^{x+\\varepsilon}{h(y) dy} &=   \\frac{1}{\\varepsilon}\\int_{x}^{x+\\varepsilon}{\\int_{y-\\varepsilon\/2}^{y+\\varepsilon\/2}{ a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}(z-y) dz dy}} = 0.\n\\end{align*}\nThis implies, using another Taylor expansion of $f$, that\n$$ \\left| \\int_{0}^{x}{ \\frac{f(y) }{\\varepsilon} \\int_{y-\\varepsilon\/2}^{y+\\varepsilon\/2}{ a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}(z-y) dz}dy dx} \\right| \\lesssim_{a(\\cdot)} \\varepsilon^2 \\|f''\\|_{L^{\\infty}}.$$\nThis shows that\n\\begin{align*}\nJ &=  \\left(  \\int_{0}^{1}{ a\\left(y\\right)^{-1} dy} \\right) \\int_{0}^{x}{ \\int_{0}^{y}{f(z) dz}dy} \\\\\n &+  \\frac{1}{\\varepsilon} \\int_{-\\varepsilon\/2}^{+\\varepsilon\/2}{ a\\left(\\frac{x}{\\varepsilon}\\right)^{-1} \\int_{0}^{x}{f(y) dy}  dx}+ \\mathcal{O}_{a(\\cdot)}(\\|f'\\|_{L^{\\infty}}\\varepsilon^2)\n\\end{align*}\n\\textit{Conclusion.} Collecting all the various estimates and terms, we now see that\n\\begin{align*}\n \\frac{1}{\\varepsilon} \\int_{x-\\varepsilon\/2}^{x+\\varepsilon\/2}{ u_{\\varepsilon}(y) dy} &=  \\left(\\int_{0}^{1}{a(y)^{-1}dy}\\right) x   \\left(\\int_{0}^{1}{ \\int_{0}^{y}{f(z)dz}} dy\\right)  -   \\left(  \\int_{0}^{1}{ a\\left(y\\right)^{-1} dy} \\right) \\int_{0}^{x}{ \\int_{0}^{y}{f(z) dz}dy}  \\\\\n& + \\mathcal{O}_{a(\\cdot)}( (\\|f'\\|_{L^{\\infty}} + |f''\\|_{L^{\\infty}} ) \\varepsilon^2) + \\ell(x)\n\\end{align*}\nwhere\n\\begin{align*}\n \\ell(x) &= \\varepsilon  \\left(\\int_{0}^{1}{f(x) dx}\\right) \\left( \\int_{0}^{1}{ a(y)^{-1}(y-1\/2)dy}\\right) x \\\\\n&+  \\varepsilon\\left(\\int_{0}^{1}{ \\int_{0}^{y}{f(z)dz} dy}\\right)    \\frac{1}{\\varepsilon}  \\int_{-\\varepsilon\/2}^{\\varepsilon\/2}{ \\int_{0}^{y}{a\\left(\\frac{z}{\\varepsilon}\\right)^{-1}  dz} dy}.\n\\end{align*}\nThe main term is exactly the solution formula for the homogenized equation and from this the desired result follows.\n\\end{proof}\n\n\n\n\n\\section{Outline of another approach}\nIn this section we outline another approach to homogenization and illustrate it in its most basic form for the problem \n\\begin{align*}\n-\\left(a\\left(\\frac{x}{\\varepsilon}\\right) u_{\\varepsilon}'(x)\\right)' &= f(x) \\quad \\mbox{for} ~x \\in (0,1)\\\\\nu_{\\varepsilon}(0) = 0& = u_{\\varepsilon}(1)\n\\end{align*}\nwhere $a:\\mathbb{R} \\rightarrow \\mathbb{R}_{+}$ is a strictly positive function with period 1, the Dirichlet\nboundary conditions $u_{\\varepsilon}(0) = 0 = u_{\\varepsilon}(1)$ are prescribed, and $f$ is assumed to be smooth.\n\n\\subsection{Two reproducing identities.}\nThe key idea behind our approach is to extend the equation in time by making it parabolic; since $u_{\\varepsilon}$ actually solves the equation, it becomes a stationary-in-time solution\nof a heat equation which can be studied with probabilistic methods (these ingredients are, of course, classical for the study of homogenization of parabolic equations, see \\cite[Section 2]{jikov}). We derive reproducing identities for both $u_{\\varepsilon}$ and $u$ and will use\nthem to bootstrap a bound. More precisely, we will study solutions of the heat equation\n\\begin{align*}\n\\frac{\\partial}{\\partial t} u_{\\varepsilon}(t,x)  -\\left(a^{}\\left(\\frac{x}{\\varepsilon}\\right) u_{\\varepsilon}'(t,x)\\right)' &= f(x) \\quad \\mbox{on} ~(0,1)\\\\\nu_{\\varepsilon}(t,0) = 0&, u_{\\varepsilon}(t,1) = 0.\n\\end{align*}\nBy construction, $u_{\\varepsilon}(t,x) = u_{\\varepsilon}(x)$ is a stationary solution in time. However, this heat equation also has a probabilistic interpretation. \nWe use $\\omega_x(t)$ to denote Brownian motion started in $x$ and running for $t$ units of time subjected to diffusivity $a^{}(x\/\\varepsilon)$. If it exits the domain $[0,1]$ before $t$ units of time\nhave passed, we assume that $\\omega_x(t)$ remains stationary at the point of exit (the boundary is 'sticky'). Then the Feynman-Kac formula implies\n$$ u_{\\varepsilon}(x) = \\mathbb{E} u_{\\varepsilon}(\\omega_x(t)) +  \\mathbb{E}\\int_{0}^{t}{ f(\\omega_x(s)) ds}.$$\nWe introduce a second Brownian motion $\\nu_x$ which is merely governed by diffusion w.r.t. to the homogenized diffusion coefficient (and 'stickiness' w.r.t. the boundary) which allows\nus to write the homogenized equation in a similar manner\n$$ u_{}(x) = \\mathbb{E} u_{}(\\nu_x(t)) +  \\mathbb{E}\\int_{0}^{t}{ f(\\nu_x(s)) ds}.$$\nWe will show that these two reproducing identities in combination with basic bounds on the two types of diffusions are sufficient to bootstrap everything into a quantitative estimate.\n\n\\subsection{Bootstrapping}\nIt will be more convenient to work with probability distributions. We will denote the distribution of $\\omega_x(t)$ in point $y$ by $k_{t}(x,y)$ and the distribution of\n$\\nu_x(t)$ in point $y$ by $\\ell_t(x,y)$. Note that both are continuous in the interval $(0,1)$ but do have atomic masses at the boundary points of the interval because these endpoints\nare sticky. In particular, they are not probability distributions on $(0,1)$ because they do not integrate to 1 in that interval.\nSince we fixed boundary conditions to be 0, this allows us to write\n$$u_{\\varepsilon}(x) = \\int_{0}^{1}{ k_t(x,y) u_{\\varepsilon}(y) dy} + \\int_{0}^{t}{ \\int_{0}^{1}{ k_s(x,y) f(y) dy} ds}$$\nas well as\n$$u_{}(x) = \\int_{0}^{1}{ \\ell_t(x,y) u_{\\varepsilon}(y) dy} + \\int_{0}^{t}{ \\int_{0}^{1}{ \\ell_s(x,y) f(y) dy} ds}.$$\nTheir difference can be written as\n\\begin{align*}\nu_{\\varepsilon}(x) - u_{}(x) &=  \\int_{0}^{t}{ \\int_{0}^{1}{ (k_s(x,y)- \\ell_s(x,y)) f(y) dy} ds} \\\\ \n&+\\int_{0}^{1}{ (k_t(x,y) - \\ell_t(x,y)) u_{\\varepsilon}(y) dy}\\\\\n&+ \\int_{0}^{1}{ \\ell_t(x,y) (u_{\\varepsilon}(y) - u(y)) dy}.\n\\end{align*}\nWe will introduce the first two terms as error estimates \n$$ \\delta =  \\max_{0 \\leq x \\leq 1}  \\left|  \\int_{0}^{t}{ \\int_{0}^{1}{ (k_s(x,y)- \\ell_s(x,y)) f(y) dy} ds}  \\right| + \\left|  \\int_{0}^{1}{ (k_t(x,y) - \\ell_t(x,y)) u_{\\varepsilon}(y) dy} \\right| $$\nand introducing $\\phi(x) = |u_{\\varepsilon}(x) - u(x)|$, this allows us to estimate\n\\begin{align*}\n \\phi(x) \\leq \\delta +  \\int_{0}^{1}{ \\ell_t(x,y) \\phi(y) dy}\n\\end{align*}\nThe interesting twist comes from interpreting the third term as the solution of yet another heat equation. Indeed, we have that\n$$ w_t(x) =   \\int_{0}^{1}{ \\ell_t(x,y) \\phi(y) dy}$$\nsolves the heat equation\n\\begin{align*}\n\\frac{\\partial}{\\partial t} w_t(x) &= \\left( \\overline{a} w_{t}'(x)\\right)' \\quad \\mbox{for} ~x \\in (0,1)\\\\\nw_{t}(0) &= 0 = w_{t}(1) \\quad \\mbox{for all}~t\\\\\nw_0(x) &= \\phi(x)\n\\end{align*}\nwith $\\overline{a}$ being the homogenized coefficient associated to the variable coefficient $a(\\cdot)$. Using $e^{t \\Delta_a}$ to denote heat propagator (with Dirichlet boundary conditions) associated to the problem, we can write our inequality as\n$$ \\phi(x) \\leq \\delta + \\left(e^{t \\Delta_a} \\phi\\right)(x).$$\nThis leads to a bound on the maximum size of $\\phi$: if $\\phi$ was very, very large, then a short application of the heat equation would diminish it quite a bit (this clearly requires Dirichlet conditions) and the inequality would fail. We now make this intuition precise.\n\\begin{lemma} If $\\phi:[0,1] \\rightarrow \\mathbb{R}_{}$ satisfies $\\phi(0) = 0 = \\phi(1) $ and, for some $0 < t \\leq 1$,\n$$ \\phi(x) \\leq \\delta + \\left(e^{t \\Delta_a} \\phi\\right)(x),$$\nthen\n$$ \\max_{0 \\leq x \\leq 1}{\\phi(x)} \\lesssim_{a} \\frac{\\delta}{t}.$$\n\\end{lemma}\n\\begin{proof} We use monotonicity of the heat equation to iterate the argument. More precisely, using the assumption twice yields\n$$ \\phi(x) \\leq \\delta + \\left(e^{t \\Delta_a} \\phi\\right)(x) \\leq \\phi(x) \\leq \\delta + \\left(e^{t \\Delta_a}  \\delta + \\left(e^{t \\Delta_a} \\phi\\right)\\right)(x).$$\nHowever, the heat flow of a constant can be bounded from above by the constant and this, together with the semigroup property, implies\n$$ \\phi(x) \\leq  2 \\delta + \\left(e^{2 t \\Delta_a} \\phi\\right)(x).$$\nIterating the argument shows that, for every integer $k \\in \\mathbb{N}$\n$$ \\phi(x) \\leq  k \\delta + \\left(e^{k t \\Delta_a} \\phi\\right)(x).$$\nThe remaining ingredient is the following fact: there exists $c_{\\overline{a}} > 0$, depending only on $\\overline{a}$, such that for all bounded functions $f: [0,1] \\rightarrow \\mathbb{R}$ and all $1 \\leq t \\leq 2$\n$$ \\max_{0 \\leq x \\leq 1}{e^{t \\Delta_a}f(x)} \\leq (1-c_{\\overline{a}}) \\max_{0 \\leq x \\leq 1}{f(x)}.$$\nOnce this statement is known, we can estimate, for $k \\sim t^{-1}$,\n\\begin{align*}\n\\max_{0 \\leq x \\leq 1} \\phi(x) &\\leq  \\max_{0 \\leq x \\leq 1}  k \\delta + \\left(e^{k t \\Delta_a} \\phi\\right)(x) \\\\\n&=  k \\delta +  \\max_{0 \\leq x \\leq 1}   \\left(e^{k t \\Delta_a} \\phi\\right)(x) \\\\\n&\\leq  k \\delta +  (1-c_{\\overline{a}})\\max_{0 \\leq x \\leq 1}    \\phi(x)\n\\end{align*}\nwhich then implies the desired result. It remains to establish the helpful fact which, in turn, follows\nfrom the fact that the exit probability of Brownian motion out of the unit interval $[0,1]$ at time $t \\sim 1$ is comparable to $\\sim 1$ uniformly on the entire interval.\n\\end{proof}\n\n\n\\subsection{Estimating the error terms.} It remains to estimate the size of the error terms. Both of these require a good understanding of the behavior of\n$k_t(x,y)$. The main ingredient is as follows (see Fig 1.): the average value of $k_t(x,\\cdot)$ over intervals of length being a multiple of $\\varepsilon$ away\nfrom $x$ and of length $\\varepsilon$ coincides \\textit{exactly} with that of the homogenized problem inducing $\\ell_t(x,\\cdot)$.\n\n\n\\begin{lemma} For all $t>0$ and any $k \\in \\mathbb{Z}$ such that $[x+k \\varepsilon,x+ (k+1) \\varepsilon] \\subset (0,1)$\n$$ \\int_{x + k\\varepsilon}^{x+ (k+1)\\varepsilon}{ k_t(x,y) dy} = \\int_{x + k\\varepsilon}^{x+ (k+1)\\varepsilon}{ \\ell_t(x,y) dy}.$$\n\\end{lemma}\n\\begin{proof} It suffices to understand diffusion induced by $a(x\/\\varepsilon)$ with diffusion induced by $\\overline{a}$. This turns out to be rather simple: instead\nof interpreting the homogenization problem as variable-strength diffusion governed by $a(x\/\\varepsilon)$, we may interpret it as classical constant-coefficient\ndiffusion on a one-dimensional manifold whose metric is determined by $a(x\/\\varepsilon)$. \n\\end{proof}\n\nWe will now use Lemma 2 to estimate the size of the error term\n$$ \\delta =   \\left|  \\int_{0}^{t}{ \\int_{0}^{1}{ (k_s(x,y)- \\ell_s(x,y)) f(y) dy} ds}  \\right| + \\left|  \\int_{0}^{1}{ (k_t(x,y) - \\ell_t(x,y)) u_{\\varepsilon}(y) dy} \\right|.$$\nSince the total weight assigned by $k_s$ and $\\ell_s$ to any interval of length $\\varepsilon$ coincides, we may argue that, for all $s > 0$,\n$$  \\left| \\int_{0}^{1}{ (k_s(x,y)- \\ell_s(x,y)) f(y) dy} \\right| \\leq \\varepsilon\\|f\\|_{L^{\\infty}} + \\varepsilon \\|f'\\|_{L^{\\infty}}$$\nas well as\n$$  \\left|  \\int_{0}^{1}{ (k_t(x,y) - \\ell_t(x,y)) u_{\\varepsilon}(y) dy} \\right| \\leq \\varepsilon \\|u_{}'\\|_{L^{\\infty}} + \\varepsilon \\|u_{\\varepsilon}'\\|_{L^{\\infty}}.$$\nAltogether, this yields the estimate\n$$\\|u_{\\varepsilon} - u\\|_{L^{\\infty}} \\lesssim  \\varepsilon\\|f\\|_{L^{\\infty}} + \\varepsilon \\|f'\\|_{L^{\\infty}}+ \\frac{ \\varepsilon \\|u_{}'\\|_{L^{\\infty}} + \\varepsilon \\|u_{\\varepsilon}'\\|_{L^{\\infty}}}{t}.$$\nSetting $t=1$ reduce the convergence rate to a priori estimates on $u$ and $u_{\\varepsilon}$.\\\\\n\n\n\n\n\n\n\\textbf{Acknowledgment.} The author is indebted to Jessica Lin for a series of helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection*{Introduction}\n\nThere is by now a significant amount of evidence that the emergence of spacetime in holographic models of quantum gravity is directly linked to the entanglement between the underlying degrees of freedom (see, for example \\cite{Maldacena:2001kr, Ryu:2006bv, Swingle:2009bg, VanRaamsdonk:2009ar,VanRaamsdonk:2010pw, Maldacena:2013xja}, or \\cite{VanRaamsdonk:2016exw} for a review). Nevertheless, this basic phenomenon is somewhat obscured by the continuous nature of the conformal field theory (CFT) systems that encode the spacetimes. In these examples, part of the spacetime structure (the fixed asymptotic behavior) is directly related to the continuous geometrical space upon which the CFT is defined. Also, the local degrees of freedom interact strongly with those around them, and completely disentangling the various parts requires an infinite amount of energy. There are interesting toy models for holography involving tensor-network states of collections of individual qubits or other discrete elementary subsystems \\cite{Swingle:2009bg, Pastawski:2015qua, Hayden:2016cfa}, but it still not clear how closely the network geometries associated to these states are related to the actual geometries in holographic models.\n\nIn this note, we introduce a new framework for holography in which the fundamental degrees of freedom are a large collection of elementary systems (boundary conformal field theories which can be thought of as ``bits'' of the original CFT) which do not interact with each other. States of a holographic CFT can be replaced by entangled states of this discrete system so that the system still describes a single connected spacetime that is close to the original one dual to the CFT state. Furthermore, in the new description, we show that there is a natural way to represent the state with arbitrary precision using a type of tensor network.\n\n\\subsubsection*{BC bits}\n\nThe motivation for our construction is the idea of cutting up a holographic conformal field theory into a large number of non-interacting pieces (e.g. using a CFT ``jigsaw'') but putting the pieces into a quantum state that approximates a state of the original CFT encoding some geometry. To make this precise, we need to define what me mean by a CFT living on a piece of space with a boundary. Roughly, we need to describe the boundary conditions for the fields at the edges. For any CFT, there are various choices of boundary conditions that are consistent. A special subset of these preserve conformal invariance\\footnote{For example, the vacuum state of the CFT on a half space with these boundary conditions at the edge preserves an $SO(d-1,2)$ of the $SO(d,2)$ conformal symmetry.} and define what is known as a boundary conformal field theory or BCFT (see, for example, \\cite{Cardy:2004hm}).\n\nA set of ``Boundary-CFT bits'' or ``BC-bits'' associated with a CFT on some spatial geometry $M$ is a collection of BCFTs defined on a set of ``sanded'' pieces $\\tilde{M}_i$ of $M$. Here, $\\{\\tilde{M}_i\\}$ are defined by cutting $M$ into a set of simply-connected pieces $\\{M_i\\}$ and ``sanding the edges,'' i.e. $\\tilde{M}_i$ is a large subset of the interior of $M_i$ with a smooth boundary. This is shown in figure \\ref{fig:SphereBits}a) - c). Each BCFT is defined from the original CFT with the same choice boundary conditions, so the BC-bit system is specified by the choice of $\\{\\tilde{M}_i\\}$ and the choice of boundary condition.\n\nOur goal below will be to consider some state of the CFT that corresponds to a smooth geometry and associate to this a state of the BC-bits that captures the same qualitative features. We will argue that the new state can also be associated with a smooth geometry that is closely related to the original one.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=140mm]{PIsimple3.eps}\n\\caption{Euclidean path integrals defining a) a state of a 2D CFT on a circle b) an entangled state of two 2D CFTs each on a spatial circle c) a state of a 2D BCFT on an interval d) an entangled state of two 2D BCFTs e) a state of a 3D CFT on a sphere f) a state of a 3D BCFT on a disk.}\n\\label{fig:PIsimple}\n\\end{figure}\n\n\\subsubsection*{Entangling BC-bits via Euclidean path integral}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=120mm]{SphereBits2.eps}\n\\caption{a)-c) BC-bits defined from a 2+1 dimensional CFT on a spatial sphere. d) Geometry $\\tilde{H}$ used to define a state of the BC-bits. e) Cross section of $\\tilde{H}$.}\n\\label{fig:SphereBits}\n\\end{figure}\nThe Euclidean path integral gives a mechanism to define a quantum state of a CFT on a spatial manifold $M$ given a spatial manifold $H$ of one higher dimension with boundary $M$. The probability amplitude in this state for a having a particular configuration of fields on $M$ is given by the CFT path integral\\footnote{This is an average over the possible field configurations in the Euclidean version of the CFT, weighted by the exponential of minus the Euclidean action.} on $H$ subject to the condition that the fields take the specified values on $M$. For holographic CFTs, this construction gives a natural way to define states that encode dual spacetimes with a good classical description (see, for example, \\cite{Skenderis:2008dh, Botta-Cantcheff:2015sav,Marolf:2017kvq} and references therein). We can change the spacetime we are describing by varying the interior geometry of $H$ and adding sources for various CFT operators on the interior of $H$; for example, arbitrary linearized perturbations about pure AdS can be obtained by choosing the right geometry and sources \\cite{Marolf:2017kvq}.\n\nWe can similarly define the state of a BCFT on a spatial manifold $M$ with boundary $B$ given a higher-dimensional manifold $H$ with boundary $M \\cup G$, where $G$ is also bounded by $B$, as shown in figures \\ref{fig:PIsimple}c and \\ref{fig:PIsimple}f. In this case, we take the boundary conditions at $G$ to be those of the BCFT we are considering.\n\nWe can use the Euclidean path integral to define natural entangled states of non-interacting CFTs or BCFTs by taking the $H$ to be a connected geometry with a disconnected boundary $M$ (see, for example \\cite{Balasubramanian:2014hda}). For example, the thermofield double state for a pair of CFTs defined on spatial spheres is defined by the path integral on a cylinder defined by the spatial sphere times an interval (figure \\ref{fig:PIsimple}b). We will use this idea to define an entangled state of our BC-bits by taking $H$ to be a connected geometry whose boundary is the collection $\\{\\tilde{M}_i\\}$.\n\nMore specifically, we would like to define a state of the BC-bits that is related to some state of the original CFT on $M$ defined by the path integral over $H$. To do so, we define a geometry $\\tilde{H}$ obtained from $H$ by removing smooth ``grooves'' at the surface\\footnote{A mathematical ``router'' is the appropriate tool here.} so that the part of the boundary remaining is $\\tilde{M}_i$. This is depicted in figure \\ref{fig:SphereBits}d for a 2+1 dimensional CFT and in figure $3 \\tilde{\\rm b}$ for a 1+1 dimensional CFT. The boundary of $\\tilde{H}$ is $\\left\\{ \\cup_i \\tilde{M}_i \\right\\} \\cup G$, where $G$ corresponds to the surface of the grooves. We can now define a state of the BC-bits by performing the Euclidean path integral for our BCFT over the geometry $\\tilde{H}$, with the appropriate boundary conditions imposed at $G$.\n\nThe state $|\\Psi_{\\tilde{H}} \\rangle$ and the possible dual geometry will depend on the details of $\\tilde{H}$. However, we will now argue that if we have been sufficiently gentle with our carpentry tools, i.e. if $\\{\\tilde{M}_i\\}$ is close enough to $\\{M_i\\}$ and $\\tilde{H}$ is close enough to $H$, that the state $|\\Psi_{\\tilde{H}} \\rangle$ of our BC-bits encodes a geometry that is closely related to the original geometry described by the CFT state $|\\Psi_{H} \\rangle$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=120mm]{PathIntD.eps}\n\\caption{Lorentzian geometries from path-integral states: a) A CFT on a circle. b) Euclidean path integral defining a holographic CFT state. c) Path integral used to compute observables in this state. Operators can be inserted on dashed lines. d) Euclidean gravity solution corresponding to this path integral. e) Spatial slice at time-symmetric point serves as initial data for Lorentzian solution. f) Lorentzian solution associated with our state. The interior of the causal diamond (dashed lines) is the part encoded by the CFT state at $t=0$. $\\tilde{a}$) - $\\tilde{f}$) Equivalent construction for the BC-bit states. Each BC-bit is a boundary CFT on an interval.}\n\\label{fig:PathInt}\n\\end{figure}\n\n\\subsubsection*{Spacetimes dual to Euclidean path integral states}\n\nLet us now recall how to understand the spacetime geometries encoded by holographic CFT states described by a Euclidean path integral. According to the AdS\/CFT correspondence, features of the encoded geometry can be deduced from the CFT by evaluating the expectation value of various local and nonlocal observables in the CFT state. For our states, these expectation values are computed using a path integral over a surface $\\bar{H}H$ that is obtained by gluing $H$ to its mirror image $\\bar{H}$ along $M$, as seen in figure \\ref{fig:PathInt}c and \\ref{fig:PathInt}$\\tilde{c}$.\\footnote{Any complex sources on $H$ should be conjugated in $\\bar{H}$, but we will restrict to the case of real sources; the resulting geometry will then have a time-reversal symmetry. We can restrict to geometries $H$ so that the resulting space $\\bar{H}H$ is smooth.} The operators of interest are inserted along the junction. By the AdS\/CFT correspondence this CFT path integral corresponds to a gravitational path integral which is dominated by a single Euclidean geometry $X_{H}$, obtained by solving the gravitational equations with boundary conditions that $X_{H}$ is asymptotically AdS with boundary geometry $\\bar{H}H$.\\footnote{The asymptotic behaviour of other fields in the geometry is fixed by the sources for the corresponding operator in the path integral action.}\n\nThe Lorentzian spacetime geometry associated with our CFT state is simply related to the Euclidean geometry $X_{H}$. The geometry $X_{H}$ has a reflection symmetry inherited from the geometry of $\\bar{H}H$. The surface left invariant under this symmetry has a geometry $(X_{H})_0$. To find the spacetime associated with our state, we use the geometry $(X_{H})_0$ (and the condition that time-derivatives of fields vanish here) as initial conditions for the real-time gravitational equations. The solution is a spacetime $X^L_{H}$ corresponding to our state. More formally, we can understand $X^L_{H}$ as an analytic continuation of $X_{H}$. The CFT state at $t=0$ strictly encodes only the region of this spacetime that is spacelike separated from the $t=0$ slice at the boundary,\\footnote{This is the domain of dependence of the region $(X_{H})_0$.} since the spacetime outside this region can be altered by changes to the CFT Hamiltonian before or after $t=0$. For example, a modification to the CFT Hamiltonian at $t=\\epsilon$ corresponds to a boundary source in the gravity picture whose effects propagate forward causally from the $t=\\epsilon$ boundary slice. Thus, the geometry directly encoded by the CFT state is this causal diamond region, also known as the Wheeler-DeWitt patch for the $t=0$ boundary time slice.\\footnote{This connection between the quantum state of a holographic theory at a particular time and the Wheeler-DeWitt patch of the dual geometry associated to that time has been emphasized at various times in the past, for example in the recent conjecture that the gravitational action integrated over the Wheeler-DeWitt patch corresponding to some boundary time provides a measure of complexity of the dual CFT state at that time \\cite{Brown:2015bva}}.\n\nThe crux of our subsequent argument will be that despite the significant differences between the original CFT and the collection of BC-bits as physical systems, the geometry $\\tilde{H}$ used to define the BC-bit state is a small perturbation to the geometry $H$ used to define the CFT state. Thus, we might expect that the procedure we have just outlined gives rise to a spacetime dual to the BC-bits that is almost the same as the spacetime dual to the CFT state.\n\n\\subsubsection*{Geometry of the BC-bit states}\n\nWe would like to understand how the Euclidean gravity solution corresponding to a BCFT on $\\bar{\\tilde{H}}\\tilde{H}$ differs from that corresponding to the CFT on $\\bar{H} H$ (figure 3b vs figure $3 \\tilde{\\rm b}$). The main obstacle here is understanding how the presence of a boundary in $\\bar{\\tilde{H}}\\tilde{H}$ (geometrically described as the space $G$ glued to a mirror image of itself along $B$) affects the gravity calculation. This question was considered originally by Karch and Randall in \\cite{Karch:2000gx} and later in more detail by Takayanagi in \\cite{Takayanagi:2011zk}. As discussed in those papers, if the BCFT state has a geometrical dual $X_{\\tilde{H}}$, this dual must itself have a boundary component in addition to the asymptotically AdS boundary with boundary geometry $\\tilde{H}$. What could this boundary be physically in a UV complete gravity theory? One possibility is that we have some compact internal dimensions that degenerate somehow, in the way that the Kaluza-Klein circle shrinks to zero size to for the edge of a bubble-of-nothing geometry. The boundary could also or alternatively involve some explicit branes of the microscopic theory. Examples of this sort have been described explicitly for higher dimensional CFTs in \\cite{Chiodaroli:2011fn, Chiodaroli:2012vc}. We expect that the specific way such a boundary is realized will depend on the boundary conditions we have chosen for our BCFT.\n\nAs a simple model, we can make use of the suggestion in \\cite{Karch:2000gx,Takayanagi:2011zk} to introduce an explicit end-of-the-world (ETW) brane with constant tension and Neumann boundary conditions. In that case, as long as a given boundary region of $\\tilde{H}$ is small compared to other geometrical features of $\\tilde{H}$ (such as the distance to other boundary components) the ETW brane ending on that boundary component stays localized to the vicinity of that boundary component, as depicted in figure \\ref{fig:PathInt}$\\tilde{d}$ and \\ref{fig:PhaseTrans}a (blue surfaces). In the limit where these boundary components become small, the ETW brane also becomes localized to the asymptotic region of the geometry near this boundary component. The ETW brane may source bulk fields and affect the rest of the geometry, but for a fixed location away from the ETW brane, these effects will become negligible in the limit where the boundary component is taken small. By contrast, if the boundary region of $\\tilde{H}$ becomes too large, we can have phase transitions where the ETW brane topology changes, as shown in figure \\ref{fig:PhaseTrans}. In this case, we might end up with a spacetime that has disconnected components.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=120mm]{PhaseTrans.eps}\n\\caption{Euclidean geometries associated with path integrals for states of two BC-bits, using the end-of-the-world brane model for holographic BCFTs. Each shows half of the Euclidean geometry, up to the slice that becomes the initial data for the Lorentzian geometry. a) When $\\tilde{H}$ is sufficiently close to $H$, the ETW branes are disconnected and localized near the corresponding boundary components. The Lorentzian initial data slice is connected and similar to that for the single CFT path integral state defined by $H$. b) As the modifications defining $\\tilde{H}$ become too severe, we can have a  transition in which the ETW brane topology changes. Here, the Lorentzian initial data slice becomes disconnected, though there will still be entanglement between the matter in the two components of spacetime. c) A path integral defining a pure state of the same two BC-bits as in b). This corresponds to two disconnected spacetimes with no entanglement between the matter in the different components.}\n\\label{fig:PhaseTrans}\n\\end{figure}\n\n\n\nFor 1+1 dimensional CFTs, we can give a more direct argument that doesn't rely on a particular holographic model for BCFTs. In this case, $\\bar{\\tilde{H}}\\tilde{H}$ is a two-dimensional Euclidean geometry, as in the example of figure \\ref{fig:PathInt}$\\tilde{c}$. Here, we can take each component of $G$ to be a small circle as displayed in that figure. If the circle is sufficiently small relative to the distance to other boundary components and operator insertions, we expect (as in the operator product expansion) that its insertion is equivalent to the insertion of a sum of operators $\\sum_i c_i {\\cal O}_i$, where $c_i$ depends on the size of the disk. To find the $c_i$, we can consider the circular boundary inserted into the path integral on a disk that defines a state of the CFT on the circular boundary of our disk. By a conformal transformation, this path integral gives an equivalent state the path integral on a finite cylinder, where one end is the circle where our CFT lives and the other end is the circle where we apply our boundary conditions. The limit where the circle in the original picture becomes small corresponds to a limit where the cylinder becomes long. But in the limit of an infinitely long cylinder, the state we define is the vacuum state. Thus, regardless which boundary conditions we are considering to define our BCFT, we can say that the coefficients $c_i$ for operators which are not the identity operator go to zero in the limit where the circle becomes small.\n\nAccording to these arguments, the Euclidean geometry associated with $\\bar{\\tilde{H}}\\tilde{H}$ should be almost the same as that associated with $\\bar{H} H$. But this is not quite true for the corresponding Lorentzian geometries. The reason is that no matter how small the boundary components of $\\bar{\\tilde{H}}\\tilde{H}$ are, they still change the asymptotic geometry of the slice that serves as the initial data for our Lorentzian evolution (see figure \\ref{fig:PathInt}$\\tilde{d}$). Thus, the state we are defining, considered as a state of the original CFT, will always have infinite energy. In the OPE language, the infinite energy is associated with the fact that we are acting with a local operator on the vacuum state; in this case, no matter how small the coefficient of this operator is in the superposition, the average energy for the full state is still infinite. The result is that in the Lorentzian picture, the initial data slice is almost the same as that for the original CFT state, but because of the differences in asymptotics at the boundary, we can have some type of shock wave evolving forward and backward from each boundary component we have introduced. In the ETW brane picture, we can understand this as a Lorentzian ETW brane whose worldvolume is part of a hyperboloid (the analytic continuation of the hemispherical ETW branes in the Euclidean picture of figure \\ref{fig:PathInt}$\\tilde{d}$. The Lorentzian spacetime is depicted in figure \\ref{fig:PathInt}$\\tilde{f}$.\n\nIn a limit where we have very many BC-bits and very many small boundary components, this shockwave or ETW brane will propagate outward from the full asymptotic boundary of the initial data slice. Thus, the BC-bit version of a holographic state faithfully reproduces the interior of the Wheeler-DeWitt patch associated with the boundary time at which the state is defined, but the geometry generally will not smoothly continue to the past and futire of this patch.\n\n\\subsubsection*{Tensor networks for holographic BC-bit states}\n\nThe BC-bit construction of holographic states bears a closer resemblance to tensor network toy models of holography \\cite{Swingle:2009bg, Pastawski:2015qua, Hayden:2016cfa} in that we have explicit multipart entangled states of a discrete set of non-interacting constituents. We will now see that as in the toy models, our states may be represented arbitrarily well by a type of tensor network, where the tensors correspond to states of small numbers of BC-bits. Our construction involves additional small changes to $\\tilde{H}$, this time carried out using a mathematical ``drill''.\n\nWe have argued previously that the introduction of a small boundary component to $\\tilde{H}$ has a vanishingly small effect on the dual Euclidean geometry in the limit that the size goes to zero, except very close to the insertion locus. Thus, if we introduce additional boundary components away from $M$, these should have a negligible effect on the BC-bit state as the new boundary components are taken to be infinitesimal.\n\nBy introducing these extra boundaries (we can imagine drilling through $\\tilde{H}$ with a narrow drill bit), it is possible to represent the new state as a tensor network, by decomposing the path integral into chunks, as shown in figures \\ref{fig:TN} and \\ref{fig:TN3D}. Here, each individual tensor corresponds to the state of a small number of BC-bits, which are the original BC-bits for the external legs or new internal BC-bits. The internal edges of the tensor network correspond in path integral language to identifying the field configurations on the two BC bits joined by the edge and integrating over these. In quantum language, this corresponds to projecting the state of the pair of BC-bits onto a maximally entangled state. The full set of these projections contracts up the tensor network, defining what is known as a projected entangled pairs state (PEPS) (see, for example \\cite{Orus:2013kga}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=120mm]{TN1.eps}\n\\caption{a) An extra circular boundary component is added to the interior of $\\tilde{H}$ in the path integral for a six BC-bit state. b) The path integral can now be decomposed into a product of path integrals defining three BC-bit states. The field configurations for the BC-bits connected by dashed lines are equated and integrated over - the path integral equivalent of the pair projection that connects a tensor network. c) The tensor network representation of the state.}\n\\label{fig:TN}\n\\end{figure}\n\nThere is clearly a great deal of freedom in how to build up a tensor network representation: we can choose where to place the new boundary components and we can choose the surfaces along which to break up $\\tilde{H}$ to give the new internal BC-bits. We can also choose the geometry of $\\tilde{H}$ from among the family of geometries related by conformal invariance. So, as expected, we can have many tensor networks that represent the same state. The networks here apparently have a closer connection to the geometry of the path-integral defining the state rather than the geometry of the space being encoded; such a connection was emphasized recently in \\cite{Milsted:2018yur}. However, the recent work \\cite{Caputa:2017urj,Caputa:2017yrh} suggests that by a particular optimization of the path-integral geometry over the geometries related by conformal invariance, the path-integral geometry actually becomes the geometry of the bulk spatial slice. Similarly, it may be that under some optimization of our tensor network description, we end up with networks similar to those in \\cite{Pastawski:2015qua, Hayden:2016cfa} where the network geometry has a close connection to the bulk spatial geometry.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=110mm]{TN3D.eps}\n\\caption{Tensor network for path-integral state of eight BC-bits. In the first step, the state is modified by drilling narrow holes through the ball along the axes, enlarging the boundary of $\\tilde{H}$. In general, for three dimensional $\\tilde{H}$ the modifications necessary in the first step involve introducing a network of narrow tunnels through $\\tilde{H}$.}\n\\label{fig:TN3D}\n\\end{figure}\n\n\\subsubsection*{The limit of many BC-bits}\n\nFor simplicity, the various examples shown in the figures involve only a small number of BC-bits. In this case, the individual BC-bits might carry information about a significant portion of the dual geometry. However, our conclusions apply equally well when the number of BC-bits is taken to be very large, so long as the modifications leading from  $H$  to $\\tilde{H}$ are kept small (e.g. each $\\tilde{M}_i$ is still a large subset of $M_i$). In the limit where the BC-bits are all small compared to any scale associated with $M$ or the original CFT state, we expect that the individual bits carry almost no information about the geometry being represented by the collection of BC-bits, apart from the asymptotic behavior of the bulk fields at a single boundary location corresponding to the bit. Thus, the spacetime geometry is almost entirely encoded in the entanglement structure of multipart BC-bit system. In this system, by disentangling the bits, it is manifestly true that the corresponding spacetime disintegrates, as suggested for continuous CFTs in \\cite{VanRaamsdonk:2010pw}. We emphasize that in the BC-bit system, the individual bits have no intrinsic location relative to one another,\\footnote{Thinking of the BC-bits as pieces of a jigsaw puzzle, we are describing the same geometry whether the pieces are spread out on the table or the puzzle is completed.} so it is not only the radial direction of the spacetime that ``emerges.''\n\n\\subsubsection*{The speculative part}\n\nSince the entanglement structure is playing such a key role here, and the individual bits carry almost no information about the interior of the spacetime, it is interesting to ask what properties of the BC-bit are really required here. If we replace the BC-bits with some other type of bit but keep the entanglement structure the same, does this still encode the same spacetime and gravitational physics? Initial inspection might suggest that this is too optimistic: the specific theory of gravity encoded in a CFT state has a certain number of dimensions and a certain set of fields in addition to the metric, and these are related to the dimensionality and the operator content of the CFT. If we try to replace BC-bits from one CFT with BC-bits from another CFT, possibly of a different dimensionality, it may seem that we can't possibly be describing the same gravitational physics since we are now dealing with a completely different theory of gravity.\n\nBut there is a way around this obstacle: the optimistic scenario could work if there is fundamentally only one theory of quantum gravity. This is in line with expectations from string theory \\cite{Polchinski:1998rr}, where the study of string dualities suggest that various UV complete gravitational theories in various dimensions can be understood as descending through compactification and dualities from eleven-dimensional M-theory. Thus, even if the gravitational physics in one region of spacetime has the fields and interactions associated with a particular low-energy theory of gravity, the fields and interactions in another region of spacetime could correspond to a different low-energy theory if we have some transition region in between where the properties of the compactification manifold or the preferred duality frame change.\n\nIn our example where we replace BC-bits from one CFT with BC-bits associated with another CFT, we will clearly have different physics in the asymptotically AdS regions very close to the BC-bits (near the diamonds in figure \\ref{fig:PathInt}f). But, moving inward in the radial directions, we may have a transition (e.g. with some compact dimensions changing shape or topology) such that inside a radial position associated with a boundary scale containing a large number of BC-bits, the physics is that of the spacetime described by the original CFT. If we replace the BC-bits with more general quantum systems (e.g. collections of qubits, or macaroni), it may be that the asymptotic region no longer has a geometrical description, but the same interior region emerges.\n\nThe idea that we could obtain a precise description of quantum gravitational physics starting from sufficiently many copies of an arbitrary quantum system is intriguing, but certainly does not follow from any of the arguments of this paper. Nevertheless, it is fascinating that the possible unity of gravitational theories as suggested by string theory leaves open the possibility of such an exact and universal entanglement-gravity duality.\\footnote{This is in line with Susskind's slogan $GR=QM$ \\cite{Susskind:2017ney}, though our discussion suggests that perhaps a better moniker would be $M=QM$ or, more simply, $Q=1$.}\n\n\\section*{Acknowledgements}\n\nThis work was supported in part by a Simons Investigator award and by a Simons Foundation collaboration grant.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nFor a real number $x$, the nearest integer of $x$ denoted by $||x||$  and it is defined as \n$$\n||x||:=\\mbox{min}\\{|x-m|:m\\in\\mathbb{Z}\\}.\n$$\n\nIn 1957, Mahler \\cite{mahler} used Ridout's theorem, which is a $p$-adic extension of  Roth's theorem to prove the following: {\\it  let $\\alpha\\in \\mathbb{Q}\\backslash\\mathbb{Z}$ and $\\varepsilon$ be a positive real number . Then there are only many finitely many $n\\in \\mathbb{N}$ satisfying  $||\\alpha^n||<2^{-\\varepsilon n}$}. Mahler also asked for which algebraic numbers $\\alpha$ the above conclusion remains hold.\n\\smallskip\n\nIn 2004, by ingenious applications of the Subspace Theorem, Corvaja and Zannier \\cite{corv} proved a `Thue-Roth' type inequality with `moving targets'. Then as an application of this result,  they answered  the  question of Mahler and  proved the following: {\\it let $\\alpha>1$ be a real algebraic number and let $\\varepsilon$ be a positive real number. Suppose that $|| \\alpha^n ||< 2^{- \\varepsilon n}$  for infinitely many $n$. \nThen, there is some integer $d\\geq 1$ such that the number $\\alpha^d$ is a Pisot number. In particular $\\alpha$ is an algebraic integer. } \n\\bigskip\n\nIn this paper, the main motivation is to prove an inhomogeneous extension of Thue-Roth's type inequality with moving targets in the same spirit as the result of Corvaja and Zannier in \\cite{corv}. Indeed, we prove the following. \n\n\\begin{theorem}\\label{maintheorem}\nLet $\\Gamma\\subset \\overline{\\mathbb Q}^{\\times}$ be a finitely generated multiplicative group of algebraic numbers.  Let $\\delta$ be a non-zero algebraic number, $\\beta\\in(0,1)$ be an algebraic  irrational, and $\\varepsilon>0$ be a fixed  real number.  Then there exist only finitely many triples $(u, q,  p)\\in\\Gamma\\times\\mathbb{Z}^2$ with $d=[\\mathbb{Q}(u):\\mathbb{Q}]$   such that  $|\\delta q u|>1$  and  \n\\begin{equation*}\\label{eq1.1}\n\\tag{1.1}\n0<|\\delta qu+\\beta-p|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}.\n\\end{equation*}\n\\end{theorem}\n\nRecently in 2019, Kulkarni, Mavraki and Nguyen \\cite{kul} generalize  Mahler problem to an arbitrary linear recurrence sequence of the form $\\{Q_1(n)\\alpha_1^n+\\cdots+Q_k(n)\\alpha^n_k:n\\in\\mathbb{N}\\}$, where $\\alpha_i$'s are non-zero algebraic number and $Q_i(x)\\in\\overline{Q}[x]\\backslash \\{0\\}$. In particular case, they proved the  following inhomogeneous extension of the problem of Mahler: {\\it let $\\alpha>1$  and $\\beta$ be real algebraic numbers and let \n$\\varepsilon$ be a positive real number.  \nSupose that $||\\alpha^n+\\beta|| < 2^{- \\varepsilon n}$  for infinitely many $n$. \nThen there is some integer $d\\geq 1$ such that the number $\\alpha^d$ is a Pisot number. \nIn particular $\\alpha$ is an algebraic integer. We recall that a real algebraic number $\\alpha > 1$ is called {\\it a Pisot number}, if  the modulus value of all the other conjugates of $\\alpha$ is $< 1$. }\n\\bigskip\n\n\n \nIn the above result, if $\\beta$ is an integer and $\\alpha$ is an algebraic number such that  $\\alpha^d$ is a Pisot number for some integer $d\\geq 1$, then clearly there are infinitely many natural numbers  $n$ satisfying $||\\alpha^{dn}+\\beta||<2^{-\\eps n}$ for some $\\eps>0$. Thus, we can conclude that   the above assertion is best possible,  if $\\beta$ is a rational number.  \nSurprisingly, when $\\beta$ is an algebraic irrational, we prove a contrast result as an application of our main theorem as follows. \n\n\\begin{theorem}\\label{maintheorem2}\nLet $\\alpha>1$ be a real number. Let $\\beta$ be an algebraic irrational and  $\\lambda$ be a non-zero real algebraic number.   For a given real number $\\varepsilon >0$, if there are infinitely many natural numbers $n$ for which  $||\\lambda\\alpha^n+\\beta|| < 2^{- \\varepsilon n}$ holds true, then  $\\alpha$ is transcendental.\n\\end{theorem}\n\nNote that Theorem \\ref{maintheorem2} strengthen   the main result of Wagner and Ziegler \\cite{wagner}. \n\n\\section{Preliminaries} \n\nLet $K\\subset \\mathbb{C}$ be a Galois extension over $\\mathbb{Q}$. Let $M_K$ be the set of all inequivalent places of $K$ and $M_\\infty$ be the set of all archimedian places of $K$. For each place $v\\in M_K$, we denote $|\\cdot |_v$ the absolute value corresponding to $v$, normalized with respect to $K$. Indeed if $v\\in M_\\infty$, then there exists an automorphism $\\sigma\\in\\mbox{Gal}(K\/\\mathbb{Q})$ of $K$ such that for all $x\\in K$, \n\\begin{equation*}\\label{eq2.1}\n\\tag{2.1}\n|x|_v=|\\sigma(x)|^{d(\\sigma)\/[K:\\mathbb{Q}]},\n\\end{equation*}\nwhere $d(\\sigma) =1$ if $\\sigma(K) = K\\subset \\mathbb{R}$ and $d(\\sigma) = 2$ otherwise. Non-archimedian absolute values are normalized accordingly so that  the product formula   $\\displaystyle\\prod_{\\omega\\in M_K}|x|_\\omega=1$ holds for any  $x\\in K^\\times$. For each automorphism $\\rho\\in \\mbox{Gal}(K\/\\mathbb{Q}),$ one defines an archimedian valuation on $K$ by the formula \n\\begin{equation*}\\label{eq2.2}\n\\tag{2.2}\n|x|_\\rho:= |\\rho^{-1}(x)|^{d(\\rho)\/[K:\\mathbb{Q}]},\n\\end{equation*}\nwhere $|\\cdot |$ denotes the complex absolute value. Note that  two distinct automorphisms, say, $\\rho_1$ and $\\rho_2$ defines the same absolute value if and only if $\\rho_1^{-1}\\circ \\rho_2$ is a complex conjugation. Thus, for each $v\\in M_\\infty$, let $\\rho_v$ be an automorphism defining the valuation $v$ according to \\eqref{eq2.2}. Then the set $\\{\\rho_v : v\\in M_\\infty\\}$ represents the left cosets of the subgroup generated by the complex conjugation in  $\\mbox{Gal}(K\/\\mathbb{Q})$. \n\nThe absolute Weil  height  $H(x)$ is defined as \n$$\nH(x):=\\prod_{\\omega\\in M_K}\\mbox{max}\\{1,|x|_\\omega\\} \\mbox{ for all } x\\in K.\n$$\nFor a vector $\\mathbf{x}=(x_1,\\ldots,x_n)\\in K^n$  and for a place $\\omega\\in M_K$, the $\\omega$-norm for  $\\mathbf{x}$ denote by $||\\mathbf{x}||_\\omega$ and given by \n$$\n||\\mathbf{x}||_\\omega:=\\mbox{max}\\{|x_1|_\\omega,\\ldots,|x_n|_\\omega\\}\n$$\nand  the projective height,  $H(\\mathbf{x})$, is defined by \n$$\nH(\\mathbf{x})=\\prod_{\\omega\\in M_K}||\\mathbf{x}||_\\omega.\n$$\nNow we are ready to present a more general version  of the Schmidt Subspace Theorem, which was  formulated by Schlickewei and Evertse.  For the reference,  see (\\cite[ Chapter 7]{bomb},  \\cite[ Chapter V, Theorem 1D$^\\prime$]{schmidt} and  \\cite[Page 16, Theorem II.2]{zannier})\n\\smallskip\n\n\\begin{theorem} (Schlickewei) \\label{schli}\n Let $K$ be an algebraic number field and $m \\geq 2$ an integer. Let $S$ be a finite set of places on $K$ containing all the archimedian  places.  For each $v \\in S$, let $L_{1,v}, \\ldots, L_{m,v}$ be linearly independent linear  forms in the variables $X_1,\\ldots,X_m$ with  coefficients in $K$.  For any $\\varepsilon>0$, the set of solutions $\\textbf{x} \\in K^m\\backslash\\{0\\}$ to the inequality \n\\begin{equation*}\n\\prod_{v\\in S}\\prod_{i=1}^{m} \\frac{|L_{i,v}(\\textbf{x})|_v}{\\|\\textbf{x}\\|_v} \\leq \\frac{1}{H(\\textbf{x})^{m+\\varepsilon}}  \n\\end{equation*}\ncontained  in finitely many proper subspaces of $K^m$.\n\\end{theorem}\n\n\nThe following lemma, established in \\cite{corv}, is used at several places in the proof of the main result\nof \\cite{corv}. \n\n\\begin{lemma}\\label{lemCZ1}\nLet $K$ be a  Galois extension over $\\mathbb{Q}$  and $S$  be a finite subset of places, containing all  the archimedean places.  Let $\\sigma_1,\\ldots,\\sigma_n$  be distinct automorphism of $K$ and let \n$\\lambda_1,\\ldots,\\lambda_n$  be non-zero elements of $K$. \nLet  $\\varepsilon>0$  be a positive real number and $\\omega\\in S$  be a distinguished place.  \nLet $c > 0$. \nLet $\\mathfrak{E}\\subset \\mathcal{O}_S^\\times$ be the set of solutions $u\\in\\mathcal{O}_S^\\times$  \nof the inequality\n\\begin{equation*}\n0< |\\lambda_1  \\sigma_1(u)+\\cdots+\\lambda_n \\sigma_n(u)|_\\omega<\n c \\max\\{|\\sigma_1(u)|_\\omega,\\ldots,|\\sigma_n(u)|_\\omega\\} H^{-\\varepsilon}(u).\n\\end{equation*}\nIf $\\mathfrak{E}$ is infinite subset of $\\mathcal{O}_S^\\times$, \nthen there exists a non-trivial linear relation of the form \n$$\na_1 \\sigma_1(u)+\\cdots+a_n \\sigma_n(u)=0,\\quad \\mbox{with }  a_i\\in K   \n$$\nwhich holds for infinitely many elements of $u$ in $\\mathcal{O}_S^\\times$.\n\\end{lemma}\n\n\nThe following application of Theorem \\ref{schli} is a slight modification of Lemma \\ref{lemCZ1}. \n\n\\begin{lemma}\\label{lem2.1}\nLet $K$ be a  Galois extension over $\\mathbb{Q}$  and $S$  be a finite subset of places, containing all  the archimedean places.  Let $\\sigma_1,\\ldots,\\sigma_n$  be distinct automorphism of $K$ and let \n$\\lambda_0, \\lambda_1,\\ldots,\\lambda_n$  be non-zero elements of $K$. \nLet  $\\varepsilon>0$  be a positive real number and $\\omega\\in S$  be a distinguished place.  Let $\\mathfrak{E}\\subset \\mathcal{O}_S^\\times\\times\\mathbb{Z}$ be the subset  defined as\n\\begin{equation*}\n\\mathfrak{E} := \\left\\{ (u,q)\\in \\mathcal{O}_S^\\times \\times\\mathbb{Z}^\\times \n \\ :  0<\\ |\\lambda_0+\\lambda_1 q\\sigma_1(u)+\\cdots+\\lambda_nq\\sigma_n(u)|_\\omega<\\frac{\\max\\{|q\\sigma_1(u)|_\\omega,\\ldots,|q\\sigma_n(u)|_\\omega\\}}{|q|^{n+\\varepsilon}H^{\\varepsilon}(u)}\\right\\}.\n\\end{equation*}\nIf $\\mathfrak{E}$ is infinite subset of $\\mathcal{O}_S^\\times\\times\\mathbb{Z}$, then there exists a non-trivial linear relation of the form \n$$\na_1 \\sigma_1(u)+\\cdots+a_n \\sigma_n(u)=0,\\quad \\mbox{with }  a_i\\in K   \n$$\nwhich holds for infinitely many elements of $u$ in $\\mathcal{O}_S^\\times$.\n\\end{lemma}\n\n\\begin{proof}\nIn order to prove this lemma, we shall apply  Theorem \\ref{schli} as in the proof \\cite[Lemma 1]{corv}. Without loss of generality, we can assume that   \n\\begin{equation*}\n|q\\sigma_1(u)|_\\omega=\\max\\{|q\\sigma_1(u)|_\\omega,\\ldots,|q\\sigma_n(u)|_\\omega\\}\n\\end{equation*}\nfor all $(u,q)\\in \\mathfrak{E}$. For $\\nu\\in S$, let us define $n+1$ linear forms $L_{0,\\nu},\\ldots,L_{n,\\nu}$ in $n+1$ variables ${\\bf  X}=(X_0,X_1,\\ldots,X_n)$ as follows: Put\n\\begin{align*}\nL_{\\omega,0}(X_0,X_1,\\ldots,X_n)&=X_0\\\\\nL_{\\omega,1}(X_0,X_1,\\ldots,X_n)&=\\lambda_0 X_0+\\lambda_1 X_1+\\cdots+\\lambda_n X_n\n\\end{align*} \nfor $2\\leq i\\leq n$, we define\n$$\nL_{\\omega,i}(X_0,X_1,\\ldots,X_n)=X_i,\n$$ \nand for each  $\\mathit{v}\\neq \\omega\\in S$,  $0\\leq j\\leq n $, we let\n$$\nL_{\\mathit{v},j}(X_0,X_1,\\ldots,X_n)=X_j.\n$$\nNow we put \n$$\n{\\bf X}=(1, q\\sigma_1(u),\\ldots,q\\sigma_n(u))\\in K^{n+1}  \n$$\nand consider the product \n\\begin{equation*}\\label{eq2.4}\n\\tag{2.3}\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{\\mathit{v},i}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}.\n\\end{equation*}\nUsing the fact that $L_{v,j}({\\bf X}) =q \\sigma_j(u)$  for $2\\leq j\\leq n $, we obtain\n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{j=2}^{n}|L_{\\mathit{v},j}(\\mathbf{X})|_\\mathit{v}\n= \\prod_{\\mathit{v}\\in S}\\prod_{j=2}^{n}|q|_\\mathit{v}\\prod_{j=2}^{n}\\prod_{\\mathit{v}\\in S}|\\sigma_j(u)|_\\mathit{v}.\n$$\nSince $\\sigma_j(u)$  are $S$-units,  by the product formula,  we obtain\n$$\n\\prod_{\\mathit{v}\\in S}|\\sigma_j(u)|_\\mathit{v}=\\prod_{\\mathit{v}\\in M_K}|\\sigma_j(u)|_\\mathit{v}=1.  \n$$\nConsequently, the above inequality gives \n\\begin{equation*}\\label{eq2.5}\n\\tag{2.4}\n\\prod_{\\mathit{v\\in S}}\\prod_{j=2}^{n}|L_{\\mathit{v},j}(\\mathbf{X})|_\\mathit{v}=\\prod_{\\mathit{v}\\in S}\\prod_{j=2}^{n}|q|_\\mathit{v}\\leq \\prod_{v\\in M_\\infty}\\prod_{j=2}^{n}|q|_\\mathit{v}= |q|^{n-1}.\n\\end{equation*} \nNow we estimate $\\prod_{\\nu\\in S}\\prod_{i=0}^n||\\bf X||_\\nu$:\n\\begin{equation*}\\label{eq2.6}\n\\tag{2.5}\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n||{\\bf X}||_\\mathit{v}=\\prod_{i=0}^n\\left(\\prod_{\\mathit{v}\\in S}||{\\bf X}||_\\mathit{v}\\right)\\geq (H({\\bf X}))^{n+1}=H^{n+1}(1,q\\sigma_1(u),\\ldots,q\\sigma_n(u))\n\\end{equation*}\nsince $||{\\bf X}||_\\mathit{v}\\leq 1$ for all $\\mathit{v}$ not in $S$.  \n\\smallskip\n\nBy re-writing \\eqref{eq2.4}, we have \n\\begin{align*}\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{\\mathit{v},i}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}&= |L_{\\omega,0}({\\bf X})|_\\omega |L_{\\omega, 1}({\\bf X})|_\\omega\\left(\\prod_{\\mathit{v}\\neq \\omega\\in S}\\prod_{i=0}^1   \n|L_{\\mathit{v},i}(\\bf X)|_\\nu\\right) \\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=2}^n|L_{\\mathit{v},i}({\\bf X})|_\\mathit{v}\\right)\\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{1}{||\\bf X||_\\mathit{v}}\\right)\n\\end{align*}\nThen by using the fact that $L_{\\mathit{v},0}({\\bf X})=1$ for $\\mathit{v}$ in $S$ and  the  product formula  \n$$\n\\prod_{\\mathit{v}\\neq \\omega\\in S}|q\\sigma_1(u)|_\\mathit{v}=\\left(\\prod_{\\mathit{v}\\neq \\omega\\in S}|q|\\right)(|\\sigma_1(u)|_\\omega)^{-1},\n$$\nwe get \n\\begin{align*}\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{\\mathit{v},i}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}&=\\frac{|L_{\\omega, 1}({\\bf X})|_\\omega}{|\\sigma_1(u)|_\\omega} \n\\prod_{\\mathit{v}\\neq \\omega\\in S} |q |_\\mathit{v}  \n\\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=2}^n|L_{\\mathit{v},i}({\\bf X})|_\\mathit{v}\\right)\\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{1}{||\\bf X||_\\mathit{v}}\\right)\\\\\n&\\leq\\frac{|L_{\\omega, 1}({\\bf X})|_\\omega}{|\\sigma_1(u)|_\\omega}\\frac{|q|}{|q|_\\omega} \\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=2}^n|L_{\\mathit{v},i}({\\bf X})|_\\mathit{v}\\right)\\left(\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{1}{||\\bf X||_\\mathit{v}}\\right), \n\\end{align*}\nsince $|q|_\\mathit{v}\\leq 1$ for all $\\mathit{v}$-non-archimedian absolute value.\n\\smallskip\nThus  from the assumption and \\eqref{eq2.5}, \\eqref{eq2.6}, we obtain\n\\begin{align*}\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{\\mathit{v},i}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}&\\leq \\frac{|\\lambda_0+\\lambda_1 q\\sigma_1(u)+\\cdots+\\lambda_n q\\sigma_n(u)|_\\omega |q|^n}{|q\\sigma_1(u)|_\\omega}\\frac{1}{H^{n+1}({\\bf X})}\\\\\n&\\leq\\frac{\\max\\{|q\\sigma_1(u)|_\\omega,\\ldots,|q\\sigma_n(u)|_\\omega\\}}{|q\\sigma_1(u)|_\\omega |(|q|H(u))^\\varepsilon}\\frac{1}{H^{n+1}({\\bf X})}, \n\\end{align*}\nsince $(u,q)\\in\\mathfrak{E}$. Using that\n$$\n|q\\sigma_1(u)|_\\omega=\\max\\{|q\\sigma_1(u)|_\\omega,\\ldots,|q\\sigma_n(u)|_\\omega\\}, \n$$\nwe obtain \n$$\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{\\mathit{v},i}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}\\leq \\frac{1}{H^{n+1}({\\bf X})}\\frac{1}{(|q|H(u))^\\varepsilon}.\n$$\nThe height of the vector ${\\bf X}=(1,q\\sigma_1(u),\\ldots,q\\sigma_n(u))$ satisfies the following inequality:\n$$\nH({\\bf X})\\leq |q|H^{K:\\mathbb{Q}}(u)=|q|H^n(u).\n$$\nHence the above estimate becomes \n$$\n\\prod_{\\mathit{v}\\in S}\\prod_{i=0}^n\\frac{|L_{i,\\mathit{v}}(\\bf X)|_\\mathit{v}}{||\\bf X||_\\mathit{v}}\\leq \\frac{1}{H^{n+1}({\\bf X})}\\frac{1}{H^{\\varepsilon\/[K:\\mathbb{Q}]}({\\bf X})}=\\frac{1}{H^{n+1+\\varepsilon\/[K:\\mathbb{Q}]}({\\bf X})}.\n$$\nThen by Theorem \\ref{schli}, there exists a non-trivial relation of the form\n\\begin{equation*}\\label{eq2.7}\n\\tag{2.6}\na_0+a_1 q\\sigma_1(u)+\\cdots+a_n q\\sigma_n(u)=0\n\\end{equation*}\nsatisfied by infinitely many elements of $(u,q)\\in \\mathfrak{E}$.\nOur next goal is to prove the following claim.\n\\bigskip\n\n\\noindent{\\bf CLAIM.~}  There exist a non-trivial relation as \\eqref{eq2.7} with vanishing coefficients $a_0$. \n\\bigskip\n\nAssume that $a_0\\neq 0$. Then by rewriting the relation \\eqref{eq2.7}, we obtain \n\\begin{equation*}\na_0=-a_1 q\\sigma_1(u)-\\cdots-a_n q\\sigma_n(u).\n\\end{equation*}\nThen from \\eqref{eq2.3}, we get \n$$\n0<\\left|-\\lambda_0\\left(\\frac{a_1}{a_0}q\\sigma_1(u)+\\cdots+\\frac{a_n}{a_0} q\\sigma_n(u)\\right)+\\lambda_1 q\\sigma_1(u)+\\cdots+\\lambda_nq\\sigma_n(u)\\right|_\\omega<\\frac{\\mbox{max}\\{|q\\sigma_1(u)|_\\omega,\\ldots,|q\\sigma_n(u)|_\\omega\\}}{|q|^{n+\\varepsilon}H^{\\varepsilon}(u)}.\n$$\nholds for infinitely many pairs $(u,q)\\in\\mathfrak{E}$.  This is equivalent to\n$$\n0<\\left|\\left(\\lambda_1-\\frac{\\lambda_0 a_1}{a_0}\\right)\\sigma_1(u)+\\cdots+\\left(\\lambda_n-\\frac{\\lambda_0 a_n}{a_0}\\right)\\sigma_n(u)\\right|_\\omega<\\frac{\\mbox{max}\\{|\\sigma_1(u)|_\\omega,\\ldots,|\\sigma_n(u)|_\\omega\\}}{|q|^{n+\\varepsilon}H^{\\varepsilon}(u)}.\n$$\nholds for infinitely many pairs $(u,q)\\in\\mathfrak{E}$. This inequality implies that  $\\beta_i=\\lambda_i-\\frac{\\lambda_0 a_i}{a_0}$ are in $K$ and not all are zero.  \nIn particular, there are infinitely many $u$ in $\\mathcal{O}_S^\\times$ such that \n$$\n0<\\left|\\left(\\lambda_1-\\frac{\\lambda_0 a_1}{a_0}\\right)\\sigma_1(u)+\\cdots+\\left(\\lambda_n-\\frac{\\lambda_0 a_n}{a_0}\\right)\\sigma_n(u)\\right|_\\omega<\\frac{\\mbox{max}\\{|\\sigma_1(u)|_\\omega,\\ldots,|\\sigma_n(u)|_\\omega\\}}{H^{\\varepsilon}(u)}.\n$$\nThen by  Lemma  \\ref{lemCZ1}, there exists a non-trivial relation of the form \n$$\na_1 \\sigma_1(u)+\\cdots+a_n\\sigma_n(u)=0 \n$$\nwhich holds for infinitely many values of $u$ in $\\mathcal{O}_S^\\times$.  \nThis proves the claim and hence the lemma.\n\\end{proof}\n\n\n\n\\section{Key lemma for the proof of Theorem \\ref{maintheorem}}\n\nThe following lemma is key to  the proof of Theorem \\ref{maintheorem}  and its proof is based on the Subspace Theorem along with the idea in \\cite{corv}, with  various modifications.\n\n\\begin{lemma}\\label{lem3.1}\nLet $K$ be a Galois extension over $\\mathbb{Q}$ of degree $n$ and $k\\subset K$  be a subfield of degree $d$  over $\\mathbb{Q}$. Let $\\delta, \\beta$ be two non-zero elements of $K$. \nLet $S$  be a finite set of places on $K$ containing all the archimedean places \nand  let $\\varepsilon>0$  be a given real number.   Let \n\\begin{equation*}\\label{eq3.1}\n\\tag{3.1}\n\\mathcal{B} = \\left\\{(u, q, p)\\in (\\mathcal{O}_S^\\times\\cap k)\\times\\mathbb{Z}^2 \\ : \\   \n0<|\\delta qu+\\beta-p|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}\\right\\}\n\\end{equation*}\nsuch that for each triple $(u,q,p)\\in\\mathcal{B}$,  $|\\delta q u|>1$. If $\\mathcal{B}$ is infinite, then there exist a proper subfield $k'\\subset k$, a non-zero element $\\delta'$ in $k$  and an infinite subset $\\mathcal{B}'\\subset \\mathcal{B}$  such that for all triples $(u, q, p)\\in\\mathcal{B}'$ \nwe have $u\/\\delta'\\in k'.$\n\\end{lemma}\n\\bigskip\n\n\\noindent{\\bf Remark.} Once hypothesis of the Lemma \\ref{lem3.1}  is satisfied,  by the  Subspace Theorem  given in the form  Theorem \\ref{schli}, we can conclude that the  infinite sequence $u$ satisfies \\eqref{eq3.1}  can not be fixed;  that is, $H(u)\\rightarrow\\infty$  as the triple varies $(u,q, p) \\in \\mathcal{B}$. \n\n\\begin{proof}\nSince $\\mathcal{B}$ is an infinite set of solutions of \\eqref{eq3.1},  we first observe that \nwe may assume that $H(u)\\rightarrow\\infty$. \n\nSuppose that $H(u)$ is bounded. Then there exists an infinite subset $\\mathcal{A}$ of $\\mathcal{B}$ such that the number $u$ is constant for all elements in  $\\mathcal{A}$, let say $u_0$ for all triples $(u,q,p)\\in\\mathcal{A}$ and $q$ is unbounded along the set $\\mathcal{A}$. Now we apply Theorem \\ref{schli} \nto the field $\\mathbb{Q}$\nwith the input $S=\\{\\infty\\}$, \nlinear forms \n$L_{1,\\infty}(X_1, X_2,X_3)=\\alpha u_0 X_1+\\beta X_2-X_3$, $L_{i,\\infty}(X_1,X_2,X_3)=X_i$ \nfor $2\\leq i\\leq 3$ \nand the points   $(q,1,p)$.  Then from \\eqref{eq3.1}, we see that there is $\\delta > 0$ such that the inequality\n$$\n\\prod_{i=1}^3|L_{j,\\infty}(q,1,p)|_\\infty\\leq \\frac{1}{(\\max\\{|q|,1,|p|\\})^{\\delta}}\n$$\nholds for infinitely many triples $(q,1,p)\\in\\mathbb{Z}^3$.  Thus by Theorem \\ref{schli}, there exists a proper subspace of $\\mathbb{Q}^3$ containing infinitely many triples $(q,1,p)$.   That is, we have a non-trivial relation of the form\n$$\na_0+a_1p+a_2q=0\n$$\nsatisfied by infinitely many triples of the form $(q,1,p)$. Since $a_i$ is in $\\mathbb{Z}$  \nand $q\\rightarrow\\infty$ along the set $\\mathcal{A}$, we conclude that $a_1\\neq 0$.  Then by substituting the value of $p$ from this above equality into the inequality \\eqref{eq3.1} along the set $\\mathcal{A}$, we get\n$$\n0<\\left|\\alpha q u_0+\\beta+\\left(\\frac{a_0}{a_1}+\\frac{a_2}{a_1}q\\right)\\right|\\leq \\frac{1}{H^{\\varepsilon}(u_0)}\\frac{1}{q^{d+\\varepsilon}}.\n$$ \nThis is equivalent to \n$$\n0<\\left|\\left(\\alpha u_0+\\frac{a_2}{a_1}\\right)q+\\beta+\\frac{a_0}{a_1}\\right|\\leq \\frac{1}{H^{\\varepsilon}(u_0)}\\frac{1}{q^{d+\\varepsilon}}, \n$$\nwhich cannot hold if $|q|$ is large enough. \nTherefore, we conclude that $H(u)\\to\\infty$  along the set $\\mathcal{B}$.\n\\bigskip\n\nLet $\\mathcal{G} = \\mbox{Gal}(K\/\\mathbb{Q})$ be the Galois group of $K$ over $\\mathbb{Q}$. Since $K$ over $k$ is Galois, we let  $\\mathcal{H}:=\\mbox{Gal}(K\/k)\\subset \\mathcal{G}$  be the subgroup fixing $k$.  Hence, $|\\mathcal{G}\/\\mathcal{H}| = [k:\\mathbb Q] = d$. Therefore, among the $n$ embedding of $K$, there are exactly $d$ embeddings, say, $\\sigma_1,\\ldots,\\sigma_d$  are a representatives for the left cosets of $\\mathcal{H}$ in $\\mathcal{G}$ with $\\sigma_1$ being identity and more precisely, we have \n$$\n\\mathcal{G}\/\\mathcal{H}:=\\{\\mathcal{H}, \\sigma_2 \\mathcal{H},\\ldots,\\sigma_d \\mathcal{H}\\}.\n$$\nFor each $j = 1, 2, \\ldots, d$,  let \n$$\nS_j = \\left\\{v\\in M_\\infty\\ : \\ \\rho_v\\vert_k = \\sigma_j: k\\rightarrow \\mathbb{C}\\right\\}\n$$\nand hence $S_1\\cup\\ldots\\cup S_d=M_\\infty$.   \nWe keep this notation throughout the paper. \nThen   for each $\\rho\\in\\mbox{Gal}(K\/\\mathbb{Q})$, by \\eqref{eq2.2},  we have \n\\begin{equation*}\\label{eq3.2}\n\\tag{3.2}\n|\\delta qu+\\beta-p|^{d(\\rho)\/[K:\\mathbb{Q}]}=|\\rho(\\delta)\\rho(qu)+\\rho(\\beta)-\\rho(p)|_\\rho=|\\rho(\\delta)q\\rho(u)+\\rho(\\beta)-p|_\\rho\n\\end{equation*}\nand hence\n\\begin{equation*}\\label{eq3.3}\n\\tag{3.3}\n\\prod_{\\mathit{v}\\in M_\\infty}|\\rho_\\mathit{v}(\\delta)\\rho_{\\mathit{v}}(qu)+\\rho_\\mathit{v}(\\beta)-p|_\\mathit{v}=\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}|\\rho_{\\mathit{v}}(\\delta)\\sigma_j(qu)+\\rho_{\\mathit{v}}(\\beta)-p|_v.\n\\end{equation*} \nBy \\eqref{eq3.2}, we see that \n$$\n\\prod_{\\mathit{v}\\in M_\\infty}|\\rho_\\mathit{v}(\\delta)\\rho_{\\mathit{v}}(qu)+\\rho_\\mathit{v}(\\beta)-p|_\\mathit{v}=\\prod_{\\mathit{v}\\in M_\\infty}|\\delta q u+\\beta-p|^{d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}=|\\delta q u+\\beta-p|^{{\\sum_{\\mathit{v}\\in M_\\infty}}d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}.\n$$\nThen, from \\eqref{eq3.3} and the well-known formula $\\displaystyle\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})=[K:\\mathbb{Q}]$, it follows that \n\\begin{equation*}\\label{eq3.4}\n\\tag{3.4}\n\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}|\\rho_{\\mathit{v}}(\\delta)\\sigma_j(qu)+\\rho_{\\mathit{v}}(\\beta)-p|_v=|\\delta q u+\\beta-p|.\n\\end{equation*}  \n\nNow, for each $\\mathit{v}\\in S$, we define $d+2$ linearly independent linear forms in $d+2$  variables as follows: For $j = 1, 2, \\ldots, d$ and for  $\\mathit{v}\\in S_j$, we let  \n\\begin{eqnarray*}\nL_{\\mathit{v},0}(x_0,x_1,\\ldots,x_{d+1})&=& \\rho_{\\mathit{v}}(\\beta) x_0 - x_1 + \\rho_{\\mathit{v}}(\\delta) x_{j+1}\\\\  \nL_{\\mathit{v},1}(x_0,x_1,\\ldots,x_{d+1})&=& x_0,\n\\end{eqnarray*}\n and for $2\\leq i\\leq d+1$, we define \n$$\nL_{v, i}(x_1,\\ldots,x_{d+1})=x_i,\n$$\nand for $\\mathit{v}\\in S\\backslash{M_\\infty}$ and for  $0\\leq i \\leq d+1$,  we let  \n$$\nL_{v,i}(x_1,\\ldots,x_{d+1})=x_i.\n$$\nLet $\\mathbf{X}$ be the point in $K^{d+2}$ which is  of the form\n$$\n\\mathbf{X}=(1,p,q\\sigma_1(u),\\ldots,q\\sigma_d(u)) \\in K^{d+2}.\n$$\nIn order to apply  Theorem \\ref{schli}, we need to calculate the following quantity \n\\begin{equation*}\\label{eq3.5}\n\\tag{3.5}\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}\\frac{|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}}{||\\mathbf{X}||_\\mathit{v}}.\n\\end{equation*}\nUsing the fact that $L_{\\mathit{v},i}(\\mathbf{X})=q\\sigma_i(u)$, for $2\\leq i\\leq d+1$, we obtain\n\\begin{eqnarray*}\n\\prod_{\\mathit{v\\in S}}\\prod_{i=2}^{d+1}|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}&=&\\prod_{\\mathit{v}\\in M_\\infty}\\prod_{i=2}^{d+1}|q\\sigma_i(u)|\\prod_{\\mathit{v}\\in S\\backslash{M_\\infty}}\\prod_{i=2}^{d+1}|q\\sigma_i(u)|_\\mathit{v}\\\\\n&=&\\prod_{\\mathit{v}\\in S}\\prod_{i=2}^{d+1}|q|_\\mathit{v}\\prod_{i=2}^{d+1}\\prod_{\\mathit{v}\\in S}|\\sigma_i(u)|_\\mathit{v}.\n\\end{eqnarray*}\nSince $\\sigma_j(u)$  are $S$-units,  by the product formula,  we obtain\n$$\n\\prod_{\\mathit{v}\\in S}|\\sigma_i(u)|_\\mathit{v}=\\prod_{\\mathit{v}\\in M_K}|\\sigma_i(u)|_\\mathit{v}=1.  \n$$\nConsequently, the above inequality gives \n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{i=2}^{d+1}|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}=\\prod_{\\mathit{v}\\in S}\\prod_{i=2}^{d+1}|q|_\\mathit{v}\\leq \\prod_{v\\in M_\\infty}\\prod_{i=2}^{d+1}|q|_\\mathit{v}=\\prod_{i=2}^{d+1}|q|^{\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}\n$$\nThen, from the formula $\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})=[K:\\mathbb{Q}]$, we get \n\\begin{equation*}\\label{eq3.6}\n\\tag{3.6}\n\\prod_{\\mathit{v\\in S}}\\prod_{i=2}^{d+1}|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}\\leq \\prod_{v\\in M_\\infty}\\prod_{i=2}^{d+1}|q|_\\mathit{v}=\\prod_{i=2}^{d+1}|q|^{\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}=|q|^d.\n\\end{equation*}\nNow we estimate the product of the denominators in \\eqref{eq3.5} as follows: consider \n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}||\\mathbf{X}||_\\mathit{v}\\geq \\prod_{\\mathit{v\\in M_K}}\\prod_{i=0}^{d+1}||\\mathbf{X}||_\\mathit{v}=\\prod_{i=0}^{d+1}\\left(\\prod_{\\mathit{v\\in M_K}}||\\mathbf{X}||_\\mathit{v}\\right) = \\prod_{i=0}^{d+1}H(\\mathbf{X}),\n$$\nsince $||\\mathbf{X}||_\\mathit{v}\\leq 1$ for all $\\mathit{v}\\not\\in S$. Thus, we get, \n\\begin{equation*}\\label{eq3.7}\n\\tag{3.7}\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}||\\mathbf{X}||_\\mathit{v}\\geq H(\\mathbf{X})^{d+2}.\n\\end{equation*}\nBy \\eqref{eq3.5}, \\eqref{eq3.6}  and \\eqref{eq3.7}, it follows that \n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}\\frac{|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}}{||\\mathbf{X}||_\\mathit{v}}\\leq \\frac{1}{H^{d+2}(\\mathbf{X})}|q|^d|\\alpha q u+\\beta-p|.\n$$\nThus, from \\eqref{eq3.1}, we have\n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}\\frac{|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}}{||\\mathbf{X}||_\\mathit{v}}\\leq \\frac{1}{H^{d+2}(\\mathbf{X})}|q|^d\\frac{1}{H^{\\varepsilon}(u)}\\frac{1}{|q|^{d+\\varepsilon}}=\\frac{1}{H^{d+2}(\\mathbf{X})}\\frac{1}{(|q|H(u))^{\\varepsilon}}.\n$$\nNotice that \n\\begin{eqnarray*}\nH(\\mathbf{X})&=&\\prod_{\\mathit{v}\\in M_K}\\mbox{max}\\{1, |p|_\\mathit{v},|q\\sigma_1(u)|_\\mathit{v},\\ldots,|q\\sigma_d(u)|_\\mathit{v}\\}\\\\\n&\\leq&\\prod_{\\mathit{v}\\in S}\\mbox{max}\\{1, |p|_\\mathit{v},|q\\sigma_1(u)|_\\mathit{v},\\ldots,|q\\sigma_d(u)|_\\mathit{v}\\}\\\\\n&\\leq& \\prod_{\\mathit{v}\\in S}\\mbox{max}\\{1,|p|_\\mathit{v},|q|_\\mathit{v} \\}\\prod_{\\mathit{v}\\in S}\\mbox{max}\\{1,|\\sigma_1(u)|_\\mathit{v},\\ldots,|\\sigma_d(u)|_\\mathit{v}\\}\\\\\n&\\leq& \\max\\{|p|, |q|\\}\\left(\\prod_{\\mathit{v}\\in S}\\mbox{max}\\{1,|\\sigma_1(u)|_\\mathit{v}\\}\\right)\\cdots \\left(\\prod_{\\mathit{v}\\in S}\\mbox{max}\\{1,|\\sigma_d(u)|_\\mathit{v}\\}\\right)\\\\\n&=&\\max\\{|p|, |q|\\}H^d(u).\n\\end{eqnarray*} \nBy using the inequality \n$\n||x|-|y||\\leq |x-y|\n$\nand the fact that $H(u)\\to \\infty$ for $(u, q, p)\\in \\mathcal{B}$, from \\eqref{eq3.1}, we conclude that $|p|\\leq |\\delta q u+\\beta|+1.$  Since $|u|^{\\frac{1}{d}}\\leq H(u)$, and  by using the fact $H(u)\\to\\infty$,   we get that\n$$\n|p|\\leq |\\delta q u+\\beta|+1\\leq |q||\\delta+\\beta|H^d(u)+1\\leq |q|H^{2d}(u)\n$$\nfor all but finitely many triples $(u, q, p)\\in \\mathcal{B}$. By combining both these above inequalities, we obtain $H(\\mathbf{X})\\leq |q| H(u)^{3d}$, and hence  we get $H(\\mathbf{X})^{1\/3d} \\leq |q|H(u)$. Therefore, we get \n$$\n\\prod_{\\mathit{v\\in S}}\\prod_{i=0}^{d+1}\\frac{|L_{\\mathit{v},i}(\\mathbf{X})|_\\mathit{v}}{||\\mathbf{X}||_\\mathit{v}}\\leq \\frac{1}{H^{d+2}(\\mathbf{X})}\\frac{1}{(|q|H(u))^\\varepsilon}\\leq \\frac{1}{H(\\mathbf{X})^{d+2+(\\varepsilon)\/3d}} = \\frac{1}{H(\\mathbf{X})^{d+2+ \\varepsilon'}},\n$$\nfor some $\\varepsilon' >0$ holds for infinitely many tuples $(1,p,q\\sigma_1(u),\\ldots,q\\sigma_d(u))$ along the triples$(u,q,p)\\in\\mathcal{B}$.  Then by  Theorem \\ref{schli}, there exists a proper subspace of $K^{d+2}$ containing infinitely many $\\mathbf{X}=(1,p,q\\sigma_1(u),\\ldots,q\\sigma_d(u))$ along the triples $(u,q,p) \\in \\mathcal{B}$.  \nThat is,  we have a non-trivial linear relation of the form\n\\begin{equation*}\\label{eq3.8}\n\\tag{3.8}\na_0+a_1 p+b_1q\\sigma_1(u)+\\cdots+b_d q \\sigma_d(u)=0,\\quad a_i, b_j\\in K,\n\\end{equation*}\nsatisfied by all the triples $(u, q, p)\\in\\mathcal{B}_1$ for an infinite subset of $\\mathcal{B}_1$.   \n\\bigskip\n\nUnder the hypotheses of the Main Theorem of \\cite{corv}, the authors establish \nthe existence of such a non-trivial linear relation with $a_0 = 0$. \nThe present situation is slightly more complicated. \nAs in \\cite{corv}, we will establish that there is \na non-trivial linear relation as above with $a_0 = a_1 = 0$, and then we will conclude exactly as in \\cite{corv}.\n\\smallskip\n\n\n\\noindent{\\bf Claim 1.}  At least one of the $b_j$'s is non-zero in the relation \\eqref{eq3.8}.\n\\bigskip\n\nIf not, suppose $b_i=0$  for all $1\\leq i\\leq d$.  Then from \\eqref{eq3.8}, we have \n\\begin{equation*}\\label{eq3.9}\n\\tag{3.9}\n0\\neq p=\\frac{-a_0}{a_1}\\in K.\n\\end{equation*}\nWe deduce from \\eqref{eq3.1} and \\eqref{eq3.9} that\n$$\n0<|\\delta q u+\\beta+\\frac{a_0}{a_1}|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}\n$$\nholds for infinitely many pairs $(u,q)$ along the set $\\mathcal{B}_1$.  Now, we apply Theorem \\ref{schli} with \n$S$ being the finite set composed of the archimedean places on $K$, the linear \nforms $L_{\\mathit{v},1}(X_1, X_2)=(\\beta+\\frac{a_0}{a_1})X_1+\\delta X_2$, $L_{\\mathit{v},2}(X_1, X_2)=X_1$~~ for $\\mathit{v}\\in S$, \nand the pairs $(1,qu)\\in K^2$. Then by Theorem \\ref{schli}, we get a  non-trivial relation of the form \n$$\nc_0+c_1 qu=0\n$$\nwhich holds for infinitely many pairs $(u,q)$ along the set $\\mathcal{B}_1$. \nSince $|qu|\\to\\infty$ as the pairs $(q,u)$ runs through the set $\\mathcal{B}_1$. \nTherefore, we conclude  that $c_0=c_1=0$, which is a contradiction.\nConsequently, at least one of the $b_j$'s is non-zero in the relation \\eqref{eq3.8}. \n\n\n\\bigskip\n\n\\noindent{\\bf Claim 2.}  There exists a non-trivial relation as \\eqref{eq3.8} with $a_0=a_1=0$.\n\\bigskip\n\nAssume that $a_0\\neq 0$. Then by rewriting the  relation \\eqref{eq3.8}, we obtain\n\\begin{equation*}\\label{eq3.10}\n\\tag{3.10}\n\\beta=-\\beta\\left(\\frac{a_1}{a_0}p+\\frac{b_1}{a_0}q\\sigma_1(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right).\n\\end{equation*}\nSubstituting the value of $\\beta$ from \\eqref{eq3.10} in \\eqref{eq3.1}, we get\n$$\n0<\\left|\\delta q u-\\beta\\left(\\frac{a_1}{a_0}p+\\frac{b_1}{a_0}q\\sigma_1(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)-p\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}. \n$$ \nThis is equivalent to write \n\\begin{equation*}\\label{eq3.11}\n\\tag{3.11}\n0<\\left|\\delta q u-(\\beta a_1\/a_0+1)p-\\beta\\left(\\frac{b_1}{a_0}q\\sigma_1(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n\\end{equation*}\n\\vspace{.1cm}\n\nNow we divide rest of the proof in two cases, according as $\\beta a_1\/a_0+1$ is $0$ or not. \n\\bigskip\n\n\\noindent{\\bf Case 1.} The quantity $\\beta a_1\/a_0+1$ vanishes.\n\\vspace{.3cm}\n\nUnder this case the relation \\eqref{eq3.10} can be view as \n\\begin{equation*}\\label{eq3.12}\n\\tag{3.12}\n\\beta-p=-\\left(\\frac{b_1}{a_1}q \\sigma_1(u)+\\cdots+\\frac{b_d}{a_1}q\\sigma_d(u)\\right).\n\\end{equation*}\nSince $K$ over $\\mathbb{Q}$ is Galois and $\\beta$  is an algebraic irrational, then there exists an automorphism $\\rho_0\\in\\mbox{Gal}(K\/\\mathbb{Q})$ such that $\\rho_0(\\beta)\\neq \\beta$. \n\\smallskip\n \nBy applying the automorphism $\\rho_0$ on both sides of the equality \\eqref{eq3.12}, we get \n$$\n\\rho_0(\\beta)-p=-\\left(\\rho_0\\left(\\frac{b_1}{a_1}\\right)q \\rho_0\\circ\\sigma_1(u)+\\cdots+\\rho_0\\left(\\frac{b_d}{a_1}\\right)q\\rho_0\\circ\\sigma_d(u)\\right).\n$$\nSince the restriction of $\\rho_0$ on $k$ is in $\\{\\sigma_1,\\ldots,\\sigma_d\\}$, we can write  this above relation as \n$$\n\\rho_0(\\beta)-p=-\\left(\\rho_0\\left(\\frac{b_1}{a_1}\\right)q \\sigma_{1,0}(u)+\\cdots+\\rho_0\\left(\\frac{b_d}{a_1}\\right)q\\sigma_{d,0}(u)\\right),\n$$\nwhere $\\sigma_{i,0}\\in\\{\\sigma_1,\\ldots,\\sigma_d\\}$ for $1\\leq i\\leq d$.\n\\vspace{.2cm}\n\nNow by subtracting this equality from \\eqref{eq3.12}, we obtain a  relation of the form \n$$\n0\\neq \\rho_0(\\beta)-\\beta=c_1 q \\sigma_1(u)+\\cdots+c_d q\\sigma_d(u),\\quad c_i\\in K\n$$\nfor all the pairs $(q,u)$  along the  triples $(u,q,p)\\in\\mathcal{B}$. This is equivalent to write \n\\begin{equation*}\n\\gamma-(c_1 q \\sigma_1(u)+\\cdots+c_d q\\sigma_d(u))=0\n\\end{equation*}\nholds for all the pairs $(q,u)$ along the set $\\mathcal{B}$, where we have set\n$$\n\\gamma=\\rho_0(\\beta)-\\beta.\n$$ \nWe can easily see that in this   relation  at least one of $c_i$'s is non-zero.  \nDividing this equality by $\\gamma$, we get the non-trivial relation of the kind \n\\begin{equation*}\n\\beta=\\beta\\left(\\frac{c_1}{\\gamma}q \\sigma_1(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right).\n\\end{equation*}\nSubstituting  value of $\\beta$ from this equality in \\eqref{eq3.1}, we get \n$$\n0<\\left|\\delta q u+\\beta\\left(\\frac{c_1}{\\gamma}q \\sigma_1(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right)-p\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n$$\nThis is equivalent to write \n\\begin{equation*}\\label{eq3.16}\n\\tag{3.13}\n0<\\left|\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_1(u)+\\beta\\left(\\frac{c_2}{\\gamma}q \\sigma_2(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right)-p\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n\\end{equation*}\nRecall that, for $j = 1, 2, \\ldots, d$,  we have set\n$$\nS_j = \\left\\{v\\in M_\\infty\\ : \\ \\rho_v\\vert_k = \\sigma_j: k\\rightarrow \\mathbb{C}\\right\\}\n$$\nand hence $S_1\\cup\\ldots\\cup S_d=M_\\infty$, where $M_\\infty$ denotes the set of all archimedean places on $K$. Then   for each $\\rho\\in\\mbox{Gal}(K\/\\mathbb{Q})$, by \\eqref{eq2.2},  we have\n\\begin{align*}\n&\\left|\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_1(u)+\\beta\\left(\\frac{c_2}{\\gamma}q \\sigma_2(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right)-p\\right|^{d(\\rho)\/[K:\\mathbb{Q}]}\\\\ &=\\left|\\rho\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\rho\\circ\\sigma_1(u)+\\left(\\rho\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\rho\\circ\\sigma_2(u)+\\cdots+\\rho\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\rho\\circ\\sigma_d(u)\\right)-p\\right|_\\rho\n\\end{align*}\nand hence \n\\begin{align*}\n&\\prod_{\\mathit{v}\\in M_\\infty}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_1(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_2(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_d(u)\\right)-p\\right|_\\mathit{v}\\\\ &=\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_1(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_2(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_d(u)\\right)-p\\right|_\\mathit{v}.\n\\end{align*}\n\nFor each $\\mathit{v}\\in M_\\infty$ and $j=1,2,\\ldots,d$, we define $\\mathit{v}(j)$  such that $\\rho_\\mathit{v}\\circ\\sigma_j=\\sigma_{\\mathit{v}(j)}$  on the field $k$, where $\\{\\mathit{v}(1), \\ldots,\\mathit{v}(d)\\}$ is a permutation of $\\{1,\\ldots,d\\}$.  Hence the above relation can be written as \n\\begin{align*}\n&\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_1(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_2(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\rho_\\mathit{v}\\circ\\sigma_d(u)\\right)-p\\right|_\\mathit{v}\\\\&=\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_{\\mathit{v}(1)}(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\sigma_{\\mathit{v}(2)}(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\sigma_{\\mathit{v}(d)}(u)\\right)-p\\right|_\\mathit{v}.\n\\end{align*}\nBy \\eqref{eq3.2}, we see that \n\\begin{align*}\n&\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_{\\mathit{v}(1)}(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\sigma_{\\mathit{v}(2)}(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\sigma_{\\mathit{v}(d)}(u)\\right)-p\\right|_\\mathit{v}\\\\ &=\\left|\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_1(u)+\\beta\\left(\\frac{c_2}{\\gamma}q \\sigma_2(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right)-p\\right|^{{\\sum_{\\mathit{v}\\in M_\\infty}}d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}.\n\\end{align*}\nThen, from \\eqref{eq3.16} and the well-known formula $\\displaystyle\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})=[K:\\mathbb{Q}]$, it follows that \n\\begin{align*}\n&\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_{\\mathit{v}(1)}(u)+\\left(\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)q \\sigma_{\\mathit{v}(2)}(u)+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)q \\sigma_{\\mathit{v}(d)}(u)\\right)-p\\right|_\\mathit{v}\\\\ &=\\left|\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)q \\sigma_1(u)+\\beta\\left(\\frac{c_2}{\\gamma}q \\sigma_2(u)+\\cdots+\\frac{c_d}{\\gamma}q \\sigma_d(u)\\right)-p\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n\\end{align*}\nNow for each $\\mathit{v}\\in S$, we define $d+1$ linearly independent linear forms in $d+1$ variables as follows:  For $j = 1, 2, \\ldots, d$ and for  $\\mathit{v}\\in S_j$, we let  \n\\begin{align*}\nL_{\\mathit{v},0}(x_0, x_1,\\ldots,x_d)&=-x_0+\\rho_{\\mathit{v}}\\left(\\delta+\\beta\\frac{c_1}{\\gamma}\\right)x_{\\mathit{v}(1)}+\\rho_\\mathit{v}\\left(\\beta\\frac{c_2}{\\gamma}\\right)x_{\\mathit{v}(2)}+\\cdots+\\rho_\\mathit{v}\\left(\\beta\\frac{c_d}{\\gamma}\\right)x_{\\mathit{v}(d)}\n\\end{align*}\nand for $1\\leq i\\leq d+1$, we define \n$$\nL_{v, i}(x_1,\\ldots,x_{d+1})=x_i,\n$$\nand for $\\mathit{v}\\in S\\backslash{M_\\infty}$ and for  $0\\leq i \\leq d+1$,  we let  \n$$\nL_{v,i}(x_1,\\ldots,x_{d+1})=x_i.\n$$\nClearly, the linear forms $L_{\\mathit{v},0},\\ldots,L_{\\mathit{v},d}$ are linearly independent for each $\\mathit{v}\\in S$. Finally, let ${}\\mathbf{X}$ be the point in $K^{d+1}$, which is of the form\n$$\n{\\mathbf{X}}=(p, q\\sigma_1(u),\\ldots,q\\sigma_d(u)).\n$$\nThen by using Theorem \\ref{schli}  similar to the first part of this lemma, we get a non-trivial relation of the form\n$$\na_1 p+b_1 q \\sigma_1(u)+\\cdots+b_d q \\sigma_d(u)=0,\\quad a_1, b_i\\in K.\n$$\nNow we prove that there exists a relation with $a_1=0$. In order to prove this, we follow the similar procedure as in \\cite[Lemma 3, Claim]{corv} along with our   Lemma \\ref{lem2.1}.  If $a_1\\neq 0$, then we have \n\\begin{equation*}\\label{eq3.17}\n\\tag{3.14}\np=-\\frac{b_1}{a_1} q \\sigma_1(u)-\\cdots-\\frac{b_d}{a_1} q\\sigma_d(u).\n\\end{equation*}\nFirst we consider the case when  $\\displaystyle\\sigma_j\\left(\\frac{b_1}{a_1}\\right) \\ne \\frac{b_j}{a_1}$ for some $j$ with $2\\leq j\\leq d$. \n\\smallskip\n\nBy applying the automorphism $\\sigma_j$  on both sides of \\eqref{eq3.16}, we get\n$$\np=-\\sigma_j\\left(\\frac{b_1}{a_1}\\right)q\\sigma_j\\circ\\sigma_1(u)-\\cdots-\\sigma_j\\left(\\frac{b_d}{a_1}\\right)q\\sigma_j\\circ\\sigma_d(u).\n$$\nBy subtracting this relation from \\eqref{eq3.17}, we get  a relation of the form\n\\begin{equation*}\nb_1 q \\sigma_1(u)+\\cdots+b_d q \\sigma_d(u)=0,\\quad  \\mbox{with~~} b_i\\in K. \n\\end{equation*}\nNow we assume that   $\\displaystyle\\frac{b_j}{a_1}= \\sigma_j\\left(\\frac{b_1}{a_1}\\right)$  for all $2\\leq j\\leq d$.  \n\\smallskip\n\nNote that $b_1 \\ne 0$. If not, then $0 =\\sigma_j(b_1\/a_1) = b_j\/a_1$ for every $j$.  Hence  $b_i=0$ for all $i$, which contradicts Claim 1.  By putting $\\lambda = -b_1\/a_1$,   we rewrite \\eqref{eq3.17} as \n\\begin{equation*}\\label{eq3.18}\n\\tag{3.15}\np=-q(\\sigma_1(\\lambda)\\sigma_1(u)+\\cdots+\\sigma_d(\\lambda)\\sigma_d(u)).\n\\end{equation*}\n Since $b_j \\in K$, it may happen that $\\lambda$ does not belong to $k$. If so,  then there exists an automorphism $\\tau\\in \\mathcal{H}$ with $\\tau(\\lambda)\\neq \\lambda$.  By applying the automorphism $\\tau$ on both sides of \\eqref{eq3.18} to eliminate $p$, we obtain the linear relation \n$$\n(\\lambda-\\tau(\\lambda))q\\sigma_1(u)+q\\sum_{i=2}^d(\\sigma_i(\\lambda)\\sigma_i(u)-\\tau \\circ\\sigma_i(u))=0.\n$$\nNote that $\\tau \\circ\\sigma_j$  coincides on $k$ with some $\\sigma_i$ and  since $\\tau\\in \\mathcal{H}$  and $\\sigma_2,\\ldots,\\sigma_d \\not\\in \\mathcal{H}$, none of the $\\tau \\circ \\sigma_j$ with $j\\geq 2$ belongs in $\\mathcal{H}$.  Hence the above relation can be view  as a linear combination of the  $\\sigma_i(u)$ with the property that the coefficients of $\\sigma_1(u)$ will remain $\\lambda-\\tau(\\lambda)$.  By our assumption that $\\lambda\\notin k$ and hence we have $\\lambda-\\tau(\\lambda)\\neq 0$. Therefore, we obtain a non-trivial relation among  $\\sigma_i(u)$ of the form \n$$\nb_1 \\sigma_1(u)+\\cdots+b_d\\sigma_d(u)=0,\\quad \\mbox{with~} b_i\\in K.\n$$ \n\\smallskip\n\nTherefore we can  assume that $\\lambda\\in k$ and write \\eqref{eq3.18} in the simpler form\n\\begin{equation*}\\label{eq3.19}\n\\tag{3.16}\np=q\\mbox{Tr}_{k\/\\mathbb{Q}}(\\lambda u)=\\mbox{Tr}_{k\/\\mathbb{Q}}(q\\lambda u).\n\\end{equation*}\nBy adding $-\\delta qu-\\beta$ on both sides of  the equality   \\eqref{eq3.19},  we get   \n$$\n0<|p-\\delta qu -\\beta|=\\left|-\\beta+(\\lambda-\\delta)q\\sigma_1(u)+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|.\n$$\nThen from \\eqref{eq3.1}, we get\n\\begin{equation*}\\label{eq3.20}\n\\tag{3.17}\n0<\\left|-\\beta+(\\lambda-\\delta)q\\sigma_1(u)+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}\n\\end{equation*}\nholds for infinitely many pairs $(u,q)$ along the triples $(u,q,p)\\in\\mathcal{B}$.\n\\smallskip\n\nNow we consider the case when  $\\lambda=\\delta$. Then from \\eqref{eq3.20}, we have\n\\begin{equation*}\\label{eq3.21}\n\\tag{3.18}\n0<\\left|-\\beta+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}.\n\\end{equation*}\nFirst we note that \n$$\n\\max\\{|\\sigma_2(q\\lambda u)|,\\ldots,|\\sigma_d(q\\lambda u)|\\}\\geq \\frac{|\\beta|}{2d}\n$$\nholds for all but finitely many   pairs $(u,q)$ satisfied \\eqref{eq3.21}. If not, we  assume that \n$$\n\\max\\{|\\sigma_2(q\\lambda u)|,\\ldots,|\\sigma_d(q\\lambda u)|\\}< \\frac{|\\beta|}{2d}\n$$\nfor all pairs $(q,u)$ satisfied \\eqref{eq3.21}. Then, we get \n$$\n\\left|-\\beta+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|\\geq \\frac{|\\beta|}{2}.\n$$\nThus from \\eqref{eq3.21}, we have \n\\begin{equation*}\\label{eq3.23}\n\\tag{3.19}\n\\frac{|\\beta|}{2}\\leq \\left|-\\beta+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}.\n\\end{equation*}\nSince $H(u)\\to\\infty$ along   infinitely many pairs $(u,q)$ satisfied \\eqref{eq3.21} and $\\beta$ is non-zero, we see that  the inequality \\eqref{eq3.23} can have only finitely many solutions in the pairs $(q,u)$. Therefore we conclude that \n$$\n\\max\\{|\\sigma_2(q\\lambda u)|,\\ldots,|\\sigma_d(q\\lambda u)|\\}\\geq \\frac{|\\beta|}{2d}\n$$\nholds for all but finitely many   pairs $(q,u)$ satisfied \\eqref{eq3.21}.\n\nThis can be re-written as \n$$\n\\max\\{|q\\sigma_2(u)|,\\ldots,|q\\sigma_d(u)|\\}\\geq \\frac{\\min\\{1,|\\beta|\\}}{2d(\\max\\{|\\sigma_2(\\lambda)|,\\ldots,|\\sigma_d(\\lambda)|\\})}.\n$$\nThen from \\eqref{eq3.21}, we conclude that \n$$\n0<\\left|-\\beta+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|<\\frac{1}{H^\\varepsilon(u)q^{d+\\varepsilon}}<\\frac{C\\max\\{|q\\sigma_2(u)|,\\ldots,|q\\sigma_d(u)|\\} }{H^\\varepsilon(u)q^{d+\\varepsilon}},\n$$\nwhere $C=\\frac{2d(\\max\\{|\\sigma_2(\\lambda)|,\\ldots,|\\sigma_d(\\lambda)|\\})}{\\min\\{1,|\\beta|\\}}$. Hence by Lemma \\ref{lem2.1}, we get a non-trivial relation as we claimed.\n\\smallskip\n\n\n\nNow we assume that $\\lambda\\neq \\delta$. In this case the term $(\\lambda-\\delta)\\sigma_1(u)$ does appear in \\eqref{eq3.20}.  Since we supposed that $|\\lambda q u|>1$,  we have \n$$\n\\max\\mbox\\{|q\\sigma_1(u)|,\\ldots,|q\\sigma_d(u)|\\}\\geq |qu|>|\\lambda|^{-1} ,\n$$\nfor infinitely many pairs $(u,q)$ along the triples $(u,q,p)\\in\\mathcal{B}$ . Then from  \\eqref{eq3.20}, we obtain\n$$\n0<\\left|-\\beta+(\\lambda-\\delta)q\\sigma_1(u)+q\\sigma_2(\\lambda)\\sigma_2(u)+\\cdots+q\\sigma_d(\\lambda)\\sigma_d(u)\\right|<\\frac{|\\lambda|\\max{|q\\sigma_1(u)|,\\ldots,|q\\sigma_d(u)|}}{H^\\varepsilon(u)q^{d+\\varepsilon}}.\n$$  \nThus by applying Lemma \\ref{lem2.1} with the inputs $\\omega$ is an archimidean places corresponding to the embedding of $K$ defined by $\\alpha\\hookrightarrow {\\alpha}$, $n=d$, $\\lambda_0=-\\beta, \\lambda_1=\\lambda-\\delta$, $\\lambda_i=\\sigma_{i}(\\lambda)$ for $2\\leq i\\leq d$, we get a non-trivial relation of the form\n\\begin{equation*}\nb_1 \\sigma_1(u)+\\cdots+b_d\\sigma_d(u)=0\n\\end{equation*}\nvalid for all triples $(u, q,p)\\in \\mathcal{B}_1$  for an  infinite subset $\\mathcal{B}_1\\subset\\mathcal{B}$.\n\\bigskip\n\n\\noindent{\\bf Case 2.~}  $\\beta a_1\/a_0+1\\neq 0$.\n\\vspace{.2cm}\n\nFirst we recall \\eqref{eq3.11},\n$$\n0<\\left|\\delta q u-\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\beta\\left(\\frac{b_1}{a_0}q\\sigma_1(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n$$ \nThe crucial point in Case 1 was the non-vanishing  of the coefficient of $p$ in the inequality \\eqref{eq3.16}. Since in this case  we also  have the  similar situation as the coefficient $(\\beta a_1\/a_0+1)$ of $p$ is non-zero in this inequality,  we  follow the similar procedure to this inequality  as we have done for the inequality \\eqref{eq3.16} in Case 1. \n\\smallskip\n\nFor each $\\rho\\in\\mbox{Gal}(K\/\\mathbb{Q})$, by \\eqref{eq2.2},  we have \n\\begin{align*}\n&\\left|\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\sigma_1(u) -\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\beta\\left(\\frac{b_2}{a_0}q\\sigma_2(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)\\right|^{d(\\rho)\/[K:\\mathbb{Q}]}\\\\&=\\left|\\rho\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\rho\\circ\\sigma_1(u)-\\rho\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\rho(\\beta b_2\/a_0q\\sigma_1(u))-\\cdots-\\rho(\\beta b_d\/a_0 q\\sigma_d(u))\\right|_\\rho\n\\end{align*}\nand hence\n\\begin{align*}\n&\\prod_{\\mathit{v}\\in M_\\infty}\\left|\\rho_\\mathit{v}\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\rho_\\mathit{v}\\circ\\sigma_1(u)-\\rho_\\mathit{v}(\\beta a_1\/a_0+1)p-\\rho_\\mathit{v}\\left(\\frac{\\beta b_2}{a_0}\\right)q\\rho_\\mathit{v}\\circ\\sigma_2(u)-\\cdots-\\rho_\\mathit{v}\\left(\\frac{b_d}{a_0}\\right) q\\rho_\\mathit{v}\\circ\\sigma_d(u)\\right|_\\mathit{v}\\\\&=\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_\\mathit{v}\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\rho_\\mathit{v}\\circ\\sigma_1(u)-\\rho_\\mathit{v}\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\cdots-\\rho_\\mathit{v}\\left(\\frac{\\beta b_d}{a_0}\\right) q\\rho_\\mathit{v}\\circ\\sigma_d(u)\\right|_\\mathit{v}.\n\\end{align*} \nFor each $\\mathit{v}\\in M_\\infty$ and $j=1,2,\\ldots,d$, we define $\\mathit{v}(j)$  such that $\\rho_\\mathit{v}\\circ\\sigma_j=\\sigma_{\\mathit{v}(j)}$  on the field $k$, where $\\{\\mathit{v}(1), \\ldots,\\mathit{v}(d)\\}$ is a permutation of $\\{1,\\ldots,d\\}$.\nBy \\eqref{eq3.2}, we see that \n\\begin{align*}\n&\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_\\mathit{v}\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\sigma_{\\mathit{v}(1)}(u)-\\rho_\\mathit{v}\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\rho_\\mathit{v}\\left(\\frac{\\beta b_2}{a_0}\\right)q\\sigma_{\\mathit{v}(2)}(u)-\\cdots-\\rho_\\mathit{v}\\left(\\frac{\\beta b_d}{a_0}\\right) q\\sigma_{\\mathit{v}(d)}(u))\\right|_\\mathit{v}\\\\&=\\left|\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right) q \\sigma_1(u) -\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\beta\\left(\\frac{b_2}{a_0}q\\sigma_1(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)\\right|^{{\\sum_{\\mathit{v}\\in M_\\infty}}d(\\rho_\\mathit{v})\/[K:\\mathbb{Q}]}.\n\\end{align*}\nThen, from \\eqref{eq3.11} and the well-known formula $\\displaystyle\\sum_{\\mathit{v}\\in M_\\infty}d(\\rho_\\mathit{v})=[K:\\mathbb{Q}]$, it follows that \n\\begin{align*}\n&\\prod_{j=1}^d\\prod_{\\mathit{v}\\in S_j}\\left|\\rho_\\mathit{v}\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)q\\sigma_{\\mathit{v}(1)}(u)-\\rho_\\mathit{v}\\left(\\frac{\\beta a_1}{a_0}+1\\right)p-\\rho_\\mathit{v}\\left(\\frac{\\beta b_2}{a_0}\\right)q\\sigma_{\\mathit{v}(2)}(u)-\\cdots-\\rho_\\mathit{v}\\left(\\frac{\\beta b_d}{a_0}\\right) q\\sigma_{\\mathit{v}(d)}(u))\\right|_\\mathit{v}\\\\&=\\left|\\left(\\delta-\\beta \\frac{b_1}{a_0}\\right) q \\sigma_1(u) -(\\beta a_1\/a_0+1)p-\\beta\\left(\\frac{b_2}{a_0}q\\sigma_2(u)+\\cdots+\\frac{b_d}{a_0}q\\sigma_d(u)\\right)\\right|<\\frac{1}{H^\\varepsilon(u)}\\frac{1}{q^{d+\\varepsilon}}.\n\\end{align*}\nNow for each $\\mathit{v}\\in S$, we define $d+1$ linearly independent linear forms in $d+1$ variables as follows: for $j=1,\\ldots,d$ and for each $\\mathit{v}\\in S_j$, we define\n\\begin{align*}\nL_{\\mathit{v},0}(x_0, x_1,\\ldots,x_d)&=-\\rho_{\\mathit{v}}\\left(\\frac{\\beta a_1}{a_0}+1\\right)x_0+\\rho_\\mathit{v}\\left(\\delta-\\beta\\frac{b_1}{a_0}\\right)x_{\\mathit{v}(1)}-\\rho_{\\mathit{v}}\\left(\\frac{\\beta b_2}{a_0}\\right)x_{\\mathit{v}(2)}-\\cdots-\\rho_{\\mathit{v}}\\left(\\frac{\\beta b_d}{a_0}\\right)x_{\\mathit{v}(d)}\n\\end{align*}\nand for $1\\leq i\\leq d+1$, we define \n$$\nL_{v, i}(x_1,\\ldots,x_{d+1})=x_i,\n$$\nand for $\\mathit{v}\\in S\\backslash{M_\\infty}$ and for  $0\\leq i \\leq d+1$,  we let  \n$$\nL_{v,i}(x_1,\\ldots,x_{d+1})=x_i.\n$$\nSince  in this case   $\\beta\\frac{a_1}{a_0}+1$ is non-zero,  we see that the linear forms $L_{\\mathit{v},0},\\ldots,L_{\\mathit{v},d}$ are linearly independent for each $\\mathit{v}\\in S$. Finally, let $\\mathbf{X}$ be the point in $K^{d+1}$, which is of the form\n$$\n{\\mathbf{X}}=(p, q\\sigma_1(u),\\ldots,q\\sigma_d(u)).\n$$\nThen by using Theorem \\ref{schli} similar to Case 1  of this lemma, we get a non-trivial relation of the form\n$$\na_1 p+b_1 q \\sigma_1(u)+\\cdots+b_d q \\sigma_d(u)=0.\n$$\nHence by  follow the  similar procedure  as we have done in Case 1, we obtain a non-trivial relation of the form \n$$\nb_1 \\sigma_1(u)+\\cdots+b_d\\sigma_d(u)=0,\\quad \\mbox{with~} b_i\\in K,\n$$ \nvalid for all triples $(u, q,p)\\in \\mathcal{B}_1$  for an  infinite subset $\\mathcal{B}_1\\subset\\mathcal{B}$. \n\\bigskip\n\n\nThus by combining all the Cases, we  obtain  a non-trivial relation of the form \n$$\nb_1 \\sigma_1(u)+\\cdots+b_d  \\sigma_d(u)=0, \\quad b_i\\in K\n$$\nholds for infinitely many   $u$ along the triples $(u,q,p)\\in\\mathcal{B}$. This proves our Claim 2.\nWe then conclude exactly as in \\cite[Lemma 3]{corv}. \n\\end{proof}\n\n\n\\section{Proofs} \n\\noindent{\\bf Proof of Theorem \\ref{maintheorem}.~} Since $\\Gamma$ is finitely generated multiplicative subgroup of $\\overline{\\mathbb{Q}}^\\times$, we can reduce to the situation where  $\\Gamma\\subset\\overline{\\mathbb{Q}}^\\times$ is the subgroup of $S$-units, namely,\n$$\n\\Gamma\\subset\\mathcal{O}^\\times_S=\\{u\\in K:\\prod_{\\mathit{v}\\in S}|u|_\\mathit{v}=1\\}\n$$\nof a suitable Galois extension $K$  over $\\mathbb{Q}$ containing  $\\delta,\\beta$  and for a suitable finite set $S$ of places of $K$ containing all the archimedean places. Also, $S$ is stable under Galois conjugation. \n\\smallskip\n\nSuppose that the conclusion of Theorem \\ref{maintheorem} is not true. Then  there exist infinitely many triples $(u,q,p)\\in \\mathcal{B} \\subset\\Gamma\\times \\mathbb{Z}^2$ satisfying \\eqref{eq1.1}.  Then by inductively,  we construct  a sequence $\\{\\delta_i\\}_{i=0}^\\infty$  elements of  $K$,  an infinite decreasing chain $\\mathcal{B}_i$ of an infinite subset of $\\mathcal{B}$  and an infinite strictly decreasing chain $k_i$ of subfields of $K$ with the following properties:\n\\bigskip\n\n{\\it For each integer $n\\geq 0$, $\\mathcal{B}_n\\subset (k_n\\times\\mathbb{Z})\\cap\\mathcal{B}_{n-1}$, $k_n\\subset k_{n-1}$, $k_n\\neq k_{n-1}$  and for all but finitely many triples $(u, q,p)\\in\\mathcal{B}_n$   satisfying  \n\\begin{equation*}\\label{eq4.1}\n\\tag{4.1}\n|\\delta_0\\cdots\\delta_n q u+\\beta-p|<\\frac{1}{H^{\\varepsilon\/(n+1)}(u)q^{d+\\varepsilon}}.\n\\end{equation*}}\nIf such a sequence exists, then  we  eventually get a contradiction from  the fact that the number field $K$ does not admit an infinite  strictly  decreasing chain of subfields.  Thus  in order to complete the proof of the theorem, it is enough to construct such a sequence.\n\\smallskip\n\nWe proceed our construction by applying induction on $n$: for $n=0$, put $\\delta_0=\\delta$,  $k_0=K$  and $\\mathcal{B}_0=\\mathcal{B}$, and we are done  in this case, since by our supposition the inequality \n$$\n|\\delta_0 q u+\\beta-p|<\\frac{1}{H^\\epsilon(u) q^{d+\\eps}}\n$$ \nhas infinitely many solutions in triples $(u, q, p)$. \nThen by the induction hypothesis, we assume that  $\\delta_n$,  $k_n$  and $\\mathcal{B}_n$  exist for an integer $n\\geq 0$ such that  \\eqref{eq4.1} holds.\nThen by  Lemma \\ref{lem3.1} to the choices $\\delta=\\delta_0\\delta_1\\cdots\\delta_n$ and $k=k_n$, we obtain  an element $\\delta_{n+1}\\in k_n$, a proper subfield $k_{n+1}$ of $k_n$  and an infinite set $\\mathcal{B}_{n+1}\\subset\\mathcal{B}_n$  such that all triples $(u, q, p)\\in\\mathcal{B}_{n+1}$  satisfy $u=\\delta_{n+1}u'$  with $u'\\in k_{n+1}$.\nNow, since $u'\\in K$, $H(\\delta_{n+1}u')\\geq H(\\delta)^{-1} H(u')$, we have in  particular that for almost for all $u'\\in K$, $H(\\delta_{n+1} u')\\geq H^{\\frac{n+1}{n+2}}(u')$.  Therefore by replacing $u$ by $\\delta_{n+1}u'$, for all but finitely many triples $(u',q,p)\\in\\mathcal{B}_{n+1}$, we have the following inequality\n$$\n|\\delta_0\\delta_1\\cdots\\delta_n q \\delta_{n+1}u'+\\beta-p|<\\frac{1}{H(u')^{\\varepsilon\/(n+2)}q^{d+\\varepsilon}}\n$$\nThis fulfil the induction and hence the theorem. $\\hfill\\Box$\n\\bigskip\n\n\\noindent{\\bf Proof of Theorem \\ref{maintheorem2}.~}  Suppose the conclusion of Theorem \\ref{maintheorem2} is not true. Then we may assume that $\\alpha$ is an algebraic number.\n\\smallskip\n\nSince $|\\alpha|>1$,  we have $|\\lambda\\alpha^n|>1$ for $n$ large enough.  Now we choose $\\epsilon'>0$ small enough so that   $\\varepsilon'<\\varepsilon\\log 2\/\\log H(\\alpha)$. Then  we get that for infinitely many $n$,\n\\begin{equation*}\\label{eq4.2}\n\\tag{4.2}\n0<\\Vert \\lambda\\alpha^n+\\beta\\Vert<H(\\alpha^n)^{-\\varepsilon'}.\n\\end{equation*}\nOn the other hand by taking  $\\Gamma$ is  the subgroup  generated by $\\alpha$ and  $q=1$, $\\delta=\\lambda$ and $u=\\alpha^n$, we  see that  the hypothesis of Theorem \\ref{maintheorem} is satisfied. Thus   from Theorem \\ref{maintheorem}, the inequality \\eqref{eq4.2} can have only finitely  many solutions in $n$. This contradiction proves that  $\\alpha$ must be a transcendental number and hence the theorem.\n\\bigskip\n\n\\noindent{\\bf Acknowledgements.} I would like to express my deep gratitude to Professor Yann Bugeaud for his valuable  suggestions and comments in the earlier draft of this article. Also, I would like to acknowledge the Department of Atomic Energy, Govt. of India, for providing the research grant.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}    \n\nOriginating with conjectures of Erd\\H{o}s and Lov\\'asz, an extensive literature has developed over the past several decades concerning the existence of particular differences within dense sets of integers. For sets $A,B\\subseteq \\Z$, we define the sum and difference sets, respectively, as usual by $A\\pm B= \\{a \\pm b: a\\in A, b\\in B\\}$, and we also define the following threshold.\n\\begin{definition}For $X\\subseteq \\Z$ and $N\\in \\N$, we define $D(X,N)=\\max \\left\\{|A|: A\\subseteq [1,N], \\ (A-A)\\cap X \\subseteq \\{0\\} \\right\\}. $  \n\\end{definition}\n\n\\noindent We use $[1,N]$ to denote $\\{1,2,\\dots,N\\}$ and $|A|$ to denote the size of a finite set $A$. To clarify, $D(X,N)$ is the threshold such that any subset of $\\{1,2,\\dots,N\\}$ with more than $D(X,N)$ elements necessarily contains two distinct elements that differ by an element of $X$. As an introductory offering prior to extensive discussions of history, motivation, notation, and terminology, a very special case of our results in this paper is the following:  \n\n\\begin{theorem} \\label{introthm} Suppose $h\\in \\Z[x,y]$ is a homogeneous polynomial of degree $k\\geq 2$. If $\\Delta(h) \\neq 0$, then \\begin{equation} \\label{2vb} D(h(\\Z^{2}),N) \\ll_h N e^{-c(\\log N)^{\\mu}}, \\end{equation} where $c=c(h)>0$ and $\\mu=[(k-1)^2+1]^{-1}$.\n\\end{theorem} \n\\noindent Here $\\Delta$ denotes the usual homogeneous discriminant, and we use $\\ll$ to denote ``less than a constant times'', with subscripts indicating on what parameters, if any, the implied constant depends. We take the same convention with subscripts on Big O notation. Theorem \\ref{introthm} follows from Corollary \\ref{2vcor} and our main result, Theorem \\ref{more}, of which we discuss various improvements and important special cases throughout Section \\ref{maindr}.\n\\subsection{Background} Lov\\'asz asked whether a set of positive upper density must contain two distinct elements that differ by a perfect square, or equivalently whether $D(S,N)=o(N)$, where $S=\\{n^2: n\\in \\N\\}$. Similarly, Erd\\H{o}s conjectured that $D(\\mathcal{P}-1,N)=o(N)$, where $\\mathcal{P}-1=\\{p-1: p \\text{ prime}\\}$. Furstenberg \\cite{Furst} verified the former using ergodic methods, specifically his correspondence principle, in the same paper in which he provided the second known proof of Szemer\\'edi's Theorem on arithmetic progressions. Independently and concurrently, S\\'ark\\\"ozy (\\cite{Sark1}, \\cite{Sark3}) verified both conjectures with a Fourier analytic density increment argument driven by the Hardy-Littlewood circle method. Further, S\\'ark\\\"ozy's results included quantitative information, showing $ D(S,N) \\ll_{\\epsilon} N(\\log N)^{-1\/3+\\epsilon}$ and $ D(\\mathcal{P}-1,N)\\ll_{\\epsilon} N (\\log\\log N)^{-2+\\epsilon}$ for every $\\epsilon>0.$ \n\nThese results have been incrementally improved and generalized in multiple ways, both through tightening of the quantitative bounds and expansion of the possibilities for the set $X$ of prohibited differences. Regarding the former, Pintz, Steiger and Szemer\\'edi \\cite{PSS} utilized a more elaborate Fourier analytic strategy to show \\begin{equation}\\label{PSSb} D(S,N)\\ll N (\\log N)^{-c\\log\\log\\log\\log N} \\end{equation} for a constant $c>0$. \n \nDramatically improving S\\'ark\\\"ozy's original bound, Ruzsa and Sanders \\cite{Ruz} showed \\begin{equation} \\label{rsb} D(\\mathcal{P}-1,N)\\ll Ne^{-c(\\log N)^{\\mu}}\\end{equation} with $\\mu=1\/4$, recently improved to $\\mu=1\/3$ by Wang \\cite{wang}. Regarding alternative choices for the set of prohibited differences, one must first consider obvious local obstructions. For example, we consider $\\mathcal{P}-1$, rather than $\\mathcal{P}$, because  $\\P\\cap 4\\Z = \\emptyset$ implies $D(\\mathcal{P},N)\\geq \\lceil N\/4 \\rceil$ by taking $A$ to be a congruence class modulo $4$. Analogously, if $h\\in \\Z[x]$ and $h(\\Z)$ contains no multiples of $q\\in \\N$, then $D(h(\\Z),N)\\geq \\lceil N\/q\\rceil$. Therefore, for even a qualitative $o(N)$ result, it is clearly necessary that $h(\\Z)$ contains a nonzero multiple of every $q\\in \\N$, in which case we say that $h$ is an \\textit{intersective polynomial}. Examples of intersective polynomials include any nonzero polynomial with an integer root or a collection of rational roots with coprime denominators. However, there are also intersective polynomials with no rational roots, such as $(x^3-19)(x^2+x+1)$.  \n\nBalog, Pelik\\'an, Pintz, and Szemer\\'edi \\cite{BPPS} extended (\\ref{PSSb}) with $S$ replaced by $\\{n^k: n\\in \\N\\}$ for any fixed $k\\in \\N$. For a general univariate intersective polynomial, Kamae and Mendes-France \\cite{KMF} established the qualitative $o(N)$ result, the first quantitative bounds were due to Lucier \\cite{Lucier}, and the second author \\cite{ricemax} fully extended (\\ref{PSSb}). In a recent preprint, Bloom and Maynard \\cite{BloomMaynard} both simplified and improved the ideas of \\cite{PSS}, using a more traditional density increment to establish \\begin{equation}\\label{BMnew} D(S,N)\\ll N (\\log N)^{-c\\log\\log\\log N} \\end{equation} for a constant $c>0$, which is currently the best-known bound for the original square difference question. Further, the methods of \\cite{BloomMaynard} are completely compatible with those of \\cite{ricemax}, so in fact (\\ref{BMnew}) should hold for the full class of intersective polynomials.  For other intermediate and related results, as well as alternative proofs, the reader may refer to (in chronological order)  \\cite{Green}, \\cite{Slip},  \\cite{Lucier2},  \\cite{LM}, \\cite{lipan},  \\cite{Lyall}, \\cite{HLR}, \\cite{Rice}, and \\cite{taoblog}. \n\nAlso in \\cite{ricemax}, the second author showed that if $g,h\\in \\Z[x]$ are intersective polynomials, then \\begin{equation} \\label{splitb} D(g(\\Z)+h(\\Z), N) \\ll_{g,h} N e^{-c(\\log N)^{\\mu}}, \\end{equation} where $c=c(g,h)>0$ and $\\mu=\\mu(\\deg(g),\\deg(h))>0$. Further, the second author \\cite{Ricebin} considered the simplest nontrivial case of a non-diagonal multivariate polynomial, showing that for a binary quadratic form $h(x,y)=ax^2+bxy+cy^2\\in \\Z[x,y]$ with $b^2-4ac\\neq 0$, we have \\begin{equation} \\label{binb} D(h(\\Z^2), N) \\ll_{h} N e^{-c\\sqrt{\\log N}}. \\end{equation} \n \n\\subsection{Motivation} As outlined in Section 2.4 of \\cite{ricemax}, the quoted upper bounds in the previous section, all of which result from adaptations of the two aforementioned Fourier analytic arguments developed in \\cite{Sark1} and \\cite{PSS}, respectively, are partially determined by the degree of decay in local exponential averages similar to \\begin{equation}\\label{gsintro} q^{-1}\\sum_{s=0}^{q-1}e^{2\\pi i h(s)a\/q}. \\end{equation}\nThe best general upper bound for (\\ref{gsintro}) is of the order $q^{-1\/k}$ where $k=\\deg(h)$, but the elaborate double iteration method developed in \\cite{PSS}, and the simplified improvement developed in \\cite{BloomMaynard}, which lead to upper bounds like (\\ref{PSSb}) and (\\ref{BMnew}), require decay at or near $q^{-1\/2}$, which we refer to as \\textit{square-root cancellation}. Inspired by \\cite{BPPS}, the second author \\cite{ricemax} eliminated this discrepancy for $k> 2$ in the general case by employing a polynomial-specific sieve to the set of considered inputs that, roughly speaking, reduced the issue to estimating (\\ref{gsintro}) at prime moduli, for which the desired square-root cancellation is a well-known result of Weil. This sieve technique can be thought of as a bridge from the integer setting to the best available exponential sum estimates over finite fields.\n \nRuzsa and Sanders \\cite{Ruz}, and later Wang \\cite{wang}, were able to adapt the more traditional density increment method to establish (\\ref{rsb}), which is a stronger type of upper bound as compared with (\\ref{PSSb}) or (\\ref{BMnew}),  based on two key factors: the high degree of decay in the relevant exponential averages, which are modifications of \\begin{equation*} \\phi(q)^{-1}\\sum_{\\substack{s=0 \\\\(s,q)=1}}^{q-1} e^{2\\pi i s\/q}=\\frac{\\mu(q)}{\\phi(q)}, \\end{equation*} and the careful analysis of the distribution of primes in arithmetic progressions, including the consideration of exceptional zeros of Dirichlet $L$-functions. In the polynomial setting, the distribution of inputs in arithmetic progressions is not as delicate of an issue, though it does rear its head when employing a sieve, but this level of local decay is out of reach with a single variable. Specifically, bounds like (\\ref{2vb}) from the density increment require decay at or near $q^{-1}$ (more specifically, $q^{-1}$ times a function of average value at most polylogarithmic in $q$, and the exponent $\\mu$ depends on the power of the logarithm), which we refer to as \\textit{q-cancellation}. \n\nWhile the image of a multivariate intersective polynomial does not necessarily contain the image of a univariate intersective polynomial, it is the case that, by only exploiting cancellation in one variable, the methods of \\cite{ricemax} and \\cite{BloomMaynard} can be adapted to show that (\\ref{BMnew}) holds for such an image, so upper bounds in the multivariate setting are only novel if they are stronger than (\\ref{BMnew}). The observation made in \\cite{ricemax} to establish (\\ref{splitb}) was a rather simple one: if we consider differences of the form $g(m)+h(n)$, then the relevant exponential sum factors into a product, our sieve gives square-root cancellation in each variable, and these combine to give $q$-cancellation. However, this observation does not fully generalize to the case of a single polynomial in several variables with nonzero cross-terms. In particular, simple examples like $h(x,y)=(x+y)^2$ make it clear that one cannot always exploit cancellation in each variable, so some sort of nonsingularity assumption is required. \n\nIn the setting of binary quadratic forms, the natural assumption is nonzero discriminant, and since sieving is not required to get square-root cancellation from each variable in degree $2$, the adaptation of the usual density increment is relatively straightforward, as done in \\cite{Ricebin} to establish (\\ref{binb}). Section 2 of \\cite{Ricebin} provides a helpful description of the density increment method in a simpler, sieve-free context. \n\nFor higher degrees, the sieve technique can indeed be adapted to the multivariate setting, which leads us toward the best available estimates on exponential sums for multivariate polynomials over finite fields, due to Deligne \\cite{Deligne} in his proof of the Weil conjectures, and their associated nonsingularity assumptions. Recall that $\\bbA^n$ and $\\bbP^n$ denote $n$-dimensional affine and projective space, respectively.\n\n\\begin{definition} Suppose $F$ is a field, $\\ell \\in \\N$, and $g\\in F[x_1,\\dots,x_{\\l}]$ is a homogeneous polynomial. We say that $g$ is \\textit{smooth} if the vanishing of $g$ defines a smooth hypersurface in $\\mathbb{P}^{\\ell-1}$ (as opposed to $\\bbA^{\\l}$). In other words, $g$ is smooth if the system $ g(\\bsx)=\\frac{\\partial g}{\\partial x_1}(\\bsx)=\\cdots=\\frac{\\partial g}{\\partial x_{\\l}}(\\bsx)=0 $ has no solution besides $x_1=\\cdots=x_{\\l}=0$ in $\\bar{F}^{\\l}$, where the bar indicates the algebraic closure. For a general polynomial $h\\in F[x_1,\\dots,x_{\\ell}]$ with  $h=\\sum_{i=0}^k h^i$, where $h^i$ is homogeneous of degree $i$ and $h^k\\neq 0$, we say that $h$ is \\textit{Deligne} if the characteristic of $F$ does not divide $k$ and $h^k$ is smooth.\n\\end{definition} \n\n\\begin{remn} For the remainder of the paper, we take the notational convention that, for a polynomial $h$, $h^i$ denotes the degree-$i$ homogeneous part of $h$, as opposed to $h$ raised to the $i$-th power.\n\n\\end{remn} \n\n\\begin{theorem}[Theorem 8.4, \\cite{Deligne}] \\label{delmain} Suppose $\\l \\in \\N$ and $p\\in \\P$. If $h\\in \\mathbb{F}_p[x_1,\\dots,x_{\\l}]$ is Deligne, then \\begin{equation*} \\left|\\sum_{\\bsx\\in \\mathbb{F}_p^{\\l}} e^{2\\pi i h(\\bsx)\/p} \\right| \\leq (\\deg(h)-1)^{\\l} p^{\\l\/2}. \\end{equation*}\n\n\\end{theorem}  \n\nThis estimate provides a guide, but additional consideration is required to develop sufficient conditions on a multivariate polynomial for an application of Theorem \\ref{delmain} that is compatible enough with the density increment procedure to establish an upper bound like (\\ref{2vb}). We explore these details in Section \\ref{maindr}. \n\n\\subsection{Lower bounds and a special case}\\label{lbsec} In all the nontrivial cases we have explored, there is a large gap in the best-known upper and lower bounds for $D(X,N)$. For an intersective polynomial $h\\in \\Z[x]$, all known lower bounds with $X=h(\\Z)$ are of order $N^c$ for some $c<1$. The greedy algorithm gives $c=1-1\/\\deg(h)$, and higher values of $c$ are known for monomials (see \\cite{Ruz2} and \\cite{Lewko}) and certain other polynomials divisible by $x^2$ (due to Younis \\cite{younis}, and explored from an algebraic number theory perspective by Wessel \\cite{wessel}). For $X=\\mathcal{P}-1$, the gap is even larger, and the best-known lower bound is of the form $N^{o(1)}$ (see \\cite{Ruz3}). Younis \\cite{younis} established lower bounds for certain homogeneous multivariate polynomials, including $D(S+S,N)\\gg \\sqrt{N}$, where $S$ is the set of squares.  All of these results are descended from methods of Ruzsa that transfer examples from the modular setting to the integer setting. In the absence of stronger lower bounds, the greedy algorithm gives $D(X,N)\\geq (N-1)\/(|X\\cap[-N,N]|+1)$ for any set $X\\subseteq \\Z$ (see \\cite{Lyall}). \n\nAs an aside, one very special case where stronger upper bounds on $D(X,N)$ are available, and where the upper and lower bounds can be relatively close, is the case when $X$ is itself, or at least contains, a difference set. Specifically, if $Y\\subseteq \\{1,\\dots, N\\}$ and $X=Y-Y$, then for a set $A\\subseteq\\{1,\\dots,N\\}$ satisfying $(A-A)\\cap X \\subseteq \\{0\\}$, we have $a+y\\neq a'+y'$ for all $a,a'\\in A$ and $y,y'\\in Y$ with $(a,y)\\neq (a',y')$. In particular, the map $(a,y)\\mapsto a+y$ into $\\{1,\\dots,2N\\}$ is an injection, so $|A||Y|\\leq 2N$, and hence $D(X,N)\\leq 2N\/|Y|$, while the greedy algorithm gives $D(X,N)\\gg N\/|X| \\geq N\/|Y|^2$. For an example relating to our discussion of multivariate polynomials, if $X$ is the set of differences of $k$-th powers for a fixed $k\\in \\N$, then we have $D(X,N)\\ll N^{1-1\/k}$, but this observation does not immediately generalize beyond the case where $X\\supseteq Y-Y$. \n\n\\section{Main definitions and results} \\label{maindr}\n\nThe density increment procedure takes as input a set $A\\subseteq \\{1,2,\\dots,N\\}$ lacking nonzero differences in the image of a polynomial $h$, and produces a new, denser subset of a slightly smaller interval lacking nonzero differences in the image of a potentially modified polynomial. The following definition keeps track of the changes in the polynomial over the course of the iteration.\n\n\\begin{definition} Fix $\\l\\in \\N$. As in the univariate setting, we say that $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is \\textit{intersective} if $h(\\Z^{\\l})$ contains a nonzero multiple of every $q\\in \\N$. Equivalently, $h$ is intersective if it is not identically zero and has a root in $\\Z_p^{\\l}$ for every prime $p$, where $\\Z_p$ denotes the $p$-adic integers.\n\\end{definition}\n\n\\noindent Suppose $h\\in \\Z[x_1,\\dots,x_{\\ell}]$ is an intersective polynomial and fix, for each prime $p$, $\\bsz_p\\in \\Z_p^{\\ell}$ with $h(\\bsz_p)=0$. All objects defined below certainly depend on this choice of $p$-adic integer roots, but we suppress that dependence in the subsequent notation. \n\n\\noindent By reducing modulo prime powers and applying the Chinese Remainder Theorem, the choice of roots determines, for each $d\\in \\N$, a unique $\\bsr_d \\in (-d,0]^{\\ell}$ with $\\bsr_d \\equiv \\bsz_p \\ \\text{mod }p^j$ for all prime powers $p^j\\mid d$.\n\n\\noindent We define a completely multiplicative function $\\lambda$ (depending on $h$ and $\\{\\bsz_p\\}$) on $\\N$ by letting $\\lambda(p)=p^{m_p}$ for each prime $p$, where $m_p$ is the multiplicity of $\\bsz_p$ as a root of $h$, that is, $$m_p=\\min\\left\\{i_1+\\cdots+i_{\\l} : \\frac{\\partial^{i_1+\\cdots+i_{\\l}}h}{\\partial x_1^{i_1}\\cdots \\partial x_{\\l}^{i_{\\l}}} (\\bsz_p) \\neq 0\\right\\}. $$\nRoughly speaking, $\\lambda(d)$ is the largest guaranteed factor of $h(\\bsn)$ for $\\bsn\\equiv \\bsr_d \\ (\\text{mod }d)$.\n\n\\begin{definition}\nWith notation as described above, for each $d\\in \\N$ we define the \\textit{auxiliary polynomial} $h_d\\in \\Z[x_1,\\dots,x_{\\l}]$ by $$h_d(\\bsx)=h(\\bsr_d+d\\bsx)\/\\lambda(d). $$\n\n\\end{definition}\n\nCombining the hypotheses of Theorem \\ref{delmain} with the technical details of the density increment iteration, the following definition captures a sufficient condition for the success of the method. \n\n\\begin{definition}\\label{defn:smoothDeligne} When considering polynomials with integer coefficients, we use the terms \\textit{smooth} and \\textit{Deligne} as previously defined by embedding the coefficients in the field of rational numbers. In particular, $h\\in \\Z[x_1,\\dots,x_{\\l}]$ of degree $k\\geq 1$ is Deligne if the system $ h^k(\\bsx)=\\frac{\\partial h^k}{\\partial x_1}(\\bsx)=\\cdots=\\frac{\\partial h^k}{\\partial x_{\\l}}(\\bsx)=0 $ has no solution besides $x_1=\\cdots=x_{\\l}=0$ in $\\bar{\\Q}^{\\l}$. In this case, there exists a finite set of primes $X=X(h)$ such that the reduction of $h$ modulo $p$ is Deligne for all $p\\notin X$: Indeed, one can take $X(h)$ to be the set of primes dividing the Macaulay resultant $\\Res\\left(h^k, \\frac{\\partial h^k}{\\partial x_1}, \\cdots, \\frac{\\partial h^k}{\\partial x_\\ell}\\right)$, which is nonzero precisely when $h$ is Deligne. (See also Prop. A.9.1.6 of \\cite{HindrySilverman}.)\n\n\\noindent Further, we say that $h$ is \\textit{strongly Deligne} if there exists a finite set of primes $X=X(h)$ and a choice $\\{\\bsz_p\\}_{p\\in \\P}$ of $p$-adic integer roots of $h$ such that the reduction of $h_d$ modulo $p$ is Deligne for all $p\\notin X$ and all $d\\in \\N$. We note that strongly Deligne polynomials are necessarily both Deligne and intersective.\n\n\\noindent To highlight some of the subtlety of this definition, we first note that  $h_d^k=\\frac{d^k}{\\lambda(d)}h^k$, so for a prime $p\\nmid d$, we have that if $h$ is Deligne modulo $p$, then $h_d$ is Deligne modulo $p$. However, complications arise when $p\\mid d$, because $h_d^i$ has a factor of $d^i\/\\lambda(d)$, and hence vanishes modulo $p$ for all $i>m_p$. For an example of a polynomial that is Deligne and intersective but not strongly Deligne, see ``the ugly''  in Section \\ref{summsec}.\n\n\n\\end{definition}\n\n\n \nFor $k,\\l\\geq 2$, we let $\\mu(k,\\l)=\\begin{cases} [(k-1)^2+1]^{-1} & \\text{if }\\l=2 \\\\ 1\/2 &\\text{if }\\l \\geq 3  \\end{cases}. $ The central result of this paper is the following:\n  \n\\begin{theorem} \\label{more} If $\\ell \\geq 2$  and $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is a strongly Deligne polynomial of degree $k\\geq 2$, then \\begin{equation}\\label{lvb1} D(h(\\Z^{\\l}),N) \\ll_h N e^{-c(\\log N)^{\\mu(k,\\l)}}, \\end{equation} where $c=c(h)>0$.\n\n\\end{theorem}  \n\n\\begin{rem} In Theorem \\ref{more}, the full image $h(\\Z^{\\l})$ is considered for ease of exposition, and to make the conclusion invariant under input translation. However, by inspection of the proof, the same upper bound can be seen to hold for $D(h([1,N^{\\epsilon}]^{\\l}),N)$ for any $\\epsilon>0$, with $c$ and the implied constant depending on $\\epsilon$. Also, in several of our results and definitions, we exclude the case $k=1$ only out of convenience due to its triviality in this context. Specifically, if $h\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(h)=1$, then $D(h(\\Z^{\\l}),N)\\ll_h 1$ if $0\\in h(\\Z^{\\l})$ and $D(h(\\Z^{\\l}),N)\\gg_h N$ otherwise.\n\\end{rem}\n\n\n \n\n\nAfter setting the stage with preliminary definitions and observations in Section \\ref{prelimsec}, we prove Theorem \\ref{more} in Section \\ref{itproof}, and then establish the needed exponential sum estimates, which we state separately as Theorem \\ref{standalonethm}, in Section \\ref{expest}. More imminently, in Sections~\\ref{sec:integer_root} and \\ref{sec:deligne_case}, we describe sufficient conditions under which $h \\in \\Z[x_1,\\ldots,x_\\ell]$ is strongly Deligne, and hence (\\ref{lvb1}) holds. Then, in Section~\\ref{sec:singular}, we explain that in many cases we may still get a bound similar to (\\ref{lvb1}) even when the strongly Deligne condition is significantly relaxed.\n\\subsection{The integer root case}\\label{sec:integer_root} The simplest sufficient condition for the intersectivity of a nonzero polynomial is the existence of an integer root. In this case, all $p$-adic integer roots can be taken to equal said integer root, which simplifies the auxiliary polynomial definition, giving rise to a pleasantly tangible sufficient condition for the strongly Deligne property, as captured with the following proposition.\n\n\\begin{proposition}\\label{prop:integer_root} Suppose $\\ell \\geq 2$ and $h\\in \\Z[x_1,\\dots,x_{\\l}]$ with $h(\\bszero)=0$. If  the highest and lowest degree homogeneous parts of $h$ are smooth, then $h$ is strongly Deligne. \n\\end{proposition}\n\n\\begin{proof} Suppose $h$ satisfies the hypotheses, let $k=\\deg(h)$, let $j$ denote the lowest degree of the nonzero terms of $h$, and let $X$ denote the finite set of primes $p$ such that $p\\mid jk$ or either $h^k$ or $h^j$ is not smooth modulo $p$. Making the natural choice of $p$-adic integer roots $\\bsz_p=\\bszero$ for all $p$, we then have $h_d(\\bsx)=h(d\\bsx)\/d^j$, hence $h^i_d(\\bsx)=d^{i-j}h^i(\\bsx)$. Fix $p\\notin X$. If $p\\nmid d$, then the highest degree part of $h_d$ modulo $p$ is a nonzero multiple of $h^k$, which is smooth modulo $p$, hence $h_d$ is Deligne modulo $p$. If $p\\mid d$, then the only nonvanishing homogeneous part of $h_d$ is precisely $h^j$, which is smooth modulo $p$, hence $h_d$ is Deligne modulo $p$.\\end{proof}\n \n\\begin{rem}\nWe note that $h(\\Z^{\\l})$, hence the threshold $D(h(\\Z^{\\l}),N)$, as well as the Deligne and strongly Deligne properties, are all invariant under translations of the form $h(\\bsx+\\bsn)$ for a fixed $\\bsn\\in \\Z^{\\l}$. In particular, Proposition \\ref{prop:integer_root} applies provided there exists $\\bsn\\in \\Z^{\\l}$ such that $h(\\bsn)=0$ and the highest and lowest degree parts of $h(\\bsx+\\bsn)$ are smooth. More generally, all of our results that hold for a polynomial $h$ also hold for the full translation equivalence class of $h$.\n\\end{rem}\n \nFor homogeneous bivariate polynomials, smoothness of the corresponding ($0$-dimensional) variety is equivalent to non-vanishing of the discriminant. Therefore, for $\\ell = 2$, we have the following, which in particular combines with Theorem \\ref{more} to yield Theorem \\ref{introthm} as a special case.\n\n\\begin{corollary}\\label{2vcor} Suppose $h\\in \\Z[x,y]$ with $h(0,0)=0$. If  the highest and lowest degree homogeneous parts of $h$ have nonzero homogeneous discriminant, then $h$ is strongly Deligne.\n\n\\end{corollary}\n\n\\subsection{The Deligne case}\\label{sec:deligne_case}\n\nTaking the next step in complexity, here we consider the case of a polynomial that is Deligne and intersective, but may not have an integer root. Recalling that if $p\\mid d$, then $h^i_d$ vanishes modulo $p$ for all $i>m_p$, we make the following definition with the hopes of exploiting the fact that a nonzero homogeneous linear polynomial is guaranteed to be smooth.\n\n\\begin{definition} For $\\ell \\in \\N$ and $h\\in \\Z[x_1,\\ldots,x_{\\l}]$ we say that $h$ is \\textit{smoothly intersective} if there exists a choice  $\\{\\bsz_p\\}_{p\\in \\mathcal{P}}$ of $p$-adic integer roots of $h$ such that $m_p=1$ for all but finitely many $p$. In other words, the variety defined by $h=0$ has at least one point over $\\Z_p$ for all $p$, and at least one nonsingular point over $\\Z_p$ for all but finitely many $p$.\n\\end{definition}\n\nFor low-hanging examples of polynomials that are intersective but not smoothly intersective, one could consider the square of any intersective polynomial, but such polynomials do not pass even our coarsest of nonsingularity filters.  For an example of a polynomial that is intersective but not smoothly intersective in a more subtle and problematic way, see our discussion of ``the ugly'' in Section \\ref{summsec}. Combining the motivation for the smoothly intersective definition with the fact that the highest degree part of a Deligne polynomial is assumed to be smooth, the following proposition provides a sufficient condition for the strongly Deligne property, and includes two notable special cases.\n\n\\begin{proposition}\\label{thm:main} Suppose $\\ell \\ge 2$ and $h \\in \\Z[x_1,\\ldots,x_{\\l}]$ is Deligne and intersective with $\\deg(h)=k\\geq 2$. If there exists a choice $\\{\\bsz_p\\}_{p\\in \\P}$ of $p$-adic integer roots of $h$ satisfying $m_p\\in \\{1,k\\}$ for all but finitely many $p$, then $h$ is strongly Deligne. In particular, if $k=2$ or $h$ is smoothly intersective, then $h$ is strongly Deligne.  \n \n\\end{proposition} \n\n\n\n\nUsing estimates on the number of  nonsingular points on irreducible varieties over finite fields, we obtain the following convenient criterion for smooth intersectivity.\n\n\\begin{proposition}\\label{prop:main}\nSuppose $\\ell \\ge 2$ and $h \\in \\Z[x_1,\\ldots,x_{\\ell}]$ is Deligne and intersective, and let $h=g_1\\cdots g_n$ be an irreducible factorization of $h$ in $\\Z[x_1,\\dots,x_{\\l}]$. If $g_i$ is geometrically irreducible for some $1\\leq i \\leq n$, then $h$ is smoothly intersective, hence strongly Deligne.\n\\end{proposition}\n\n\\begin{rem} \nThe conclusion of Proposition~\\ref{prop:main} remains true under weaker assumptions on the factorization of $h$. We give this cleaner statement here, but prove the more general statement in Corollary~\\ref{prop:main_stronger}.\n\\end{rem}\n\n\n\n\n\n\nFor $\\ell \\ge 3$, the Deligne condition actually implies geometric irreducibility, yielding the following:\n\n\\begin{corollary}\\label{3var} Suppose $\\l \\geq 3$ and $h\\in \\Z[x_1,\\dots,x_{\\l}]$. If $h$ is Deligne and intersective, then $h$ is smoothly intersective, hence strongly Deligne.\n\n\\end{corollary} \n\n\\begin{proof}\nBy Proposition~\\ref{prop:main}, it suffices to show that if $h$ is a Deligne polynomial in $\\ell \\ge 3$ variables, then $h$ is geometrically irreducible. Suppose to the contrary that $h = g_1g_2$ with $g_1,g_2 \\in \\QQbar[x_1,\\ldots,x_\\ell]$ nonconstant of degrees $d$ and $k - d$, respectively. In particular, we have $h^k = g_1^dg_2^{k-d}$. Each of $\\{g_1^d = 0\\}$ and $\\{g_2^{k-d} = 0\\}$ has codimension $1$ in $\\bbP^{\\ell -1}$ (since they are hypersurfaces) and dimension at least $1$ (since we assumed $\\ell \\ge 3$). In particular, $\\{g_1^d = 0\\}$ and $\\{g_2^{k-d} = 0\\}$ have nontrivial intersection, and any intersection point must be a singular point of the union $\\{h^k = 0\\}$, contradicting the fact that $h$ is Deligne.\n\\end{proof}\n\nIn Section \\ref{AG1}, we collect some crucial tools from algebraic geometry, which are followed by the proofs of both Proposition \\ref{thm:main} and the aforementioned generalization of Proposition \\ref{prop:main}.\n\n \n\\subsection{The singular case}\\label{sec:singular}\n\nWhile the Deligne condition is required to apply Theorem \\ref{delmain} to get the desired cancellation in our exponential sums, brief consideration reveals that the condition is not strictly necessary for a bound like (\\ref{lvb1}) to hold, provided the failure of the Deligne condition is in balance with the freedom of extra variables.  For a particularly simple example, consider $h(x,y,z)=(x+z)^4+(x+z)y^3+y^4.$ This is a homogeneous degree-$4$ polynomial, and the variety $\\widehat{V}\\subseteq \\bbP^2$ defined by its vanishing has a unique singular point, namely $(1:0:-1)$. In particular, $h$ is not Deligne. However, by fixing $z=0$, we can define $g(x,y)=h(x,y,0)=x^4+xy^3+y^4$, which is a bivariate homogeneous polynomial of nonzero discriminant. In particular, $g$ is strongly Deligne, so Theorem \\ref{more} applies, and moreover $g(\\Z^2)=h(\\Z^3)$, so (\\ref{lvb1}) holds for $h$ as well, applied as if $\\l=2$ as opposed to $\\l=3$.\n\n\nThis example hints at a less black-and-white consideration of the singularity of a projective variety. For $h\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(h)=k\\geq 1$, $h$ is Deligne precisely when the variety $\\widehat{V}\\subseteq \\bbP^{\\l-1}$ defined by $h^k=0$ is nonsingular. The example above indicates that we should really only need to avoid this variety being ``too singular'', which leads to the following definition.\n\n\n\\begin{definition} For $\\ell \\in \\N$ and a nonconstant homogeneous polynomial $g\\in \\Z[x_1,\\dots,x_{\\ell}]$, let $\\widehat{V}\\subseteq \\mathbb{P}^{\\ell-1}$ be the variety defined by $g=0$, and let $\\widehat{V}^\\sing$ be the singular locus of $\\widehat{V}$. We define the \\textit{rank} of $g$ to be the codimension of $\\widehat{V}^\\sing$ in $\\mathbb{P}^{\\ell-1}$, with the convention that the empty set has dimension $-1$, hence the codimension of the empty set in $\\bbP^{\\l-1}$ is $\\l$. This is a notion of rank developed by Birch in \\cite{birch} and utilized, for example, in \\cite{cookmagyar}.\n\n\\end{definition}\n\nFor $h\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(h)=k\\geq 1$, the rank of $h^k$ should, roughly speaking, encode the number of variables $r$ such that $g(\\Z^r)\\subseteq h(\\Z^{\\l})$ for some Deligne polynomial $g\\in \\Z[x_1,\\dots,x_{r}]$. In particular, $h$ is Deligne if and only if the rank of $h^k$ is $\\l$. In Section \\ref{dimlowsec}, using careful dimension-lowering arguments, we successfully expand the class of polynomials for which a result analogous to Theorem \\ref{more} holds, generalizing our efforts from Sections \\ref{sec:integer_root} and \\ref{sec:deligne_case} as follows.\n \n\n\n\\begin{theorem} \\label{dimlowrootthm} Suppose $\\ell \\geq 2$ and $h \\in \\Z[x_1,\\dots,x_{\\ell}]$ with $h(\\bszero)=0$ and $\\deg(h)=k\\geq 2$. Let $r$ be the minimum rank of the highest and lowest degree homogeneous parts of $h$. If $r\\geq 2$, then \\begin{equation}\\label{lvb} D(h(\\Z^{\\ell}),N) \\ll_h N e^{-c(\\log N)^{\\mu(k,r)}},\\end{equation} where $c=c(h)>0$.\n\\end{theorem}\n\n\\begin{theorem} \\label{dimlowthm} Suppose $\\ell \\geq 2$ and $h \\in \\Z[x_1,\\dots,x_{\\ell}]$ is intersective of degree $k\\geq 2$. Let $r$ be the rank of $h^k$. If $r\\geq 3$, OR if $r=2$ and there exists a choice $\\{\\bsz_p\\}_{p\\in \\P}$ of $p$-adic integer roots of $h$ satisfying $m_p\\in \\{1,k\\}$ for all but finitely many $p$, then (\\ref{lvb}) holds. \n\\end{theorem}\n \n \n \n\\begin{rem} To shed light on the hypotheses of Theorems \\ref{dimlowrootthm} and \\ref{dimlowthm}, we note that, for $\\ell \\geq 2$ and a nonconstant homogeneous polynomial $g\\in \\Z[x_1,\\dots,x_{\\l}]$ of rank $r$, we have $r\\geq 2$ if and only if $g$ is squarefree---in other words, if and only if $g = 0$ defines a reduced variety.\n\\end{rem}\n\n\n\\subsection{Summary of results} \\label{summsec} For this section, we suppose $k,\\l\\geq 2$ and $h\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(h)=k$, and we let $r$ denote the rank of $h^k$. We assume $h$ is  intersective, as otherwise $D(h(\\Z^{\\l}),N)\\gg_h N$. The following bullet points summarize the reach and limitations of our results.\n\n\\begin{itemize} \\item \\textbf{The good:} In addition to previously known results on sums of univariate intersective polynomials (Theorems 1.2 and 5.7 of \\cite{ricemax}), we now have that (\\ref{lvb}) holds provided $h$, or in the case of (\\ref{rootitem}) any translation of $h$, meets any of the following criteria: \n\n \\begin{enumerate}[(i)] \\item $r\\geq 3$ (including Deligne with $\\l \\geq 3$)  \\item \\label{quaditem} $r=k=2$ (including Deligne with $\\l=k=2$)  \\item \\label{rootitem} $r=2$ (including Deligne with $\\ell=2$), $h(\\bszero)=0$, and the lowest degree homogeneous part of $h$ has rank at least $2$. This includes as a special case bivariate homogeneous polynomials with nonzero discriminant, which is Theorem \\ref{introthm} from the introduction.   \\item \\label{smoothitem} $r=2$ (including Deligne with $\\ell=2$) and $h$ is smoothly intersective, the latter of which in particular holds if any irreducible (over $\\Z$) factor of $h$ is geometrically irreducible. Parts of this item can be made slightly more general, as seen in the hypotheses of Proposition \\ref{prop:main} and Corollary \\ref{prop:main_stronger}.  \n\n\\noindent An interesting example of (\\ref{smoothitem}) that does not fit into any other category is $h(x,y)=x^3+y^3-q$, where $q$ is a prime congruent to $1$ modulo $90090$ that is not expressable as the sum of two integer cubes, of which there are plenty. This polynomial has no rational root, and it cannot be decomposed into a sum of two univariate intersective polynomials, but it is Deligne and it has simple roots in $\\Z_p^2$ for all primes $p$. This example was discussed in a remark following Theorem 1.2 in \\cite{ricemax} to illustrate a limitation of that result. \\end{enumerate}\n\n\n\n\\item \\textbf{The bad:} The methods utilized here fail to improve on univariate results in the case that $r=1$, or equivalently the case that $h^k$ has a repeated factor. It should be noted that we can only definitively say that it is impossible to reach beyond the cutting edge of the univariate setting if $h=f\\circ g$ for some $g\\in \\Z[x_1,\\dots,x_{\\l}]$ and $f\\in \\Z[x]$ with $\\deg(f)\\geq 2$, because in this case $h(\\Z^{\\l})\\subseteq f(\\Z)$. This was hinted at in the introduction with the example $h(x,y)=(x+y)^2$. In this situation, $h^k$ is a proper power of the highest-degree part of $g$, so we definitely have $r=1$. While it is certainly possible to have $r=1$ without $h$ being given as a composition of this sort, our current methods cannot distinguish between the two.\n\n\\item \\textbf{The ugly:} A more subtle remaining hurdle is the case where $r=2$ (including Deligne with $\\l=2$), $k\\geq 3$, and $h$ does not meet either of the criteria described in items  (\\ref{rootitem}) or (\\ref{smoothitem}). Focusing on the $\\l=2$ Deligne case, such a polynomial must satisfy $\\Delta(h^k)\\neq 0$, must be intersective and hence have roots in $\\Z_p^{2}$ for all primes $p$, but by Proposition \\ref{thm:main}, for infinitely many $p$, all roots in $\\Z_p^{2}$ must satisfy $2\\leq m_p\\leq k-1$. In particular, by Proposition \\ref{prop:main}, at least one coefficient in every geometrically irreducible factor of $h$ must fail to be an integer. Finally, by Corollary \\ref{2vcor}, if $h$ satisfies $h(0,0)=0$, then the lowest degree part of $h$ must have discriminant $0$.\n\n\\noindent One example is   $$ h(x,y) = x^4 - 2y^4 + 2x^2(x + y) + (x + y)^2 = \\left(x^2 - \\sqrt{2}y^2 + (x + y)\\right)\\left(x^2 + \\sqrt{2}y^2 + (x + y)\\right).$$ \nFor any  prime $p$ such that $2$ is not a square in $\\Q_p$, the only $\\Q_p$ roots of $h$ are $(0,0)$ and $(-1,0)$. With these choices for $\\bsz_p$, the highest degree nonvanishing part of $h_p$ modulo $p$ is either $(x+y)^2$ or $(x - y)^2$, respectively. In both cases $\\Delta(h_p^2)=0$, and hence $h_p$ is not Deligne at this infinite collection of primes. In other words, $h$ is not strongly Deligne, and we cannot claim that (\\ref{lvb}) holds.\n \n\\item \\textbf{The future:} The issue in the previous bullet point may represent an avoidable artifact of the method, in which case the upper bound (\\ref{lvb}) could be shown to hold for all intersective polynomials satisfying $r\\geq 2$. Regarding improved bounds, as noted in Section 2.3 of \\cite{Ricebin}, and as implicitly referenced in \\cite{Ruz} when noting that the exponent $\\mu$ in (\\ref{rsb}) could be increased to $1\/2$ conditioned on the Generalized Riemann Hypothesis, an upper bound of order $Ne^{-c\\sqrt{\\log N}}$ appears to be the limit of a Fourier analytic $L^2$ density increment. More specifically, if $(\\delta,N)\\mapsto (\\delta',N')$ represents the change in density and interval size at each step of the iteration, then any further improvement would require either $N'\/N$ to decay more slowly than any power of $\\delta$, or $\\delta'\/\\delta$ to tend to infinity, as $\\delta\\to 0$, both of which appear incompatible with the method.  To be clear, this is not at all to say that much stronger upper bounds do not hold, even in the univariate polynomial setting. As discussed in Section \\ref{lbsec}, this question is rather murky. However, to achieve such a goal would likely require a fundamentally different proof strategy. \n\\end{itemize}\n\n\n\n\n\n\n\n\n\\section{Preliminaries} \\label{prelimsec}\n\nIn this section we make some preliminary definitions and observations required to execute the sieve-powered $L^2$ density increment strategy utilized to prove Theorem \\ref{more}.\n\n\\subsection{Fourier analysis and the circle method on $\\Z$} We embed our finite sets in $\\Z$, on which we utilize an unnormalized discrete Fourier transform. Specifically, for a function $F: \\Z \\to \\C$ with finite support, we define $\\widehat{F}: \\T \\to \\C$, where $\\T$ denotes the circle  parameterized by the interval $[0,1]$ with $0$ and $1$ identified, by \\begin{equation*} \\widehat{F}(\\alpha) = \\sum_{x \\in \\Z} F(x)e^{-2 \\pi ix\\alpha}. \\end{equation*}\n\\noindent Given $N\\in \\N$ and a set $A\\subseteq [1,N]$ with $|A|=\\delta N$, we examine the Fourier analytic behavior of $A$ by considering the \\textit{balanced function}, $f_A$, defined by\n$ f_A=1_A-\\delta 1_{[1,N]}.$\n\n\\noindent As is standard, we decompose the frequency space into two pieces: the points of $\\T$ that are close to rational numbers with small denominator, and the complement.\n\n\\begin{definition}Given $\\gamma>0$ and $Q\\geq 1$, we define, for each $a,q\\in \\N$ with $0\\leq a \\leq q-1$,\n$$\\mathbf{M}_{a\/q}(\\gamma)=\\left\\{ \\alpha \\in \\T : \\Big|\\alpha-\\frac{a}{q}\\Big| < \\gamma \\right\\},  \\ \\mathbf{M}_q(\\gamma)=\\bigcup_{(a,q)=1} \\mathbf{M}_{a\/q}(\\gamma), \\text{ and }\\mathbf{M}'_q(\\gamma)=\\bigcup_{r\\mid q} \\mathbf{M}_r(\\gamma)=\\bigcup_{a=0}^{q-1} \\mathbf{M}_{a\/q}(\\gamma).$$  We then define the \\textit{major arcs} by\n$$ \\mathfrak{M}(\\gamma,Q)=\\bigcup_{q=1}^{Q} \\mathbf{M}_q(\\gamma),$$ \nand the \\textit{minor arcs} by  \n$\\mathfrak{m}(\\gamma,Q)=\\T\\setminus \\mathfrak{M}(\\gamma,Q).\n$ \nWe note that if $2\\gamma Q^2<1$, then \\begin{equation} \\label{majdisj}\\mathbf{M}_{a\/q}(\\gamma)\\cap\\mathbf{M}_{b\/r}(\\gamma)=\\emptyset \\end{equation}whenever  $a\/q\\neq b\/r$ and  $q,r \\leq Q$. \n\\end{definition} \n\n\\subsection{Inheritance proposition} As previously noted, we defined auxiliary polynomials to keep track of an inherited lack of prescribed differences at each step of a density increment iteration. The following proposition makes this inheritance precise.\n\n\\begin{proposition} \\label{inh} Suppose $\\l\\in \\N$, $h\\in\\Z[x_1,\\dots,x_{\\ell}]$ is intersective, $d,q\\in \\N$, and $A\\subseteq \\N$.\n\n\\noindent If $(A-A)\\cap h_d(\\Z^{\\ell})\\subseteq \\{0\\}$ and $A'\\subseteq \\{a: x+\\lambda(q)a \\in A\\}$ for some $x\\in \\Z$, then $(A'-A')\\cap h_{qd}(\\Z^{\\ell})\\subseteq \\{0\\}$.\n\\end{proposition}\n\\begin{proof} Suppose that $A\\subseteq \\N$, $A'\\subseteq \\{a : x+\\lambda(q)a \\in A\\}$, and $$a-a'=h_{qd}(\\bsn)=h(\\bsr_{qd}+qd\\bsn)\/\\lambda(qd)\\neq 0$$ for some $\\bsn\\in \\Z^{\\ell}$, $a, a' \\in A'$. By construction we have that $\\bsr_{qd}\\equiv \\bsr_d$ mod $d$, so there exists $\\boldsymbol{s}\\in \\Z^{\\ell}$ such that $\\bsr_{qd}=\\bsr_d+d\\boldsymbol{s}$. Further, $\\lambda$ is completely multiplicative, and therefore $$0\\neq h_d(\\boldsymbol{s}+q\\bsn)=h(\\bsr_d+d(\\boldsymbol{s}+q\\bsn))\/\\lambda(d)=\\lambda(q)h_{qd}(\\bsn)=\\lambda(q)a-\\lambda(q)a' \\in A-A.$$ Since $a-a'\\neq 0$, we have $(A-A)\\cap h_d(\\Z^{\\ell})\\not\\subseteq \\{0\\}$, and the contrapositive is established. \n\\end{proof} \n\n\\subsection{Sieve definitions and observations} \\label{sievesec} As in \\cite{ricemax}, we apply a polynomial-specific sieve to our set of considered inputs in order to, roughly speaking, reduce our analysis of local exponential averages to the case of prime moduli, which in the multivariate setting allows for the application of Theorem \\ref{delmain}. To this end, for $\\l\\in \\N$, an intersective polynomial $h\\in\\Z[x_1,\\dots,x_{\\ell}]$, and each prime $p$ and $d\\in \\N$, we define $\\gamma_{d}(p)$ to be the smallest power such that $\\grad h_d$ modulo $p^{\\gamma_{d}(p)}$ does not vanish identically {\\it as a function on $(\\Z\/p^{\\gamma_d(p)}\\Z)^{\\l}$}, and we let $j_d(p)$ denote the number of solutions to $\\grad h_d = \\bszero$ in $(\\Z\/p^{\\gamma_{d}(p)}\\Z)^{\\l}$. Then, for $d\\in \\N$ and $Y>0$ we define $$W_d(Y)=\\left\\{\\bsn\\in \\N^{\\l}: \\grad h_d(\\bsn) \\not\\equiv \\bszero \\text{ mod } p^{\\gamma_{d}(p)} \\text{ for all } p\\leq Y \\right\\}.$$\nIn the absence of a subscript $d$ in the usage of $\\gamma(p), j(p),$ and $W(Y)$, we assume $d=1$, in which case the definitions make sense even for non-intersective polynomials. Further, for any $g\\in \\Z[x_1,\\dots,x_{\\ell}]$ and $q\\in \\N$, we define $$W^{q}(Y)=\\left\\{\\bsn\\in \\N^{\\l}: \\grad g(\\bsn) \\not\\equiv \\bszero \\text{ mod }p^{\\gamma(p)} \\text{ for all } p\\leq Y, \\ p^{\\gamma(p)}\\mid q \\right\\}.$$ \nUnlike in the univariate case, the size of $W(Y)$ here can be estimated with a straightforward application of the inclusion-exclusion principle, as opposed to a Brun sieve truncation thereof (see Proposition 2.4 in \\cite{ricemax}). To achieve this goal, however, we must first look forward and invoke an estimate established in Section \\ref{gv2}. For the following two statements, we assume $\\ell \\geq 2$ and $g\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(g)=k\\geq 1$.\n\n\\begin{lemma} \\label{gradconst} If $p$ is prime and $g$ is Deligne modulo $p$, then $j(p)\\ll_{k,\\l} 1$.\n\n\\end{lemma} \n\n\\begin{proposition}\\label{brunprop} For any $x_1,\\dots,x_{\\l},Y>0$ we have \\begin{equation}\\label{sieve} \\left|B\\cap W(Y)\\right| = x_1x_2\\cdots x_{\\l}\\prod_{p\\leq Y} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right)+E, \\end{equation} where $B=[1,x_1]\\times\\cdots \\times [1,x_{\\l}]$, $$E=\\begin{cases} O(X^{\\l-1}\\log^C(Y)) & \\text{if }\\l=2 \\\\ O(X^{\\l-1})& \\text{if }\\l \\geq 3 \\end{cases}, $$ $X=\\max\\{x_1,\\dots,x_{\\l}\\}$, $C=C(k,\\l)$, and the implied constants depend only on $k$, $\\l$, the moduli at which $\\grad g$ identically vanishes, and the primes $p\\leq Y$ modulo which $g$ is not Deligne.\n \n\\end{proposition}\n\n\\begin{proof} Fix $x_1,\\dots,x_{\\l},Y>0$ and let $X=\\max\\{x_1,\\dots,x_{\\l}\\}$. For primes $p_1<p_2<\\cdots<p_s$, we let \\begin{equation*}\\mathcal{A}_{p_1\\cdots p_s}=\\mathcal{A}_{p_1\\cdots p_s}(x_1,\\dots,x_{\\l}) = \\left|\\left\\{\\bsn \\in B: \\grad g (\\bsn) \\equiv \\bszero\\text{ mod } p_i^{\\gamma(p_i)} \\ \\text{for all } 1\\leq i \\leq s  \\right\\}\\right|.\\end{equation*} \n \n\\noindent Fixing $Y>0$ and letting $r$ denote the number of primes that are at most $Y$, we have by the Chinese Remainder Theorem and the inclusion-exclusion principle that \\begin{equation}\\label{brunalt1} \\left|B \\cap  W(Y)\\right|= \\sum_{s=0}^r  (-1)^s \\sum_{p_1<\\dots < p_s\\leq Y} \\mathcal{A}_{p_1\\cdots p_s}.\\end{equation} Further,  \\begin{equation}\\label{AP1} \\mathcal{A}_p = \\frac{j(p)}{p^{\\gamma(p)\\ell}}x_1\\cdots x_{\\l} + R_p, \\end{equation} where $|R_p|\\ll_{\\l} j(p)(X\/p^{\\gamma(p)})^{\\ell-1}$.  We trivially have $j(p) \\leq p^{\\gamma(p)\\ell}$, while if $g$ is Deligne modulo $p$, then $\\gamma(p)=1$ and, by Lemma \\ref{gradconst}, $j(p)\\leq C=C(k,\\l)$. In particular, we can apply the Chinese Remainder Theorem again and extend (\\ref{AP1}) to \\begin{equation} \\label{AP2}  \\mathcal{A}_{p_1\\cdots p_s} = x_1\\cdots x_{\\l} \\prod_{i=1}^s\\frac{j(p_i)}{p_i^{\\gamma(p_i)\\ell}} + R_{p_1\\cdots p_s}, \\end{equation} where $|R_{p_1\\cdots p_s}| \\leq KC^s(X\/p_1\\cdots p_s)^{\\l-1}$, where $K$ depends only on the moduli at which $\\grad g$ identically vanishes and the primes $p\\leq Y$ modulo which $g$ is not Deligne.  Now, by (\\ref{brunalt1}) and (\\ref{AP2}) we have \n\\begin{align*}  \\left|B \\cap  W(Y)\\right|&= \\sum_{s=0}^r  (-1)^s \\sum_{p_1<\\dots < p_s\\leq Y} \\mathcal{A}_{p_1\\cdots p_s}\\\\ &=\\sum_{s=0}^r  (-1)^s \\sum_{p_1<\\dots < p_s\\leq Y}\\left(x_1\\cdots x_{\\l} \\prod_{i=1}^s\\frac{j(p_i)}{p_i^{\\gamma(p_i)\\ell}} + R_{p_1\\cdots p_s}\\right) \\\\ &= x_1x_2\\cdots x_{\\l}\\prod_{p\\leq Y} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right)+E, \\end{align*} where \\begin{align*}|E| &\\leq KX^{\\l-1}\\sum_{s=0}^r \\sum_{p_1<\\dots < p_s\\leq Y} \\frac{C^s}{(p_1\\cdots p_s)^{\\l-1}} = KX^{\\l-1}\\prod_{p\\leq Y}\\left(1+\\frac{C}{p^{\\l-1}}\\right),\n\t\\end{align*} and the estimate follows. \\end{proof} \n\n\\subsection{Control over gradient vanishing: Part I} A potential hazard of the density increment method is the possibility that, as $d$ grows, $\\grad h_d$ could identically vanish at a larger and larger collection of moduli. This section is dedicated to establishing that, for strongly Deligne polynomials, this does not occur. We begin by noting that the collection of moduli at which a polynomial identically vanishes is firmly controlled in terms of its degree and the gcd of its coefficients. Throughout this section we assume $k,\\l\\in \\N$.\n\n\\begin{definition} We define a \\textit{multi-index} to be an $\\ell$-tuple $\\bsi=(i_1,\\dots,i_{\\ell})$ of nonnegative integers. We let $|\\bsi|=i_1+\\dots+i_{\\ell}$, we let $\\bsi ! =i_1!\\cdots i_{\\l}!$, and for $\\bsx=(x_1,\\dots,x_{\\ell})$, we let $\\bsx^{\\bsi}=x_1^{i_1}\\cdots x_{\\ell}^{i_{\\ell}}$. Finally, for a polynomial $g(\\bsx)$, we let $\\partial^{\\bsi}g=\\frac{\\partial^{|\\bsi|} g}{\\partial x_1^{i_1}\\cdots \\partial x_{\\ell}^{i_{\\ell}}}$.\n\n\\end{definition}\n\n\\begin{proposition} \\label{idzero} If  $g(\\bsx)= \\sum_{|\\boldsymbol{i}|\\leq k}a_{\\boldsymbol{i}}\\bsx^{\\boldsymbol{i}}  \\in \\Z[x_1,\\dots,x_{\\l}]$ is identically zero modulo $q\\in \\N$, then $$q \\mid k!\\gcd(\\{a_{\\bsi}\\}). $$\n\\end{proposition} \n\n\\begin{proof} We first note that $g$ is identically zero as a function on $\\Z\/q\\Z$ if and only if the polynomial $g\/q$ is integer-valued. In this case, since products of binomial coefficients $$\\binom{\\bsx}{\\bsi}=\\binom{x_1}{i_1}\\cdots \\binom{x_{\\l}}{i_{\\l}}=\\frac{x(x-1)\\dots(x-i_1+1)}{i_1!} \\cdots \\frac{x(x-1)\\dots(x-i_{\\l}+1)}{i_{\\l}!}$$form a $\\Z$-basis for integer-valued polynomials in $\\Q[x_1,\\dots,x_{\\l}]$, we can write $g(x)=\\sum_{|\\bsi|\\leq k}qb_{\\bsi}\\binom{\\bsx}{\\bsi}$ for $b_{\\bsi} \\in \\Z$. In particular, by clearing denominators we see that the coefficients of $k!g$ are all divisible by $q$, and the proposition follows.\n\\end{proof}\n\n\\noindent Further, we note that the gcd of the coefficients of each partial derivative of a polynomial $h\\in \\Z[x_1,\\dots,x_{\\l}]$ divides $k!$ times the gcd of the nonconstant coefficients of $h$. With this in mind, the following definition and proposition complete the task at hand. \n\n\\begin{definition} For    $h(\\bsx)= \\sum_{|\\boldsymbol{i}|\\leq k}a_{\\boldsymbol{i}}\\bsx^{\\boldsymbol{i}}  \\in \\Z[x_1,\\dots,x_{\\l}]$, we define $$\\cont(h)=\\gcd(\\{a_{\\boldsymbol{i}}: |\\boldsymbol{i}|>0\\}).$$ \n\n\\noindent We note that our use of $\\cont(h)$ does not precisely align with the standard notion of the \\textit{content} of a polynomial, as we exclude the constant coefficient.  \n\n\n\\end{definition}\n\n\n\\begin{proposition}\\label{content} If $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is a strongly Deligne polynomial of degree $k$, then  $$\\cont(h_d) \\ll_h 1. $$\n\n\\end{proposition}\n \n\\begin{proof} Suppose  $d\\in \\N$ and $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is a strongly Deligne polynomial of degree $k$. Let $\\{\\bsz_p\\}_{p\\in \\P}$ and $X$ denote the choice of $p$-adic integer roots and the finite set of primes, respectively, guaranteed by the strongly Deligne condition. In particular, $h_d$ is Deligne modulo $p$ for all $p\\notin X$. Because constant polynomials are not Deligne, $\\cont(h_d)$ can only be divisible by primes in $X$.\n\n\\noindent Recalling that $h_d(\\bsx)=h(\\bsr_d+d\\bsx)\/\\lambda(d)$, we make the trivial note that for any multi-index $\\bsi$ with $|\\bsi|=k$, the $\\bsx^{\\bsi}$ coefficient of $h_d$ is precisely $d^k\/\\lambda(d)$ times the corresponding coefficient $a_{\\bsi}$ of $h$. In particular, \\begin{equation} \\label{dk} \\cont(h_d) \\mid \\frac{d^k}{\\lambda(d)}a_{\\bsi} \\ \\text{whenever} \\ |\\bsi|=k.\n\\end{equation}\n\n\\noindent Now fix $p\\in X$. By definition of the multiplicity $m_p$, there exists a multi-index $\\bsi$ with $|\\bsi|=m_p$ and $\\partial^{\\bsi}h(\\bsz_p)\\neq 0$, so in particular $\\partial^{\\bsi}h(\\bsz_p)$ has some finite $p$-adic valuation $v_1(p)$. \n\n\\noindent If $p^{v_1(p)+1}\\nmid d$, then by (\\ref{dk}), we have that $p^{kv_1(p)+v_2(p)+1}\\nmid \\cont(h_d)$, where $v_2(p)$ is the minimum $p$-adic valuation amongst the degree-$k$ coefficients of $h$. Now suppose that $p^{v_1(p)+1}\\mid d$. \n \n\\noindent Let $b_{\\bsi}$ denote the $\\bsx^{\\bsi}$ coefficient of $h_d$. By Taylor's formula, we have that $$b_{\\bsi}=\\frac{d^{m_p}}{\\lambda(d)}\\frac{\\partial^{\\bsi}h(\\bsr_{d})}{\\bsi !}.$$ By definition of $\\lambda$ we have $p\\nmid (d^{m_p}\/\\lambda(d))$, and since $\\bsr_d\\equiv \\bsz_p \\ \\text{mod }p^{v_1(p)+1}$ and $p^{v_1(p)+1}\\nmid\\partial^{\\bsi}h(\\bsz_{p})$, we have that $p^{v_1(p)+1}\\nmid b_{\\bsi}$. In either case, we have that $p^{kv_1(p)+v_2(p)+1}\\nmid \\cont(h_d)$, and hence $$\\cont(h_d)\\leq \\prod_{p\\in X}p^{kv_1(p)+v_2(p)+1} \\ll_h 1,  $$ as required.\\end{proof}\n\n\\noindent For strongly Deligne $h\\in\\Z[x_1,\\dots,x_{\\ell}]$ with $\\deg(h)=k$, we have now established control over not only the error term in the size of $W_d(Y)$, but also the main term, since Lemma \\ref{gradconst}, Proposition \\ref{idzero}, and Proposition \\ref{content}  give \\begin{equation}\\label{logk} \\prod_{p\\leq Y} \\left(1-\\frac{j_d(p)}{p^{\\gamma_d(p)\\ell}} \\right) \\gg_h \\prod_{C=C(h)\\leq p\\leq Y} \\left(1-\\frac{C}{p^{\\l}} \\right) \\gg_h 1\\end{equation} for all $d \\in \\N$ and $Y\\geq 2$. \n\n\\subsection{Summary of new exponential sum estimates} \\label{standalone} In Section \\ref{expest}, we combine new and old techniques to establish the sieved multivariate exponential sum estimates necessary to prove Theorem \\ref{more}. These estimates are obtained through a sequence of lemmas presented in the context of the larger proof, so we separately present a summary here in case the estimates are of independent interest to the reader.\n\nFor the following theorem, a multivariate generalization of Theorem 2.7 in \\cite{ricemax}, we utilize all the sieve-related notation and definitions from Section \\ref{sievesec}. Further, we use $\\tau$ and $\\omega$ to denote the divisor and distinct prime divisor counting functions, respectively, as well as $\\phi$ to denote the Euler totient function.\n\n\\begin{theorem}\\label{standalonethm} For $k,\\ell \\geq 2$,  $g(\\bsx)=\\sum_{|\\bsi|\\leq k} a_{\\bsi} \\bsx^{\\bsi} \\in \\Z[x_1,\\dots,x_{\\ell}]$, $J=\\sum_{|\\bsi|\\leq k} |a_{\\bsi}|$, and $a,q\\in \\N$, the following estimates hold:\n\n\\begin{enumerate}[(i)]  \\item \\label{majitem} \\textnormal{\\textbf{Major arc estimate:}} If $X,Y > 0$ and  $\\alpha=a\/q+\\beta$, then \\begin{align*}\\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)}e^{2\\pi i g(\\bsn)\\alpha}&=q^{-\\ell} \\prod_{\\substack{ p\\leq Y \\\\ p^{\\gamma(p)}\\nmid q}}\\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}}\n\\right)\\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q}\\int_{[0,X]^{\\ell}}e^{2\\pi i g(\\bsx)\\beta}d\\bsx\\\\\\\\&\\qquad + O_{k,\\l}\\left(qE(1+JX^{k}|\\beta|)^{\\ell}\\right)  ,\\end{align*} where $E$ is as in Proposition \\ref{brunprop}. \\item \\label{locitem} \\textnormal{\\textbf{Local cancellation:}}  If $(a,q)=1$ and $Y>0$, then $$\\left| \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q} \\right| \\leq C_1\\begin{cases} (k-1)^{\\l \\omega(q)}\\Phi(q,\\l) q^{\\ell\/2} &\\text{if }q\\leq Y \\\\ C_2^{\\omega(q)}\\tau(q)^{\\l}q^{\\ell-1\/k} &\\text{for all }q \\end{cases},$$ where $C_2=C_2(k)$, $\\Phi(q,2)=(q\/\\phi(q))^{C_2}$, $\\Phi(q,\\l)\\ll_{k,\\l} 1$ for $\\l\\geq 3$, and $C_1$ depends only on the moduli at which $\\grad g$ identically vanishes and the primes $p\\leq Y$ dividing $q$ modulo which $g$ is not Deligne.  \\item \\label{minitem} \\textnormal{\\textbf{Minor arc estimate:}} If $X,Y,Z\\geq 2$, $YZ\\leq X$, $(a,q)=1$, and $|\\alpha-a\/q|<q^{-2}$, then\n\\end{enumerate}\n$$\\left|\\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)} e^{2\\pi \\i g(\\bsn)\\alpha} \\right| \\ll_{k,\\l} \\textnormal{cont}(g)^6(\\log Y)^{ek} X^{\\ell}\\left(e^{-\\frac{\\log Z}{\\log Y}}+\\left(J\\log^{k^2}(JqX)\\left(q^{-1}+\\frac{Z}{X}+\\frac{qZ^k}{X^k}\\right) \\right)^{2^{-k}} \\right). $$\n\n\\end{theorem} \n\n \n\n\\section{Proof of Theorem \\ref{more}} \\label{itproof}\nIn this section, we exploit the estimates enumerated in Theorem \\ref{standalonethm} and apply a Fourier analytic $L^2$ density increment, essentially an improved, streamlined version of S\\'ark\\\"ozy's \\cite{Sark1} original method, in order to prove Theorem \\ref{more}. The core of this method has been utilized in \\cite{Lucier}, \\cite{LM}, \\cite{Ruz}, and \\cite{Rice}, among others. Most specifically, this section very closely follows Section 5 of \\cite{ricemax}.\n\n\n\\subsection{Main iteration lemma and proof of Theorem \\ref{more}}\n\nFor the remainder of Section \\ref{itproof} we fix $k,\\l\\geq 2$, a strongly Deligne polynomial $h\\in \\Z[x_1,\\dots,x_{\\ell}]$ with $\\deg(h)=k$,  and positive constants $C_0=C_0(h)$ and $c_0=c_0(h)$ that are appropriately large and small, respectively. For $N\\in \\N$ we let $$\\cQ=\\cQ(N,h)=e^{c_0\\sqrt{\\log N}}.$$   For a density $\\delta\\in (0,1]$, we define $\\theta(k,\\ell,\\delta)$ by $\\theta(k,\\ell,\\delta)=1$ if $\\ell\\geq 3$ and $\\theta(k,2,\\delta)=\\log^{-k(k-2)}((c_0\\delta)^{-1})$.\n\n We deduce Theorem \\ref{more} from the following iteration lemma, which makes precise the aforementioned passage from a set lacking nonzero differences in the image of a polynomial to a new, denser subset of a slightly smaller interval lacking nonzero differences in the image of an appropriate auxiliary polynomial. \n\n\\begin{lemma} \\label{mainit} Suppose $A\\subseteq [1,N]$ with $|A|=\\delta N$. If $(A-A)\\cap h_d(\\Z^{\\ell})\\subseteq \\{0\\}$, $C_0,\\delta^{-1}\\leq \\cQ$, and $d\\leq N^{c_0}$, then there exists $q\\ll_{h} \\delta^{-2}$ and $A'\\subseteq [1,N']$ such that \n$N'\\gg_{h}  \\delta^{4k}N$, \n\\begin{equation*}|A'|\\geq (1+c\\theta(k,\\ell,\\delta))\\delta N'   , \\end{equation*} where $c=c(h)>0$, and $$(A'-A')\\cap h_{qd}(\\Z^{\\ell})\\subseteq \\{0\\}. $$   \n\n\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem~\\ref{more}]\nThroughout this proof, we let $C$ and $c$ denote sufficiently large or small positive constants, respectively, which we allow to change from line to line, but which depend only on  $h$.  \n\n\\noindent Suppose $A \\subseteq [1,N]$ with $|A|=\\delta N$ and $$(A-A)\\cap h(\\Z^{\\ell})\\subseteq \\{0\\}.$$ Setting $A_0=A$, $N_0=N$, $d_0=1$, and $\\delta_0=\\delta$, Lemma \\ref{mainit} yields, for each $m$, a set $A_m \\subseteq [1,N_m]$ with $|A_m|=\\delta_mN_m$ and $(A_m-A_m)\\cap h_{d_m}(\\Z^{\\ell})\\subseteq \\{0\\}.$ Further, we have that\n\\begin{equation} \\label{NmI} N_m \\geq c\\delta^{4k}N_{m-1} \\geq (c\\delta)^{4km} N,\n\\end{equation}\n\\begin{equation} \\label{incsizeI} \\delta_m \\geq (1+c\\theta(k,\\ell,\\delta))\\delta_{m-1}, \n\\end{equation}\nand\n\\begin{equation}\\label{dmI} d_m \\leq (c\\delta)^{-2} d_{m-1} \\leq (c\\delta)^{-2 m}, \n\\end{equation}\nas long as \n\\begin{equation} \\label{delmI} C,  \\delta_m^{-1} \\leq e^{c\\sqrt{\\log N_m}}, \\quad d\\leq N_m^{c} .\n\\end{equation}\nBy (\\ref{incsizeI}), the density $\\delta_m$ will exceed $1$, and hence (\\ref{delmI}) must fail, for $m=M=M(h,\\delta)$, where $$M(h,\\delta)=\\begin{cases} C\\log (C\\delta^{-1})& \\text{if }\\ell \\geq 3 \\\\ C\\log^{(k-1)^2}(C\\delta^{-1}) & \\text{if }\\ell=2 \\end{cases}. $$ However, by (\\ref{NmI}), (\\ref{incsizeI}), and (\\ref{dmI}), (\\ref{delmI}) holds for $m=M$ if \\begin{equation}\\label{endgame} (c\\delta)^{4kM}=e^{C\\log^{\\mu(k,\\l)^{-1}}(C\\delta^{-1})}\\leq N^{c}.\\end{equation} Therefore, (\\ref{endgame}) must fail, or in other words $\\delta \\ll_{h} e^{-c(\\log N)^{\\mu(k,\\l)}},$ as claimed.\n\\end{proof}\n\n\\subsection{$L^2$ Fourier concentration and proof of Lemma \\ref{mainit}} The philosophy behind the proof of Lemma \\ref{mainit} is that the condition $(A-A)\\cap h_d(\\Z^{\\l})\\subseteq \\{0\\}$ represents highly nonrandom behavior, which should be detectable in the Fourier analytic behavior of $A$. Specifically, we locate one small denominator $q$ such that $\\widehat{f_A}$ has $L^2$ concentration around rationals with denominator $q$, then invoke a standard lemma stating that $L^2$ concentration of  $\\widehat{f_A}$ implies the existence a long arithmetic progression on which $A$ has increased density. \n\n\\begin{lemma}  \\label{L2I} Suppose $A\\subseteq [1,N]$ with $|A|=\\delta N$, $\\eta=c_0\\delta$, and  $\\gamma=\\eta^{-2k}\/N$. If $(A-A)\\cap h_d(\\Z^{\\l})\\subseteq \\{0\\}$, $C_0,\\delta^{-1}\\leq \\cQ$, $d\\leq N^{c_0}$, and  $|A\\cap(N\/9,8N\/9)|\\geq 3\\delta N\/4$, then there exists $q\\leq \\eta^{-2}$ such that \n\\begin{equation*} \\int_{\\mathbf{M}'_q(\\gamma)} |\\widehat{f_A}(\\alpha)|^2d\\alpha \\gg_{h} \\theta(k,\\ell,\\delta) \\delta^{2} N.  \n\\end{equation*}  \n\\end{lemma}\n\n\\noindent Lemma \\ref{mainit} follows from Lemma \\ref{L2I} and the following standard $L^2$ density increment lemma.\n\n\\begin{lemma}[Lemma 2.3 in \\cite{thesis}, see also \\cite{Lucier}, \\cite{Ruz}] \\label{dinc} Suppose $A \\subseteq [1,N]$ with $|A|=\\delta N$. If  $0< \\theta \\leq 1$, $q \\in \\N$, $\\gamma>0$, and\n\\begin{equation*} \\int_{\\mathbf{M}'_q(\\gamma)}|\\widehat{f_A}(\\alpha)|^2d\\alpha \\geq \\theta\\delta^2 N,\n\\end{equation*} \nthen there exists an arithmetic progression \n\\begin{equation*}P=\\{x+\\ell q : 1\\leq \\ell \\leq L\\}\n\\end{equation*}\nwith $qL \\gg \\min\\{\\theta N, \\gamma^{-1}\\}  $ and $|A\\cap P| \\geq (1+\\theta\/32)\\delta L$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{mainit}]\nSuppose $A\\subseteq [1,N]$, $|A|=\\delta N$, $(A-A)\\cap h_d(\\Z^{\\ell})\\subseteq \\{0\\}$, $C_0, \\delta^{-1}\\leq \\cQ$, and $d\\leq N^{c_0}$. If $|A\\cap (N\/9,8N\/9)| < 3\\delta N\/4$, then $\\max \\{ |A\\cap[1,N\/9]|, |A\\cap [8N\/9,N]| \\} > \\delta N\/8$. In other words, $A$ has density at least $9\\delta\/8$ on one of these intervals. \n\n\\noindent Otherwise, Lemmas \\ref{L2I} and \\ref{dinc} apply, so in either case, letting $\\eta=c_0\\delta$, there exists $q\\leq \\eta^{-2}$ and an arithmetic progression \n\\begin{equation*}P=\\{x+\\ell q : 1\\leq \\ell \\leq L\\}\n\\end{equation*}\nwith $qL\\gg_{h} \\delta^{2k} N$ and $$|A\\cap P| \\geq (1+c\\theta(k,\\ell,\\delta))\\delta L  .$$ Partitioning $P$ into subprogressions of step size $\\lambda(q)$, the pigeonhole principle yields a progression \n\\begin{equation*} P'=\\{y+a \\lambda(q) : 1\\leq a \\leq N'\\} \\subseteq P\n\\end{equation*}\nwith $N'\\geq qL\/2\\lambda(q)$ and $|A\\cap P'|\/N' \\geq |A\\cap P|\/L$. This allows us to define a set $A' \\subseteq [1,N']$ by \\begin{equation*} A' = \\{a \\in [1,N'] : y+a \\lambda(q) \\in A \\},\n\\end{equation*} which satisfies $|A'|=|A\\cap P'|$ and $N'\\gg_{h} \\delta^{2k}N\/\\lambda(q) \\gg_{h} \\delta^{4k}N$. Moreover, $(A-A)\\cap h_d(\\Z^{\\ell}) \\subseteq \\{0\\}$ implies $(A'-A')\\cap h_{qd}(\\Z^{\\ell})\\subseteq \\{0\\}$ by Proposition \\ref{inh}.\n\\end{proof}\n\nOur task for this section is now completely reduced to a proof of Lemma \\ref{L2I}.\n\n\\subsection{Preliminary notation for proof of Lemma \\ref{L2I}} Before delving into the proof of Lemma \\ref{L2I}, we take the opportunity to define some relevant sets and quantities, depending on our strongly Deligne polynomial $h\\in \\Z[x_1,\\dots,x_{\\ell}]$, scaling parameter $d$, a parameter $Y>0$, and the size $N$ of the ambient interval. In all the notation defined below, we suppress all of the aforementioned dependence, as the relevant objects will be fixed in context.  \n\nWe define $W_d$, $\\gamma_d$, and $j_d$ in terms of $h$ as in Section \\ref{sievesec}. We then let $M=\\left(\\frac{N}{9J}\\right)^{1\/k}$,  where $J$ is the sum of the absolute value of all the coefficients of $h_d$, and hence $h_d([1,M]^{\\ell})\\subseteq [-N\/9,N\/9]$. We let $$w=\\prod_{p\\leq Y}\\left(1-\\frac{j_{d}(p)}{p^{\\gamma_{d}(p)\\ell}}\\right), $$ and we let $T=wM^{\\ell}$. \n\nWe let $Z=\\{\\bsn\\in \\Z^{\\ell}: h_d(\\bsn)=0\\}$, and we let $H=\\left([1,M]^{\\ell}\\cap W_{d}(Y)\\right)\\setminus Z$. We note that the hypothesis $\\cQ\\geq C_0$ allows us to assume at any point that $\\cQ$, and hence also $N$, are sufficiently large with respect to $h$, which we take as a perpetual assumption moving forward. Under this assumption, it follows from (\\ref{sieve}), (\\ref{logk}), and the estimate \\begin{equation} \\label{Zest} |Z\\cap[1,M]^{\\ell}|\\ll_h M^{\\ell-1}\\end{equation} that \\begin{equation}\\label{Hsize} \\left| H \\right|\\geq T\/2. \\end{equation} \n\n\n\\subsection{Proof of Lemma \\ref{L2I}} \\label{massproof} Suppose $A\\subseteq [1,N]$ with $|A|=\\delta N$, $(A-A)\\cap h_d(\\Z^{\\ell}) \\subseteq \\{0\\}$, $C_0,\\delta^{-1}\\leq \\cQ$, and $d\\leq N^{c_0}$.  Further, let $\\eta=c_0\\delta$, let $Q=\\eta^{-2}$, and let $Y=\\eta^{-2k}$.  Since $h_d(H) \\subseteq [-N\/9,N\/9]\\setminus \\{0\\}$, we have\n\\begin{align*} \\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} f_A(x)f_A(x+h_d(\\bsn))&=\\sum_{\\substack{x \\in \\Z \\\\ \\bsn \\in H}} 1_A(x)1_A(x+h_d(\\bsn)) -\\delta\\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} 1_A(x)1_{[1,N]}(x+h_d(\\bsn)) \\\\ &\\qquad -\\delta \\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} 1_{A}(x+h_d(\\bsn))1_{[1,N]}(x)+\\delta^2\\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} 1_{[1,N]}(x)1_{[1,N]}(x+h_d(\\bsn))  \\\\&\\leq \\Big(\\delta^2N -2\\delta|A\\cap (N\/9,8N\/9)|\\Big)|H|. \n\\end{align*}\nTherefore, if $|A \\cap (N\/9, 8N\/9)| \\geq 3\\delta N\/4$, then by (\\ref{Hsize}) we have\n\\begin{equation}\\label{neg} \\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} f_A(x)f_A(x+h_d(\\bsn)) \\leq -\\delta^2NT\/4.\n\\end{equation} \nWe see from (\\ref{Zest}) and orthogonality of characters that \n\\begin{equation}\\label{orth} \n\\sum_{\\substack{x \\in \\Z \\\\ \\bsn\\in H}} f_A(x)f_A(x+h_d(\\bsn))=\\int_0^1 |\\widehat{f_A}(\\alpha)|^2 S(\\alpha)d\\alpha +O_h(NM^{\\ell-1}),\n\\end{equation} \nwhere \n\\begin{equation*}S(\\alpha)= \\sum_{\\bsn \\in [1,M]^{\\ell} \\cap W_{d}(Y)}e^{2\\pi i h_d(\\bsn)\\alpha}.\n\\end{equation*} \nCombining (\\ref{neg}) and (\\ref{orth}), we have  \n\\begin{equation} \\label{mass}\n\\int_0^1 |\\widehat{f_A}(\\alpha)|^2|S(\\alpha)|d\\alpha \\geq \\delta^2NT\/8.\n\\end{equation} Letting $\\gamma=\\eta^{-2k}\/N$, Theorem \\ref{standalonethm} yields that for $\\alpha \\in \\mathbf{M}_q(\\gamma), \\ q\\leq Q $, we have \\begin{equation} \\label{SmajII} |S(\\alpha)| \\ll_{h} (k-1)^{\\l\\omega(q)}\\Phi(q,\\l)q^{-\\ell\/2}T,\n\\end{equation} where $\\Phi(q,2)=(q\/\\phi(q))^C$ for $C=C(k)$ and $\\Phi(q,\\l)\\ll_{k,\\l} 1$ for $\\l\\geq 3$. Further, for $\\alpha \\in \\mathfrak{m}(\\gamma,Q)$ we have \n\\begin{equation} \\label{SminII} |S(\\alpha)| \\leq  \\delta T\/16.\n\\end{equation} The proof of the estimates in Theorem \\ref{standalonethm} and the subsequent deduction of (\\ref{SmajII}) and (\\ref{SminII}) can be found in Section \\ref{expest}. From (\\ref{SminII}) and Plancherel's Identity, we have \\begin{equation*}  \\int_{\\mathfrak{m}(\\gamma,Q)} |\\widehat{f_A}(\\alpha)|^2|S(\\alpha)|d\\alpha \\leq \\delta^2NT\/16, \\end{equation*} which together with (\\ref{mass}) yields \\begin{equation}\\label{majmass}  \\int_{\\mathfrak{M}(\\gamma,Q)}|\\widehat{f_A}(\\alpha)|^2|S(\\alpha)|d\\alpha \\geq \\delta^2 NT\/16. \\end{equation} \nFrom (\\ref{SmajII}) and (\\ref{majmass}) , we have \n\\begin{equation} \\label{majmassII} \\sum_{q=1}^Q  (k-1)^{\\l\\omega(q)}\\Phi(q)q^{-\\ell\/2} \\int_{\\mathbf{M}_q(\\gamma)}|\\widehat{f_A}(\\alpha)|^2 {d}\\alpha \\gg_{h} \\delta^2N.\n\\end{equation}\nFor $\\ell=2$, the function $b(q)=(k-1)^{2\\omega(q)}(q\/\\phi(q))^C$ satisfies $b(qr)\\geq b(r)$, and we make use of the following proposition, which is based on a trick that originated in \\cite{Ruz}.\n\\begin{proposition}[Proposition 5.6, \\cite{ricemax}] \\label{rstrick} For any $\\gamma,Q>0$ satisfying $2\\gamma Q^2<1$, and for any function $b: \\N \\to [0,\\infty)$ satisfying $b(qr)\\geq b(r)$ for all $q,r\\in \\N$, we have $$\\max_{q\\leq Q} \\int_{\\mathbf{M}'_q(\\gamma)}|\\widehat{f_A}(\\alpha)|^2 {d}\\alpha \\geq Q \\Big(2\\sum_{q=1}^Q b(q)\\Big)^{-1} \\sum_{r=1}^Q \\frac{b(r)}{r}\\int_{\\mathbf{M}_r(\\gamma)}|\\widehat{f_A}(\\alpha)|^2 {d}\\alpha. $$\n\\end{proposition}\n\n\n\\noindent Because $b$ is a multiplicative function, $b(p^v)=(k-1)^2(1+1\/(p-1))^C\\ll_k 1$ for all prime powers $p^v$, and $$\\sum_{q=1}^Q \\frac{b(q)}{q}\\leq \\prod_{p\\leq Q} \\left(1+\\frac{b(p)}{p}+\\frac{b(p)}{p^2}+\\cdots\\right)=\\prod_{p\\leq Q} \\left(1+\\frac{(k-1)^2}{p}+O_k(1\/p^2)\\right)\\ll_k \\log^{(k-1)^2}Q, $$ it follows from Theorem 01 of \\cite{HallTen} that \n\\begin{equation*} \\sum_{q=1}^Q b(q) \\ll_k Q\\log^{(k-1)^2-1} Q, \n\\end{equation*}\nand the lemma for $\\ell=2$ follows from (\\ref{majmassII}) and Proposition \\ref{rstrick}. For $\\l \\geq 3$, since $(k-1)^{\\l\\omega(q)}\\ll_{k,\\l,\\epsilon} q^{\\epsilon}$ for any $\\epsilon>0$, the sum $\\sum_{q=1}^{\\infty}  (k-1)^{\\l\\omega(q)}q^{-\\ell\/2}$ is convergent, and hence (\\ref{majmassII}) immediately yields $$\\max_{q\\leq Q} \\int_{\\mathbf{M}_q(\\gamma)}|\\widehat{f_A}(\\alpha)|^2 {d}\\alpha \\gg_h \\delta^2 N.$$ Since $\\mathbf{M}_q(\\gamma)\\subseteq \\mathbf{M}'_q(\\gamma), $ this establishes the lemma for $\\ell \\geq 3$. \\qed\n\n\\section{Criteria for strongly Deligne polynomials} \\label{AG1}\nIn this section, we prove Proposition~\\ref{thm:main} and a stronger version of Proposition~\\ref{prop:main}. We begin, though, by collecting a few facts from algebraic geometry that will be useful in subsequent sections. Throughout this section, for a variety $V$, we let $V^\\sing$ denote the singular locus of $V$, and we let $V^\\ns=V\\setminus V^\\sing$.\n\n\\subsection{Results from algebraic geometry}\n\nWe first state a classical version of B\\'ezout's Theorem; see \\cite[Example 8.4.6]{Fulton}.\n\n\\begin{lemma}[B\\'ezout's Theorem]\\label{lem:bezout}\nLet $V_1,\\ldots,V_k$ be subvarieties of $\\bbP^\\ell$. Then $\\deg\\bigcap_{i=1}^k V_i \\le \\prod_{i=1}^k \\deg V_i.$\nIn particular, if the intersection is finite, then $\\left| \\bigcap_{i=1}^k V_i \\right| \\le \\prod_{i=1}^k \\deg V_i.$\n\\end{lemma}\n\nWe now record estimates due to Lang and Weil \\cite{LangWeil} on the number of points on varieties over finite fields. The following is a well known consequence of Theorem 1 of \\cite{LangWeil} (see, for example, Theorem 5.1 of \\cite{PoonenSlavov}), but we give the short proof for completeness.\n\n\\begin{lemma} \\label{langweillem}\nLet $k$, $\\ell$, $m$, and $r$ be positive integers, and let $q$ be a prime power. Let $V$ be a (reduced) closed subvariety of $\\bbP^\\ell$, defined over $\\F_q$ (the field with $q$ elements), of degree $k$ and dimension $r$. Let $m \\ge 1$ be the number of geometrically irreducible components of $V$ which are defined over $\\F_q$. Then\n\t\\begin{equation}\\label{eq:LangWeil}\n\t\t|V(\\F_q)|,\\ |V^\\ns(\\F_q)| = mq^r + O_{k,\\ell,r}(q^{r-1\/2}).\n\t\\end{equation}\nMoreover, the same is true if we replace $V$ with a closed subvariety $W \\subseteq \\bbA^\\ell$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on $r$, noting that the case $r = 0$ is elementary, and amounts to considering the following observations.\n\t\\begin{enumerate}[1.]\n\t\\item If $P \\in Z(\\F_q)$ for a component $Z \\subset V$ not defined over $\\F_q$, then $P = P^\\sigma \\in Z^\\sigma \\ne Z$ for nontrivial $\\sigma \\in \\Gal(\\overline{\\F_q}\/\\F_q)$, hence $P \\in Z \\cap Z^\\sigma$, which has dimension strictly less than $r$. Thus, the number of points on components not defined over $\\F_q$ is absorbed into the error term.\n\t\n\t\\item Each component of $V$ defined over $\\F_q$ has $q^r + O_{k,\\ell,r}(q^{r-1\/2})$ by Theorem 1 of \\cite{LangWeil}. Summing the number of points on each component is an overcount, but the surplus is due to points on pairwise intersections of components, which again is absorbed into the error term. (Note that $m \\le k$, so even after multiplying the error by $m$, the implied constant still depends only on $k$, $\\ell$, and $r$.) Thus $|V(\\F_q)|$ has the claimed magnitude.\n\t\n\t\\item We have $V^\\ns := V \\setminus V^\\sing$; since $V^\\sing$ has dimension at most $r - 1$ and degree controlled by $k$, $r$, and $\\ell$ (by B\\'ezout's Theorem), the size of $V^\\sing(\\F_q)$ is included in the error term. Thus, $|V^\\ns(\\F_q)|$ also has the desired magnitude.\n\t\n\t\\item Finally, if we let $V$ be the projective closure of $W$, then $W = V \\setminus (V \\cap H)$, where $H$ is the hyperplane at infinity. Since $V \\cap H$ has lower dimension and degree $k$, we are once again removing a set whose cardinality is subsumed by the error term, so $W(\\F_q)$ (and, similarly, $W^\\ns(\\F_q)$) has the appropriate cardinality. \\qedhere\n\t\\end{enumerate}\n\\end{proof}\n\n\n\\subsection{A key equivalence} The following equivalence observation yields a strengthening of Proposition \\ref{prop:main} as a corollary, and is also instrumental in subsequent proofs.\n\n\\pagebreak\n\n\\begin{lemma}\\label{lem:equiv}\nLet $V$ be a variety (reduced, but not necessarily irreducible) of dimension $d \\ge 1$ defined over $\\Z$. For a sufficiently large (with respect to $V$) prime $p$, the following are equivalent:\n\n\t\\begin{enumerate}[(a)]\n\t\\item $V^\\ns(\\F_p) \\ne \\emptyset$.\n\t\\item $V^\\ns(\\Z_p) \\ne \\emptyset$.\n\t\\item At least one of the geometric components of $V$ is defined over $\\Z_p$.\n\t\\end{enumerate}\n\\end{lemma} \n \n\\begin{proof}\\mbox{}\n\n\\noindent((a) $\\implies$ (b)) Suppose $V^\\ns(\\F_p) \\ne \\emptyset$, and let $Q \\in V^\\ns(\\F_p)$. By Hensel's lemma, there exists $P \\in V(\\Z_p)$ such that $\\widetilde{P} = Q$. Since $\\widetilde{P}$ is nonsingular, so must be $P$.\n \n\\noindent ((b) $\\implies$ (c))\nLet $P \\in V^\\ns(\\Z_p)$, and let $Z$ be a geometric component of $V$ containing $P$. As in part 1 of the proof of Lemma~\\ref{langweillem}, if $Z$ were not defined over $\\F_q$, then $P$ would lie in the intersection of two components, hence would be a singular point on $V$, contradicting our assumption on $P$.\n\n\n\\noindent ((c) $\\implies$ (a)) \nLet $Z_1,\\ldots,Z_m$ be the irreducible components of $V$. By Lemma \\ref{langweillem}, for each $1 \\le i \\le m$ there exists a bound $B_i$ such that for all $p \\ge B_i$ with $Z_i$ defined over $\\Z_p$, $Z_i^\\ns(\\F_p)$ contains a point that does not lie on $Z_j$ for any $j \\ne i$. Letting $B = \\max\\{B_1,\\ldots,B_m\\}$, we have that for $p \\ge B$, the existence of $Z_i$ defined over $\\Z_p$ implies the existence of $Q \\in Z_i^\\ns(\\F_p) \\setminus \\bigcup_{j\\ne i} Z_j(\\F_p)$. Since $Q$ is nonsingular on $Z_i$ and is not a point of intersection with any other component $Z_j$, we have $Q\\in V^\\ns(\\F_p)$. \n\\end{proof}  \n\nAs previously noted, if $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is Deligne, then $h=0$ defines a reduced variety. Further, a nonsingular point over $\\Z_p$ on this variety corresponds precisely to a root $\\bsz_p\\in \\Z_p^{\\l}$ of $h$ satisfying $m_p=1$, hence Lemma \\ref{lem:equiv} establishes the following sufficient condition for smooth intersectivity. Here we let $\\ZZbar$ denote the ring of algebraic integers.\n \n\\begin{corollary}\\label{prop:main_stronger}\nSuppose $\\ell \\ge 2$ and $h \\in \\Z[x_1,\\ldots,x_{\\ell}]$ is Deligne and intersective, and let $h=g_1\\cdots g_n$ be an irreducible factorization of $h$ in $\\ZZbar[x_1,\\dots,x_{\\l}]$. If, for all but finitely many $p\\in \\P$, $g_i$ has coefficients in $\\Z_p$ for some $1\\leq i \\leq n$, then $h$ is smoothly intersective, hence strongly Deligne.\n\\end{corollary}\n \nNote that Proposition~\\ref{prop:main} is an immediate consequence of Corollary~\\ref{prop:main_stronger}, since the hypotheses of the proposition imply that one of the factors over $\\ZZbar$ is defined over $\\Z$, hence over $\\Z_p$ for all $p$. We now complete this section by using Lemma \\ref{lem:equiv} to prove Proposition \\ref{thm:main}.\n\n\\begin{proof}[Proof of Proposition~\\ref{thm:main}]\nLet $\\l\\geq 2$, and suppose $h\\in \\Z[x_1,\\dots,x_{\\l}]$ is Deligne and intersective with $\\deg(h)=k\\geq 2$. Let $\\{\\bsz_p\\}_{p\\in \\P}$ be a choice of $p$-adic integer roots of $h$ satisfying $m_p\\in \\{1,k\\}$ for all but finitely many $p$. \nLet $X$ denote the finite set of primes such that \\\\[-20pt]\n\n\t\\begin{itemize}\n\t\\item $m_p\\notin \\{1,k\\}$, or \\\\[-18pt]\n\t\\item $p\\mid k$, or \\\\[-18pt]\n\t\\item $h^k$ is not smooth modulo $p$, or \\\\[-18pt]\n\t\\item the equivalence in Lemma \\ref{lem:equiv} fails.\n\t\\end{itemize}\n\\noindent We note that the first item is assumed to be finite, the second item is clearly finite, the fourth item is proven to be finite in Lemma \\ref{lem:equiv}, and the third item is finite because $h$ is Deligne (see Definition~\\ref{defn:smoothDeligne}). \n\n\\noindent Fix $d\\in \\N$ and $p\\notin X$. If $p\\nmid d$ or $m_p=k$, then $p\\nmid \\frac{d^k}{\\lambda(d)}$, so $h_d^k=\\frac{d^k}{\\lambda(d)}h^k$ is a nonzero scalar multiple of $h^k$, hence remains smooth modulo $p$. Therefore, $h_d$ is Deligne modulo $p$.\n\n\\noindent The remaining case is $p\\mid d$ and $m_p=1$. In this case, since $h^i_d$ has a factor of $\\frac{d^i}{\\lambda(d)}$, the definition of $\\lambda$ assures that the polynomial $h^i_d$ identically vanishes modulo $p$ for all $i>1$. Since nonzero homogeneous linear polynomials are automatically smooth, we need only argue that $$h_d^1(\\bsx)=\\frac{d}{\\lambda(d)}\\sum_{i=1}^{\\l}\\frac{\\partial h}{\\partial x_i} (\\bsr_d)x^i$$ does not identically vanish modulo $p$. We know that $p\\nmid \\frac{d}{\\lambda(d)}$ by definition of $\\lambda$. Further, the fact that $h$ is Deligne ensures that $h=0$ defines a reduced variety, so by Lemma \\ref{lem:equiv}, we can choose $\\bsz_p$ to reduce to a nonsingular point over $\\F_p$. Since $\\bsr_d\\equiv \\bsz_p \\ (\\text{mod }p)$, we have that $\\frac{\\partial h}{\\partial x_i} (\\bsr_d)\\equiv \\frac{\\partial h}{\\partial x_i} (\\bsz_p)\\not\\equiv 0 \\ (\\text{mod }p)$ for some $1\\leq i \\leq \\l$, as required. Therefore, $h_d$ is Deligne modulo $p$ for all $p\\notin X$, hence $h$ is strongly Deligne.\n\\end{proof}\n\n\n\n\n\n\n\n\n \n\\section{Dimension lowering argument}\\label{dimlowsec}\n\nIn this section, we generalize the phenomenon exemplified at the beginning of Section \\ref{sec:singular}, establishing Theorems \\ref{dimlowrootthm} and \\ref{dimlowthm} by reducing to the case covered in Theorem \\ref{more}. In the integer root setting, this reduction is very direct, as Theorem \\ref{dimlowrootthm} follows immediately from Theorem \\ref{more} and the following proposition.\n\n\\begin{proposition}\\label{prop:dim_lower_0}\nSuppose $\\l \\ge 2$ and $h \\in \\Z[x_1,\\ldots,x_{\\l}]$ with $h(\\boldsymbol{0}) = 0$. Let $r$ be the minimum rank of the highest and lowest degree homogeneous parts of $h$. If $r\\geq 2$, then there exists a strongly Deligne polynomial $g \\in \\Z[x_1,\\ldots,x_r]$  such that $g(\\Z^r) \\subseteq h(\\Z^{\\l})$.\n\\end{proposition}\n\nBefore delving into the proof of this proposition, we state a version of Bertini's theorem that will allow us to eliminate the singularity in the top-degree parts of our polynomials, one dimension at a time. \nThroughout this section we let $(\\bbP^n)^*$ denote the dual space of $\\bbP^n$, that is, the space of hyperplanes in $\\bbP^n$. Note that $(\\bbP^n)^*$ is isomorphic to $\\bbP^n$, with the hyperplane $\\{a_0x_0 + \\cdots + a_nx_n = 0\\} \\in (\\bbP^n)^*$ corresponding to the point $(a_0 : \\cdots : a_n) \\in \\bbP^n$. A {\\it linear system of hyperplanes in $\\bbP^n$} is a linear subspace of $(\\bbP^n)^*$.\n\n\\pagebreak\n\n\\begin{theorem}[Bertini's Theorem]\\label{thm:bertini}\nLet $V$ be a (quasi-projective) subvariety of $\\bbP^n$ with irreducible components $V_1,\\ldots,V_m$ of equal dimension $d \\ge 1$, and let $\\calL \\subseteq (\\bbP^n)^*$ be a linear system. After a change of coordinates if necessary, we may assume that there exists $k \\in \\{0,\\ldots,n\\}$ such that $\\calL$ is the space of all hyperplanes of the form $\\{a_kx_k + \\cdots + a_nx_n = 0\\}$. Assume that the coordinates $x_k, \\ldots, x_n$ do not simultaneously vanish at any point on $V$ (i.e., the linear system $\\calL$ has no base-points in $V$), so that\n\t\\begin{align*}\n\t\\Phi_\\calL : V &\\longrightarrow \\bbP^{n-k}\\\\\n\t(z_0 : \\cdots : z_n) &\\longmapsto (z_k : \\cdots : z_n)\n\t\\end{align*}\ndefines a morphism.\nThen there exists a nonempty open subset $U \\subseteq \\calL$ such that for all hyperplanes $H \\in U$,\n\t\\begin{enumerate}[(a)]\n\t\\item $V^\\ns \\cap H$ is nonsingular, and\n\t\\item $\\dim \\left(V^\\sing \\cap H\\right) < \\dim V^\\sing$ (if $V^\\sing \\neq \\emptyset$).\n\t\\end{enumerate}\n\n\\noindent Moreover, if $\\dim \\Phi_\\calL(V) \\ge 2$, then $U$ may be chosen so that for all $H \\in U$ we have\n\n\t\\begin{enumerate}[(a)]\n\t\\setcounter{enumi}{2}\n\t\\item for all $1 \\le i \\le m$, the intersection $V_i \\cap H$ is either empty or geometrically irreducible.\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{rem}\nTheorem~\\ref{thm:bertini} is stated somewhat more generally than we need, so we specify the two situations for which we will actually need the result:\n\n\t\\begin{enumerate}[1.]\n\t\\item Let $V$ be a closed hypersurface in $\\bbP^n$ and let $\\calL = (\\bbP^n)^*$. Then $\\Phi_\\calL$ is just the inclusion map of $V$ into $\\bbP^n$, and the hypotheses of Theorem~\\ref{thm:bertini} are satisfied. Moreover, since each component $V_i$ is a closed subvariety of $\\bbP^n$ of positive dimension, each intersection $V_i \\cap H$ is nonempty; thus, if $d = \\dim V = \\dim \\Phi_\\calL(V) \\ge 2$, then $V_i \\cap H$ is irreducible for all $1 \\le i \\le m$ and all $H \\in U$.\n\t\\item Identify $\\bbA^n$ with the Zariski open subset $\\{x_0 \\ne 0\\} \\subset \\bbP^n$. Let $V$ be a closed hypersurface in $\\bbA^n$ {\\it not containing the origin $\\boldsymbol{0} = (0,\\ldots,0)$}, and let $\\calL$ be the space of all hypersurfaces of the form $\\{a_1x_1 + \\cdots + a_nx_n = 0\\}$. Then the conditions of Theorem~\\ref{thm:bertini} are satisfied once again. A fiber of $\\Phi_\\calL$ is precisely the intersection of $V$ with a line in $\\bbA^n$ passing through $\\boldsymbol{0}$. Since $V$ is closed in $\\bbA^n$ and does not contain $\\boldsymbol{0}$, $V$ cannot contain a line through $\\boldsymbol{0}$, hence each such intersection is finite. In particular, this means the map $\\Phi_\\calL$ has finite fibers, so $\\dim \\Phi_\\calL(V) = \\dim V = d$. Moreover, the failure of a hyperplane $H \\in \\calL$ to intersect every $V_i$ is a proper Zariski closed condition.\n\n\tTherefore, removing such hyperplanes from $U$ if necessary, we again have that $V_i \\cap H$ is nonempty for all $1 \\le i \\le n$ and $H \\in U$, thus $V_i \\cap H$ is irreducible for all $1 \\le i \\le n$ and $H \\in U$ as long as $d \\ge 2$.\n\t\\end{enumerate}\n\\end{rem}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bertini}]\nConsider the set $\\calX$ of hyperplanes $H \\in \\calL$ satisfying the following conditions:\n\n\t\\begin{enumerate}[(a$'$)]\n\t\\item $V_i^\\ns \\cap H$ is nonsingular for all $1 \\le i \\le m$;\n\t\\item $H$ does not contain any components of $V_i^\\sing$ nor $(V_i \\cap V_j)$ for $1 \\le i,j \\le m$ with $i \\ne j$; and\n\t\\item for all $1 \\le i \\le m$, the intersection $V_i \\cap H$ is either empty or geometrically irreducible (if $d \\ge 2)$.\n\t\\end{enumerate}\n\n\\noindent We begin by showing that if $H \\in \\calX$, then $H$ satisfies properties (a), (b), and (c). Indeed, condition (c$'$) is exactly condition (c), so we need only show that $H$ also satisfies (a) and (b).\n\n\\noindent For (a), note that a point $P \\in V$ is nonsingular if and only if $P$ is a nonsingular point on $V_i$ for some $1 \\le i \\le m$ and $P \\notin V_j$ for all $j \\ne i$. Thus $V^\\ns$ is a {\\it disjoint} union $V^\\ns = \\bigsqcup_{i=1}^m W_i$, where each $W_i$ is a subset of $V_i^\\ns$. Then (a) follows from (a$'$) since $V^\\ns \\cap H = \\bigsqcup_{i=1}^m (W_i \\cap H)$ and each $W_i \\cap H \\subseteq V_i^\\ns \\cap H$ is nonsingular. Finally, (b$'$) implies that $H$ intersects each component of $V^\\sing$ properly (assuming $V^\\sing \\ne\\emptyset$), so (b) follows.\n\n\\noindent It remains to show that $\\calX$ contains a Zariski open subset of $\\calL$.\nBy the standard form of Bertini's Theorem (see Corollaire 6.11 of \\cite{Jouanolou}, or Corollary 10.9 and Remark 10.9.1 of \\cite{Hartshorne}), since $\\calL$ has no base-points in $V$, the set of hyperplanes $H \\in \\calL$ satisfying (a$'$) and (c$'$) contains a nonempty open subset of $\\calL$. Moreover, $H$ containing any of a finite collection of nonempty subvarieties of $\\bbP^n$ is a proper closed condition on $H$, so condition (b$'$) is a nonempty open condition; therefore, $\\calX$ contains a nonempty open subset of $\\calL$.\n\\end{proof}\nArmed with Theorem~\\ref{thm:bertini}, the proof of Proposition \\ref{prop:dim_lower_0} is pleasingly straightforward.\n\\begin{proof}[Proof of Proposition~\\ref{prop:dim_lower_0}]\nSuppose that  $\\l \\ge 2$ and $h \\in \\Z[x_1,\\ldots,x_{\\l}]$ with $h(\\boldsymbol{0}) = 0$. Let $k$ and $j$ denote the highest and lowest degrees, respectively, of the terms appearing in $h$, and let $r$ denote the minimum rank of $h^k$ and $h^j$. Let $\\Vhat_k,\\Vhat_j\\subseteq \\bbP^{\\l-1}$ denote the varieties defined by $h^k=0$ and $h^j=0$, respectively. By \nTheorem~\\ref{thm:bertini} (see also case 1 of the remark that follows)\napplied to the linear system $\\calL = (\\bbP^{\\l - 1})^*$ and the varieties $\\Vhat_k$ and $\\Vhat_j$, respectively, the set of hyperplanes $H$ in $\\bbP^{\\l - 1}$ satisfying \\\\[-20pt]\n\t\\begin{itemize}\n\t\\item $H \\cap \\Vhat_k^\\ns$ and $H \\cap \\Vhat_j^\\ns$ are nonsingular, and \\\\[-18pt]\n\t\\item $\\dim (H \\cap \\Vhat_k^\\sing) < \\dim \\Vhat_k^\\sing$, if $V^\\sing \\neq \\emptyset$, and $\\dim (H \\cap \\Vhat_j^\\sing) < \\dim \\Vhat_j^\\sing$, if $\\Vhat_j^\\sing\\neq \\emptyset$, \n\t\\end{itemize}\ncontains a nonempty open subset $U \\subseteq (\\bbP^{\\l-1})^*$. Thus, we can choose $H \\in U$ defined by the vanishing of $l(x_1,\\ldots,x_{\\l}) = a_1x_1 + \\cdots + a_{\\l-1}x_{\\l-1} - x_{\\l}$ with $a_1,\\ldots,a_{\\l-1} \\in \\Z$. Here, we're using the fact that the set of integer points is Zariski dense in the affine space $\\bbA^{\\l-1} \\subset \\bbP^{\\l-1} \\cong (\\bbP^{\\l-1})^*$.\n\n\\noindent Let $\\mu(x_1,\\ldots,x_{\\l-1}) := a_1x_1 + \\cdots + a_{n-1}x_{\\l-1}$, and set\n\t\\[\n\t\tg_1(x_1,\\ldots,x_{\\l-1}) := h(x_1,\\ldots,x_{\\l-1},\\mu(x_1,\\ldots,x_{\\l-1})).\n\t\\]\nNote that, by construction, $g_1(\\Z^{\\l-1}) \\subseteq h(\\Z^\\l)$, $g_1(\\bszero)=0$, and the highest and lowest degrees of the nonzero terms of $g_1$ are still $k$ and $j$, respectively.\n\n\\noindent Now, the subvariety $\\What_k$ (resp., $\\What_j$) of $\\bbP^{\\l-2}$ defined by $g_1^k = 0$ (resp., $g_1^j = 0$) is isomorphic to $H \\cap \\Vhat_k$ (resp., $H \\cap \\Vhat_j$). In particular, the minimum rank of $g_1^k$ and $g_1^j$ can only drop below $r$ if both singular loci were originally empty, which would imply $r=\\l$. Thus, repeating this process $(\\l - r)$ times yields a sequence of polynomials $\\left(g_i(x_1,\\ldots,x_{\\l-i})\\right)_{i=0}^{\\l-r}$, with $g_0 := h$, satisfying \\\\[-20pt]\n\t\\begin{itemize}\n\t\\item $g_i(\\Z^{\\l-i}) \\subseteq g_{i-1}(\\Z^{\\l-i+1})$ for all $1 \\le i \\le \\l - r$,  \\\\[-18pt]\n\t\\item $g_i(\\bszero)=0$ for all $0 \\le i \\le \\l - r$, \\\\[-18pt]\n\t\\item the highest and lowest degrees of the nonzero terms of each $g_i$ are $k$ and $j$, respectively, and \\\\[-18pt]\n\t\\item the rank of each $g_i^k$ (resp., $g_i^j$) is at least $r$.\n\t\\end{itemize}\nFinally, let $g := g_{\\l-r}\\in \\Z[x_1,\\dots,x_r]$, so the rank for each of $g^k$ and $g^j$ is $r$. In other words, $g^k$ and $g^j$ are smooth, and thus by Proposition \\ref{prop:integer_root}, $g$ is strongly Deligne.\n\\end{proof}\n\n\\begin{rem} \nThe conclusion of Proposition~\\ref{prop:dim_lower_0} technically holds for $r = 1$ as well, since nonconstant univariate polynomials are necessarily Deligne; however, this case is not useful for our purposes.\n\\end{rem}\n\n\\subsection{Proof of Theorem \\ref{dimlowthm}} We now proceed with the more elaborate of our two dimension-lowering arguments, in which we cannot exploit the existence of an integer root. Throughout this section, we fix $\\l\\geq 2$ and a polynomial $h\\in \\Z[x_1,\\dots,x_{\\l}]$ satisfying all hypotheses of Theorem \\ref{dimlowthm}, and we recall that $k=\\deg(h)$ and $r$ denotes the rank of $h^k$. Note that the hypotheses of Theorem~\\ref{dimlowthm} imply that $r \\ge 2$. We assume without loss of generality that $h(\\bszero)\\neq 0$, which is permissible because $h(\\Z^{\\l})$ is invariant under input translation. We let $V\\subseteq \\bbA^{\\l}$ and $\\Vhat \\subseteq \\bbP^{\\l-1}$ denote the varieties defined by $h=0$ and $h^k=0$, respectively. The following crucial lemma says that we can eliminate the singularity in the top-degree part of $h$, one dimension at a time, while maintaining the existence of nonsingular $\\F_p$-points.\n\n\n\n\\begin{lemma}\\label{lem:lin_poly}\nSuppose $\\ell \\ge 3$. Then there exists a homogeneous linear polynomial $l \\in \\Z[x_1,\\ldots,x_{\\l}]$, monic in $x_{\\l}$, for which the following holds: Letting $\\Lhat$ and $L$ denote the hyperplanes in $\\bbP^{\\l - 1}$ and $\\bbA^\\l$, respectively, defined by $l = 0$, we have\n\t\\begin{enumerate}[(i)]\n\t\\item $\\dim (\\Vhat \\cap \\Lhat)^\\sing < \\dim \\Vhat^\\sing$, if $\\Vhat^\\sing\\neq \\emptyset$; and\n\t\\item For sufficiently large $p$, $V^\\ns(\\F_p)\\neq \\emptyset$ implies $(V \\cap L)^\\ns(\\F_p) \\ne\\emptyset$.\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\calLhat$ and $\\calL$ denote the linear systems of {\\it hyperplanes in $\\bbP^{\\ell-1}$} and {\\it hyperplanes in $\\bbA^{\\ell}$ passing through $\\bszero$}, respectively. We identify each of $\\calLhat$ and $\\calL$ with $\\bbP^{\\ell-1}$, with the point $\\bsa = (a_1 : \\cdots : a_\\ell) \\in \\bbP^{\\ell-1}$ corresponding to the hyperplanes $\\{a_1x_1 + \\cdots + a_\\ell x_\\ell = 0\\}$ in $\\bbP^{\\ell - 1}$ and $\\bbA^\\ell$, respectively.\n\n\\noindent \nThe hypotheses of Theorem~\\ref{thm:bertini} are satisfied by $\\Vhat$ and $\\calLhat$ (resp., $V$ and $\\calL$), as explained in case 1 (resp., case 2) of the remark immediately following the theorem. Thus, there is a nonempty open set $U \\subseteq \\bbP^{\\ell-1}$ such that for all $\\bsa = (a_1 : \\cdots : a_\\ell) \\in U$, the hyperplanes $\\Lhat_\\bsa \\subset \\bbP^{\\ell - 1}$ and $L_\\bsa \\subset \\bbA^\\ell$ defined by $l_\\bsa := a_1x_1 + \\cdots + a_\\ell x_\\ell = 0$ satisfy the conclusion of Theorem~\\ref{thm:bertini} (intersected with $\\Vhat \\subset \\bbP^{\\ell - 1}$ and $V \\subset \\bbA^\\ell$, respectively). Similar to the proof of Proposition~\\ref{prop:dim_lower_0}, we may choose $\\bsa \\in U$ of the form $\\bsa = (a_1 : \\cdots : a_{\\ell - 1} : 1)$ with $a_1,\\ldots,a_{\\ell-1} \\in \\Z$. Set $l := l_\\bsa$ for such a choice of $\\bsa \\in U$, hence also $\\Lhat = \\Lhat_\\bsa$ and $L = L_\\bsa$. By construction, we immediately have that $l \\in \\Z[x_1,\\ldots,x_\\ell]$, $l$ is monic in $x_\\ell$, and property (i) holds, so it remains only to show that (ii) holds.\n\n\\noindent Let $V_1,\\ldots,V_m$ be the geometrically irreducible components of $V$. Since $\\dim V = \\l - 1 \\ge 2$, our choice of $\\bsa$ guarantees that the geometrically irreducible components of $V \\cap L$ are $V_i \\cap L$ with $1 \\le i \\le m$. (We are again using Theorem~\\ref{thm:bertini} and case 2 of the remark that follows.) By Lemma~\\ref{lem:equiv}, if $p$ is sufficiently large, then $V^\\ns(\\F_p) \\ne\\emptyset$ implies that $V_i$ is defined over $\\Z_p$ for some $1 \\le i \\le m$. Since $L$ is defined over $\\Z$, hence over $\\Z_p$, the intersection $V_i \\cap L$ is also defined over $\\Z_p$. Appealing to Lemma~\\ref{lem:equiv} once more implies that $(V \\cap L)^\\ns(\\F_p)$ is nonempty.\n\\end{proof}\n\nThe hyperplane produced by Lemma \\ref{lem:lin_poly} quickly yields a suitable polynomial with one fewer variable.\n\n\\begin{corollary}\\label{cor:one_step}\nSuppose $\\l \\ge 3$. Then there exists $g' \\in \\Z[x_1,\\ldots,x_{\\l-1}]$ with $\\deg(g')=k$  such that\n\t\\begin{enumerate}[(i)]\n\t\\item $g'(\\bszero) \\ne 0$;\n\t\\item $g'(\\Z^{\\l-1}) \\subseteq h(\\Z^\\l)$; \n\t\\item $\\dim (\\What')^\\sing < \\dim \\Vhat^\\sing$, if $\\Vhat^\\sing \\neq \\emptyset$; and\n\t\\item for sufficiently large $p$, $V^\\ns(\\F_p)\\neq \\emptyset$ implies $(W')^\\ns(\\F_p) \\ne\\emptyset$;\n\t\\end{enumerate}\nwhere $\\What' \\subset \\bbP^{\\l-2}$ and $W' \\subset \\bbA^{\\l-1}$ are the varieties defined by $(g')^k = 0$ and $g' = 0$, respectively.\n\\end{corollary}\n\n\\begin{proof}\nLet $L = L_\\bsa$ be as in Lemma~\\ref{lem:lin_poly}, and write $l = l_\\bsa = a_1x_1 + \\cdots + a_{\\l-1}x_{\\l-1} + x_{\\l}$. To ease notation, we also set $\\mu = \\mu_\\bsa := -(a_1x_1 + \\cdots + a_{\\l-1}x_{\\l-1})$. Now, define\n\t\\[\n\t\tg'(x_1,\\ldots,x_{\\l-1}) := h(x_1,\\ldots,x_{\\l-1},\\mu(x_1,\\ldots,x_{\\l-1})).\n\t\\]\nClearly $g'(\\Z^{\\l-1}) \\subseteq h(\\Z^{\\l})$ and, since $\\mu$ is homogeneous, $g'(\\bszero) = h(\\bszero) \\ne 0$.\nFinally, since $V \\cap L \\cong W'$ and $\\Vhat \\cap \\Lhat \\cong \\What'$, properties (iii) and (iv) follow immediately from Lemma~\\ref{lem:lin_poly}.\n\\end{proof}\n\nRecall our assumption that the rank satisfies $r \\ge 2$. Repeated application of Corollary~\\ref{cor:one_step} yields the following:\n\n\\begin{corollary}\\label{cor:all_steps}\nThere exists $g \\in \\Z[x_1,\\ldots,x_{r}]$ with $\\deg(g)=k$ such that\n\t\\begin{enumerate}[(i)]\n\t\\item $g(\\Z^r) \\subseteq h(\\Z^{\\l})$;\n\t\\item $g$ is Deligne; and\n\t\\item for sufficiently large $p$, $V^\\ns(\\F_p)\\neq \\emptyset$ implies $W^\\ns(\\F_p) \\ne\\emptyset$;\n\t\\end{enumerate}\nwhere $W \\subseteq \\bbA^r$ is the variety defined by $g = 0$.\n\\end{corollary}\n\n\\begin{proof}\nWhen $\\l \\ge 3$, this follows immediately by applying Corollary~\\ref{cor:one_step} recursively $(\\l - r)$ times. The fact that $r \\ge 2$ ensures that at each step we are applying Corollary~\\ref{cor:one_step} to a polynomial in at least $3$ variables.\n\n\\noindent When $\\l = 2$, the statement is trivial, since $r = 2$ implies that $h$ is already Deligne, so we can take $g = h$.\n\\end{proof}\n\n\\begin{rem}\nUsing the construction from the proof of Corollary~\\ref{cor:one_step}, the polynomial $g$ of Corollary~\\ref{cor:all_steps} may be written in the form\n\t\\[\n\t\tg(x_1,\\ldots,x_r) = h(x_1,\\ldots,x_r,\\mu_{r+1}(x_1,\\ldots,x_r), \\ldots, \\mu_{\\l}(x_1,\\ldots,x_r)),\n\t\\]\nwhere each $\\mu_j$ is a homogeneous linear polynomial. We will use this precise form in our proof of Theorem~\\ref{dimlowthm}, which we are now ready to begin.\n\\end{rem}\n\n\\begin{proof}[Proof of Theorem \\ref{dimlowthm}]\n\n\\noindent Let $g \\in \\Z[x_1,\\ldots,x_r]$ be as in Corollary~\\ref{cor:all_steps}, and let $W \\subseteq \\bbA^r$ be the variety defined by $g = 0$. Throughout this proof we use the notation $\\tilde{\\bsx}=(x_1,\\dots,x_r)$ and $\\bsx=(x_1,\\dots,x_{\\l})$.\n\n\\noindent As mentioned in the remark above, $g$ may be given by $g(\\tilde{\\bsx})=h(M\\tilde{\\bsx})$, where $$M(x_1,\\dots,x_r)=(x_1,\\dots,x_r,\\mu_{r+1}(x_1,\\dots,x_r), \\dots, \\mu_{\\l}(x_1,\\dots,x_r)) $$ for linear forms $\\mu_{r+1},\\dots,\\mu_{\\l}$. Note that $g$ and the linear forms have been constructed once and for all from $h$, so any quantities depending on them implicitly depend only on $h$. \n\n\\noindent Let $X = X(h)$ be the set of primes $p$ for which \n\t\\begin{itemize} \n\t\\item $p\\mid k$;\n\t\\item $g^k$ is not smooth modulo $p$; or\n\t\\item $W^\\ns(\\F_p) = \\emptyset$ and $m_p\\neq k$ for all $\\bsz_p\\in V(\\Z_p)$. \n\t\\end{itemize}\n The first item clearly defines a finite set, the second item defines a finite set because $g$ is Deligne (see Definition~\\ref{defn:smoothDeligne}). If $r\\geq 3$, then the third item defines a finite set by Lemma \\ref{lem:equiv} and the fact that Deligne polynomials in $r\\geq 3$ variables are geometrically irreducible, as seen in the proof of Corollary \\ref{3var}. If $r=2$, then item (iii) of Corollary \\ref{cor:all_steps}, Lemma \\ref{lem:equiv}, and the hypotheses of Theorem \\ref{dimlowthm} ensure that the third item defines a finite set. Thus, $X$ is finite.\n \n\\noindent In order to construct auxiliary polynomials $h_d$ for $d \\in N$, we first choose $\\Z_p$-roots of $h$ as follows: If $p \\in X$, then choose a point $\\bsz_p \\in V(\\Z_p)$ arbitrarily; such points exist because $h$ is intersective. For $p \\notin X$ with $W^\\ns(\\F_p)\\neq \\emptyset$, choose $\\tilde{\\bsz}_p \\in W(\\Z_p)$ to be a Hensel lift of a nonsingular point on $W(\\F_p)$, then set $\\bsz_p=M\\tilde{\\bsz}_p\\in V(\\Z_p).$ Finally, for all remaining $p\\notin X$, fix $\\bsz_p \\in V(\\Z_p)$ with $m_p=k$. \n\n\\noindent For each prime $p$, by definition of multiplicity, we have a decomposition of the form \\begin{equation}\\label{hdecomp} h(\\bsx+\\bsz_p)=\\sum_{m_p\\leq |\\bsi| \\leq k} b_{\\bsi}\\bsx^{\\bsi} \\end{equation} for $b_{\\bsi}\\in \\Z_p$. However, the substitution $\\bsx=M\\tilde{\\bsx}$ could cause some homogeneous parts to identically vanish, so we define $\\tilde{m}_p$ to be the multiplicity of $\\bszero$ as a root of $h(M\\tilde{\\bsx}+\\bsz_p)$, so in particular \\begin{equation}h(M\\tilde{\\bsx}+\\bsz_p)=\\sum_{m_p\\leq |\\bsi| \\leq k} b_{\\bsi}(M\\tilde{\\bsx})^{\\bsi}=\\sum_{\\tilde{m}_p\\leq |\\bsi|\\leq k} a_{\\bsi} \\tilde{\\bsx}^{\\bsi},\n\\end{equation} where $a_{\\bsi}\\neq 0$ for some $\\bsi$ with $|\\bsi|=\\tilde{m}_p$. We quickly note that $\\tilde{m}_p=m_p$ for all $p\\notin X$. If $p\\notin X$ with $m_p=k$, the degree-$k$ part of $h(M\\tilde{\\bsx}+\\bsz_p)$ is the same as the degree-$k$ part of $g$. If $p\\notin X$ and $\\bsz_p=M\\tilde{\\bsz}_p$ as above, then $h(M\\tilde{\\bsx}+\\bsz_p)$ is precisely $g(\\tilde{\\bsx}+\\tilde{\\bsz}_p)$, and in particular the linear part does not vanish modulo $p$.\n\n\\noindent To account for this possible increase in multiplicity for primes $p\\in X$, we define a completely multiplicative function $\\tilde{\\lambda}(d)$ by setting $\\lambda(p)=p^{\\tilde{m}_p}$ for all primes $p$. We define $\\{\\bsr_d\\}_{d\\in \\N}$ from $\\{\\bsz_p\\}_{p\\in\\P}$ as usual from the Chinese remainder theorem, then define the slightly modified auxiliary polynomials $\\{\\tilde{h}_d\\}_{d\\in \\N}$ by $$\\tilde{h}_d(\\bsx)=h(\\bsr_d+d\\bsx)\/\\tilde{\\lambda}(d).$$ We note that $\\tilde{h}_d$ can potentially have non-integer coefficients, with denominators divisible by primes in $X$. However, the analog of Proposition \\ref{inh}, and the deduction of Lemma \\ref{mainit} from Lemma \\ref{L2I} and Proposition \\ref{inh}, still hold because $d\\mid \\tilde{\\lambda}(d)$ and $\\tilde{\\lambda}$ is completely multiplicative. \n\n\\noindent We now let $d'=\\prod_{p\\mid d} p^{(\\tilde{m}_p-m_p+1)\\text{ord}_p(d)}\\leq d^{k},$ and we define $$g_d(\\tilde{\\bsx})=\\tilde{h}_d(\\bss_d+M\\tilde{\\bsx})=h(\\bsr_{d'}+dM\\tilde{\\bsx})\/\\tilde{\\lambda}(d),$$ where $\\bss_d$ satisfies $\\bsr_{d'}=\\bsr_d+d\\bss_d$. We will establish the following properties of $g_d$: \n\n\\begin{enumerate}[(i)]\n\\item $g_d(\\Z^r)\\subseteq \\tilde{h}_d(\\Z^{\\l})$,\n\n\\item $g_d$ has integer coefficients,  \n\n\\item $g_d$ is Deligne modulo $p$ for all $p\\notin X$,\n\n\\item The coefficients of $g_d$ are of size $O_h(d^{k^2})$,\n\n\\item $\\text{cont}(g_d)\\ll_h 1$.\n\n\\end{enumerate} \n\n\\noindent Unlike Proposition \\ref{prop:dim_lower_0}, these efforts cannot be applied ``externally'' to immediately yield Theorem \\ref{dimlowthm} because the family $\\{g_d\\}_{d\\in \\N}$ is not necessarily the set of auxiliary polynomials of a single intersective polynomial. However, the enumerated properties of this family make it perfectly suited for us to apply our efforts ``internally'', using the estimates enumerated in Theorem \\ref{standalonethm}, as follows: \n\n\\begin{itemize} \\item[(1)] Replace all occurrences of $h_d$ in the proof of Theorem \\ref{more} with $\\tilde{h}_d$. The fact that $\\tilde{h}_d$ potentially has non-integer coefficients is not a problem, as the analog of Proposition \\ref{inh} still holds, and as explained in the next step.\n\n\\item[(2)] When proving Lemma \\ref{L2I} (the only piece of the proof of Theorem \\ref{more} that requires integer coefficients or a nonsingularity condition), use that $(A-A)\\cap g_d(\\Z^r) \\subseteq (A-A)\\cap \\tilde{h}_d(\\Z^{\\l}) \\subseteq\\{0\\},$ then do the remainder of the proof with $h_d$ replaced by $g_d$. For this purpose, properties (ii)-(v) above assure that $g_d$ functions as if it were the auxiliary polynomial of a strongly Deligne polynomial in $r$ variables. In particular, the conclusion of Lemma \\ref{L2I} holds with $\\theta(k,\\l,\\delta)$ replaced by $\\theta(k,r,\\delta)$.\n\n\\item[(3)] The remainder of the argument is identical, and Theorem \\ref{dimlowthm} follows. \n\\end{itemize}  \n\n\n\\noindent Our task is now reduced to verifying properties (i)-(v). Properties (i) and (iv) are immediate from the definition of $g_d$ and $\\tilde{h}_d$. We next simultaneously establish (ii) and the property \n\\begin{equation}\\label{ordbound} \\text{ord}_p(\\text{cont}(g_d))\\ll_{h,p} 1  \\text{ for all } p\\in \\P. \\end{equation}\nWhen we later establish (iii), it immediately combines with (\\ref{ordbound}) to yield (v), because $p\\nmid \\text{cont}(g_d)$ if $g_d$ is Deligne modulo $p$. We fix $p\\in \\P$ and set $j=\\text{ord}_p(d)$. By (\\ref{hdecomp}), we have  \n\\begin{equation}\\label{longeq} g_d(\\tilde{\\bsx})=\\tilde{h}_d(\\bss_d+M\\tilde{\\bsx}) =\\frac{1}{\\tilde{\\lambda}(d)}h(\\bsr_{d'}+dM\\tilde{\\bsx}) =\\frac{1}{\\tilde{\\lambda}(d)}\\sum_{m_p\\leq |i| \\leq k}b_{\\bsi}(dM\\tilde{\\bsx}+\\bsr_{d'}-\\bsz_p)^{\\bsi}. \n\\end{equation}\n Since $p^j\\mid d$ and $p^{(\\tilde{m}_p-m_p+1)j}$ divides all coordinates of $\\bsr_{d'}-\\bsz_p$, all terms in the summation apart from \\begin{equation}\\label{vanish} \\sum_{m_p\\leq|\\bsi|\\leq \\tilde{m}_p-1}b_{\\bsi}(dM\\tilde{\\bsx})^{\\bsi} \\end{equation} have coefficients divisible by $p^{j\\tilde{m}_p}$, and the polynomial (\\ref{vanish}) identically vanishes by definition of $\\tilde{m}_p$.  Since $\\text{ord}_p(\\tilde{\\lambda}(d))=j\\tilde{m}_p$, all coefficients of $g_d$ have nonnegative $p$-adic valuation. Since $p\\in \\P$ was arbitrary, it follows that $g_d$ has integer coefficients. \n\n\\noindent Further,  we see in (\\ref{longeq}) that all degree-$\\tilde{m}_p$ terms have a factor of $p^j$ apart from those arising from  $$\\frac{d^{\\tilde{m}_p}}{\\tilde{\\lambda}(d)}\\sum_{|\\bsi|=\\tilde{m}_p} b_{\\bsi} (M\\tilde{\\bsx})^{\\bsi}= \\frac{d^{\\tilde{m}_p}}{\\tilde{\\lambda}(d)}\\sum_{|\\bsi|=\\tilde{m}_p} a_{\\bsi} \\tilde{\\bsx}^{\\bsi}, $$\nwhere $a_{\\bsi}\\neq 0$ for some $\\bsi$ with $|\\bsi|=\\tilde{m}_p$. \n\n\\noindent Since $p\\nmid (d^{\\tilde{m}_p}\/\\tilde{\\lambda}(d))$, we have that $$\\text{ord}_p(\\text{cont}(g_d))\\leq v:=\\min_{|\\bsi|=\\tilde{m}_p}\\text{ord}_p(a_{\\bsi}),$$ provided $j>v$. Alternatively, if $j\\leq v $, then $\\text{ord}_p(\\text{cont}(g_d))$ is at most $kv$ plus the minimum $p$-adic valuation of the degree-$k$ coefficients of $g$, which establishes (\\ref{ordbound}). \n\n\n\n\n\\noindent Our task is now reduced to verifying property (iii), for which we fix $p\\notin X$, and proceed similarly to the proof of Proposition \\ref{thm:main}. Since $g_d^k$ is precisely $\\frac{d^k}{\\tilde{\\lambda}(d)}g^k$, we know that if $p\\nmid d$ or $m_p=k$, then $g_d^k$ modulo $p$ is a nonzero multiple of $g^k$, hence remains smooth. Therefore, $g_d$ is Deligne modulo $p$. \n\n\\noindent The remaining case is when $p\\mid d$ and $\\bsz_p=M\\tilde{z}_p$, where $\\tilde{z}_p\\in W(\\Z_p)$ is a Hensel lift of a nonsingular point of $W(\\F_p)$, so in particular the linear part of $g(\\tilde{\\bsx}+\\tilde{z}_p)=h(M\\tilde{\\bsx}+\\bsz_p)$ does not identically vanish modulo $p$. \n\n\\noindent Using (\\ref{hdecomp}), letting $j=\\text{ord}_p(d)$, we note that $\\text{ord}_p(\\tilde{\\lambda}(d))=j$ and $p^j$ divides all coordinates of $ \\bsr_{d'}-\\bsz_p$, and we have $$g_d(\\tilde{\\bsx})=\\frac{1}{\\tilde{\\lambda}(d)}\\sum_{1\\leq |\\bsi|\\leq k} b_{\\bsi}(dM\\tilde{\\bsx}+\\bsr_{d'}-\\bsz_p)^{\\bsi}=p^jf(\\tilde{\\bsx})+\\frac{d}{\\tilde{\\lambda}(d)}\\sum_{|\\bsi|=1}b_{\\bsi}(M\\tilde{\\bsx})^{\\bsi} +C $$ for some $f\\in \\Z_p[x_1,\\dots,x_r]$ and constant $C$. In particular, modulo $p$, the highest-degree part of $g_d$ is a nonzero multiple of the nonvanishing linear part of $g(\\tilde{\\bsx}+\\tilde{\\bsz}_p)$, hence $g_d$ is Deligne modulo $p$. All five properties of $g_d$ are now verified and the proof of Theorem \\ref{dimlowthm} is complete.\n \\end{proof}\n  \n\\section{Exponential sum estimates} \\label{expest} In this final section, we establish the exponential sum estimates claimed in Theorem \\ref{standalonethm}, which we then use to deduce (\\ref{SmajII}) and (\\ref{SminII}). This effort consists primarily of careful multivariate adaptations of the tools used to prove Theorem 2.7 in \\cite{ricemax}, but we begin with another foray into varieties over finite fields. \n\n\\subsection{Control over gradient vanishing: Part II} \\label{gv2} Since we are sieving away inputs at which the gradient of our polynomial vanishes, but then appealing to Theorem \\ref{delmain}, which is a complete exponential sum estimate, it is important for us to have an upper bound on the number of points our sieve might be throwing away. With this in mind, we make the following definition. \n\n\n\\begin{definition} For a field $F$ and $g\\in F[x_1,\\dots,x_{\\ell}]$ we define the \\textit{gradient locus} of $g$ to be the variety $$\\mathcal{G}_g =\\{\\bsx\\in \\mathbb{A}^{\\ell}: \\grad g(\\bsx)=\\bszero\\}\\subseteq \\mathbb{A}^{\\ell}. $$\n\n\\end{definition}\n\nThe following proposition establishes firm control over the gradient locus of a Deligne polynomial. \n\n\\begin{proposition}\\label{gradcodim}\nSuppose $F$ is a field, $\\ell \\in \\N$, and $g \\in F[x_1,\\ldots,x_\\ell]$ with $\\deg(g)=k \\ge 1$. If $g$ is Deligne, then $\\calG_g=\\emptyset$ or $\\dim \\calG_g = 0$.\n\\end{proposition}\n\n\\begin{proof}\nFirst, assume $g$ is not homogeneous. Let $G(x_0,x_1,\\ldots,x_\\ell)$ be the homogenization of $g$. Thus, we have\n\t\\[\n\t\tg(x_1,\\ldots,x_\\ell) = G(1,x_1,\\ldots,x_\\ell) \\quad\\text{and}\\quad g^k(x_1,\\ldots,x_\\ell) = G(0,x_1,\\ldots,x_\\ell).\n\t\\]\nThe variety\n\t\\[\n\t\t\\What := \\{G = 0\\} \\cap \\{x_0 = 0\\} \\subset \\bbP^\\ell\n\t\\]\nis isomorphic to $\\{g^k = 0\\}$, hence is nonsingular since $g$ is Deligne. Thus, the Jacobian matrix\n\t\\[\n\t\t\\begin{pmatrix}\n\t\t\\dfrac{\\partial G}{\\partial x_0} & \\dfrac{\\partial G}{\\partial x_1} & \\cdots & \\dfrac{\\partial G}{\\partial x_\\ell}\\\\\n\t\t\\\\\n\t\t1 & 0 & \\cdots & 0\n\t\t\\end{pmatrix}\n\t\\]\nhas rank $2$ at every point on $\\What$. In other words, the system\n\t\\begin{align*}\n\t\tG=x_0=\\frac{\\partial G}{\\partial x_1} = \\cdots = \\frac{\\partial G}{\\partial x_\\ell} &= 0\n\t\\end{align*}\nhas no solutions in $\\bbP^\\ell$. The equation $G = 0$ is actually superfluous here; by Euler's theorem on homogeneous functions, we have\n\t\\[\n\t\tkG(x_0,x_1,\\ldots,x_\\ell) = x_0\\dfrac{\\partial G}{\\partial x_0} + x_1\\dfrac{\\partial G}{\\partial x_1} + \\cdots + x_\\ell\\dfrac{\\partial G}{\\partial x_\\ell},\n\t\\]\nso the vanishing of $x_0$ and the $x_1$- through $x_\\ell$-partials would guarantee the vanishing of $G$. Here we use the fact that the characteristic of $F$ does not divide $k$, as included in the definition of the Deligne property. It follows that the system\n\t\\begin{align*} \n\t\tx_0=\\frac{\\partial G}{\\partial x_1} = \\cdots = \\frac{\\partial G}{\\partial x_\\ell} &= 0\n\t\\end{align*}\nhas no solutions in $\\bbP^\\ell$, so the subvariety of $\\bbP^\\ell$ defined by\n\t\\begin{equation}\\label{eq:gradient_vanishing}\n\t\t\\frac{\\partial G}{\\partial x_1} = \\cdots = \\frac{\\partial G}{\\partial x_\\ell} = 0 \n\t\\end{equation}\nis contained in $\\{\\bsx \\in \\bbP^\\ell \\mid x_0 \\ne 0\\} \\cong \\bbA^\\ell$ and has dimension $0$. But, for $\\boldsymbol{\\alpha} = (\\alpha_1,\\ldots,\\alpha_\\ell) \\in \\bbA^\\ell$, we have\n\t\\[\n\t\\frac{\\partial g}{\\partial x_i}(\\boldsymbol{\\alpha}) = \\frac{\\partial G}{\\partial x_i}(1, \\boldsymbol{\\alpha})\n\t\\]\nfor all $1 \\le i \\le \\ell$. Thus, $\\calG_g$ is (isomorphic to) the zero-dimensional subvariety of $\\bbP^\\ell$ given by \\eqref{eq:gradient_vanishing}, concluding the proof in the case that $g$ is not homogeneous.\n\nFinally, suppose $g$ is homogeneous. Again using Euler's theorem on homogeneous functions, we write\n\t\\[\n\t\tkg(x_1,\\ldots,x_\\ell) = x_1\\frac{\\partial g}{\\partial x_1} + \\cdots + x_\\ell\\frac{\\partial g}{\\partial x_\\ell}.\n\t\\]\nThus, if all partials of $g$ vanish at $\\bsx$, then $g(\\bsx) = 0$ as well. By hypothesis, $g = g^k$ is smooth, so there are no common zeroes of $g, \\frac{\\partial g}{\\partial x_1}, \\ldots, \\frac{\\partial g}{\\partial x_\\ell}$ in $\\bbP^\\ell$, so in $\\bbA^\\ell$ the only possible common zero is the origin. Therefore, $\\calG_g$ contains at most one point.\n\\end{proof}\n\nProposition \\ref{gradcodim} combines with B\\'ezout's Theorem (Lemma \\ref{lem:bezout}) to yield the following estimate on the size of the gradient vanishing locus for a Deligne polynomial over a finite field, which yields Lemma \\ref{gradconst} as a special case. \n  \n\\begin{corollary}\\label{gradcor} If $\\l \\geq 1$ and $g\\in \\F_q[x_1,\\dots,x_{\\l}]$ is a Deligne polynomial of degree $k\\geq 1$, then $|\\mathcal{G}_g|$ is bounded by a constant depending only $k$ and $\\ell$.\n\\end{corollary}\n\n\n\\subsection{Major arc estimates} In this section we establish item (\\ref{majitem}) of Theorem \\ref{standalonethm}. Derivations of asymptotic formulas of this type typically rely on partial summation, so we begin with a multivariate version thereof, proven by induction from the usual formula.\n \n\\begin{lemma}[Multivariable Partial Summation] \\label{mps}\n\nSuppose $\\ell\\in \\N$ and $a:\\N^{\\ell}\\to \\C$. Suppose further that $\\psi: \\R^{\\ell}\\to \\C$ is $C^{\\ell}$. For any $X>0$, we have \\begin{align*}\\sum_{\\bsn \\in [1,X]^{\\ell}} a(\\bsn)\\psi(\\bsn)&= A(X,\\dots,X)\\psi(X,\\dots,X) \\\\ &\\qquad +\\sum_{i=1}^{\\ell} (-1)^i\\sum_{1\\leq j_1<\\cdots<j_i\\leq \\ell} \\int_{[0,X]^{i}} A(\\star)\\frac{\\partial^i \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_i}}(\\star) \\ dx_{j_1}\\cdots dx_{j_i},\\end{align*} where $$A(x_1,\\dots,x_{\\ell})=\\sum_{\\bsn \\in [1,x_1]\\times \\cdots \\times [1,x_{\\ell}]} a(\\bsn)$$ and $\\star=(X,\\dots, x_{j_1},\\dots ,x_{j_{i}}, \\dots, X),$ with $x_{j_1},\\dots,x_{j_i}$ plugged into coordinate positions $j_1,\\dots,j_i$ and all other coordinates evaluated at $X$.\n\n\\end{lemma}\n\n\\begin{proof} We induct on $\\ell$. The base case $\\ell=1$ is the usual partial summation formula $$\\sum_{1\\leq n \\leq X} a(n)\\psi(n)= A(X)\\psi(X)-\\int_0^X A(x)\\psi'(x) \\ dx. $$ Fix $\\ell\\geq 2$ and assume the formula holds for $\\ell-1$. Defining some notation before proceeding, let  $$\\tilde{A}(x_1,\\dots,x_{\\ell-1},n_{\\ell})=\\sum_{\\bsn \\in [1,x_1]\\times \\cdots \\times [1,x_{\\ell-1}]} a(\\bsn,n_{\\ell}), $$ let $$\\tilde{I}(j_1,\\dots,j_i,n_{\\ell})=\\int_{[0,X]^{i}} \\tilde{A}(\\star,n_{\\ell})\\frac{\\partial^i \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_i}}(\\star,n_{\\ell}) \\ dx_{j_1}\\cdots dx_{j_i}, $$ and let $$I(j_1,\\dots,j_i)=\\int_{[0,X]^{i}} A(\\star)\\frac{\\partial^i \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_i}}(\\star) \\ dx_{j_1}\\cdots dx_{j_i}, $$ where $A$ and $\\star$ are as defined in the statement of the lemma. By our inductive hypothesis, we have \\begin{align*} \\sum_{\\bsn \\in [1,X]^{\\ell}} a(\\bsn)\\psi(\\bsn)&= \\sum_{1\\leq n_{\\ell}\\leq X} \\sum_{\\bsn \\in [1,X]^{\\ell-1}} a(\\bsn, n_{\\ell})\\psi(\\bsn, n_{\\ell}) \\\\ &=\\sum_{1\\leq n_{\\ell}\\leq X} \\Big( \\tilde{A}(X,\\dots,X,n_{\\ell})\\psi(X,\\dots,X,n_{\\ell}) +\\sum_{i=1}^{\\ell-1} (-1)^i\\sum_{1\\leq j_1<\\cdots<j_i\\leq \\ell-1} \\tilde{I}(j_1,\\dots,j_i,n_{\\ell})\\Big) . \\end{align*} We now apply the standard single-variable formula to the first term and each individual integral, yielding \\begin{equation}\\label{fterm} \\sum_{1\\leq n_{\\ell}\\leq X} \\tilde{A}(X,\\dots,n_{\\ell})\\psi(X,\\dots,n_{\\ell}) = A(X,\\dots,X)\\psi(X,\\dots,X)-\\int_0^X A(X,\\dots,x_{\\ell})\\frac{\\partial\\psi}{\\partial x_{\\ell}}(X,\\dots,x_{\\ell}) dx_{\\ell},\\end{equation} and \\begin{align*}\\sum_{1\\leq n_{\\ell}\\leq X} & (-1)^i\\tilde{I}(j_1,\\dots,j_i,n_{\\ell}) \\\\ &= (-1)^i\\int_{[0,X]^{i}} \\left(A(\\star,X)\\frac{\\partial^i \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_i}}(\\star,X)-\\int_0^X A(\\star,x_{\\ell})\\frac{\\partial^{i+1} \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_i}\\partial x_{\\l}}(\\star,x_{\\ell})dx_{\\ell}\\right)\\ dx_{j_1}\\cdots dx_{j_i} \\\\ &=(-1)^i I(j_1,\\dots,j_i)+(-1)^{i+1}I(j_1,\\dots,j_i,\\ell). \\end{align*} Summing this final expression over $1\\leq i \\leq \\ell-1$ and over all choices of $1\\leq j_1<\\cdots<j_i\\leq \\ell-1$ accounts for all required terms with $1\\leq i \\leq \\ell$ and $1\\leq j_1<\\cdots<j_i\\leq \\ell$, with the single exception of $i=1$ and $j_1=\\ell$, which is precisely the integral present in (\\ref{fterm}), and the induction is complete. \\end{proof} \n\nWe use Lemma \\ref{mps} and the same calculation as in Proposition \\ref{brunprop} to establish our asymptotic formula for sieved multivariate exponential sums near rationals with small denominator. \n\n\\begin{lemma}\\label{Sasym} Suppose $\\ell,k\\in \\N$, $g(\\bsx)=\\sum_{|\\bsi|\\leq k} a_{\\bsi} \\bsx^{\\bsi} \\in \\Z[x_1,\\dots,x_{\\ell}]$, and let $J=\\sum_{|\\bsi|\\leq k} |a_{\\bsi}|$. If $X,Y > 0$, $a,q\\in \\N$, and $\\alpha=a\/q+\\beta$, then \\begin{align*}\\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)}e^{2\\pi i g(\\bsn)\\alpha}&=q^{-\\ell} \\prod_{\\substack{ p\\leq Y \\\\ p^{\\gamma(p)}\\nmid q}}\\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}}\n\\right)\\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q}\\int_{[0,X]^{\\ell}}e^{2\\pi i g(\\bsx)\\beta}d\\bsx\\\\\\\\&\\qquad + O_{k,\\l}\\left(qE(1+JX^{k}|\\beta|)^{\\ell}\\right)  ,\\end{align*} where $E$ is as in Proposition \\ref{brunprop}.\n\n\\end{lemma}\n\n\\begin{proof} We begin by noting that for any $a,q \\in \\N$ and $0\\leq x_1,\\dots,x_{\\ell} \\leq X$, letting $$B=[1,x_1]\\times\\cdots\\times [1,x_{\\ell}],$$ we have\n\\begin{align*}T(x_1,\\dots,x_{\\ell})&:=\\sum_{\\bsn \\in B\\cap W(Y)}e^{2\\pi i g(\\bsn)a\/q}\\\\ &\\phantom{:}=\\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell}} e^{2\\pi i g(\\boldsymbol{s})a\/q} \\left| \\left\\{\\bsn \\in B \\cap W(Y) :  \\bsn \\equiv \\bss \\ (\\text{mod }q) \\right\\} \\right|.\n\\end{align*}\nFor $s\\in W^q(Y)$ we have by the same calculation as Proposition \\ref{brunprop} that \\begin{equation*} \\label{count} \\left| \\left\\{\\bsn \\in B \\cap W(Y) :  \\bsn \\equiv \\bss \\ (\\text{mod }q) \\right\\} \\right|=\\frac{x_1\\cdots x_{\\ell}}{q^{\\ell}}\\prod_{\\substack{p\\leq Y\\\\ p^{\\gamma(p)} \\nmid q}} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right)+E\/q^{\\l-1} ,\\end{equation*} where $E$ is as in Proposition \\ref{brunprop}, whereas for $s\\notin W^q(Y)$ the set is empty.\n\n\\noindent  Therefore, \\begin{equation} \\label{Tx} T(x_1,\\dots,x_{\\ell})=\\frac{x_1\\cdots x_{\\ell}}{q^{\\ell}}\\prod_{\\substack{p\\leq Y\\\\ p^{\\gamma(p)} \\nmid q}} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right)\\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1 \\}^{\\ell}\\cap W^q(Y)} e^{2\\pi i g(\\boldsymbol{s})a\/q}+O\\left(qE\\right). \\end{equation} Letting $\\psi(\\bsn)=e^{2\\pi i g(\\bsn) \\beta}$, we now decompose our sum as $$\\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)}e^{2\\pi i g(\\bsn)\\alpha}=\\sum_{\\bsn \\in [1,X]^{\\ell}} \\left(1_{W(Y)}(\\bsn)e^{2\\pi i g(\\bsn)a\/q}\\right) \\psi(\\bsn)  $$ and apply Lemma \\ref{mps}, yielding \n\\begin{align*} \\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)}e^{2\\pi i g(\\bsn)\\alpha}&=T(X,\\dots,X)\\psi(X,\\dots,X) \\\\ &\\qquad+\\sum_{m=1}^{\\ell} (-1)^m\\sum_{1\\leq j_1<\\cdots<j_m\\leq \\ell} \\int_{[0,X]^{m}} T(\\star)\\frac{\\partial^m \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_m}}(\\star) \\ dx_{j_1}\\cdots dx_{j_m},\n\\end{align*} where $\\star$ is as in Lemma \\ref{mps}. Substituting (\\ref{Tx}) gives the main term \\begin{align*}&q^{-\\ell}\\prod_{\\substack{p\\leq Y\\\\ p^{\\gamma(p)} \\nmid q}} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right)\\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1 \\}^{\\ell}\\cap W^q(Y)} e^{2\\pi i g(\\boldsymbol{s})a\/q} \\Big( X^{\\ell}\\psi(X,\\dots,X) \\\\ &\\qquad + \\sum_{m=1}^{\\ell} (-1)^m\\sum_{1\\leq j_1<\\cdots<j_m\\leq \\ell} X^{\\ell-m}\\int_{[0,X]^{m}} x_{j_1}\\cdots x_{j_m}\\frac{\\partial^m \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_m}}(\\star) \\ dx_{j_1}\\cdots dx_{j_m}\\Big). \\end{align*} By iteratively applying integration by parts, this equals \\begin{equation*}q^{-\\ell}\\prod_{\\substack{p\\leq Y\\\\ p^{\\gamma(p)} \\nmid q}} \\left(1-\\frac{j(p)}{p^{\\gamma(p)\\ell}} \\right) \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1 \\}^{\\ell}\\cap W^q(Y)} e^{2\\pi i g(\\boldsymbol{s})a\/q} \\int_{[0,X]^{\\ell}} \\psi(\\bsx) d\\bsx, \\end{equation*} as desired. It remains to bound the error term that results from our substitution of (\\ref{Tx}). This error term is the sum of a first term of order $qE$ and   $2^{\\ell}-1$ terms of the form \\begin{equation*}qE\\left(\\int_{[0,X]^{m}} \\frac{\\partial^m \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_m}}(\\star) \\ dx_{j_1}\\cdots dx_{j_m} \\right). \\end{equation*} Iteratively applying the product rule, we see that $\\frac{\\partial^m \\psi}{\\partial x_{j_1}\\cdots \\partial x_{j_m}}$ is the sum of less than $m!$ terms bounded in absolute value by $(2\\pi k^{m}J|\\beta|)^jX^{jk-m}$ for some $1\\leq j \\leq m$. In particular, each integral is bounded by $$\\ell!\\max_{1\\leq j \\leq \\ell} (2\\pi k^{\\ell}JX^k|\\beta|)^j \\leq \\ell!(1+2\\pi k^{\\ell}JX^k|\\beta|)^{\\ell}, $$ and the error bound follows.\\end{proof}\n\n\\subsection{Local cancellation} In this section, we apply Theorem \\ref{delmain} to establish the necessary cancellation in our sieved local exponential sums, yielding item (\\ref{locitem}) in Theorem \\ref{standalonethm}. We begin by invoking a multivariate version of Hensel's Lemma that allows us to reduce to the case of prime moduli. This statement in particular follows from Theorem~1.1 of \\cite{Conrad}.\n\n\n\n\\begin{lemma}[Multivariable Hensel's Lemma] \\label{hensel} Suppose $\\ell\\in \\N$, $g\\in \\Z[x_1,\\dots,x_{\\ell}]$, $p$ is prime, $\\bsn \\in \\Z^{\\l}$, and $\\gamma,v\\in \\N$ with $v\\geq 2\\gamma-1$. If $$g(\\bsn)\\equiv 0 \\ \\textnormal{mod} \\ p^{2\\gamma-1}$$ and $\\grad g (\\bsn) \\not\\equiv \\bszero$ \\textnormal{mod} $p^{\\gamma}$, then there exists $\\bsm\\in \\Z^{\\l}$ with $g(\\bsm) \\equiv 0$ \\textnormal{mod} $p^v$.\n\\end{lemma}\n\nWe now prove the following multivariate generalization of Lemma 4.3 in \\cite{ricemax}.\n\n\\begin{lemma}\\label{locgen} Suppose $\\ell\\in \\N$, $g\\in \\Z[x_1,\\dots, x_{\\l}]$ with $\\deg(g)=k\\geq 2$, and $Y>0$. If $q\\in \\N$ has prime factorization $q=p_1^{v_1}\\cdots p_r^{v_r}$ with $p_1<\\cdots< p_t\\leq Y < p_{t+1}< \\cdots < p_r$, and $(a,q)=1$, then $$\\left| \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q} \\right| \\leq C_1 \\prod_{i=1}^t \\left((k-1)^{\\l}p_i^{\\l\/2}+j(p_i)\\right)\\prod_{i=t+1}^r C_2(v_i+1)^{\\ell} p_i^{v_i(\\l-1\/k)}, $$ where $C_2=C_2(k)$ and $C_1$ depends only on the moduli at which $\\grad g$ identically vanishes and the primes $p\\leq Y$ dividing $q$ modulo which $g$ is not Deligne. Further, the sum is $0$ if $v_i\\geq 2\\gamma(p_i)$ for some $1\\leq i \\leq t$.\n\n\\end{lemma}\n\n\\begin{proof} Factor $q=p_1^{v_1}\\cdots p_r^{v_r}$ as in the lemma. By the Chinese Remainder Theorem, we have \\begin{equation*} \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\bss)a\/q}= \\prod_{m=1}^{r} \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,p_m^{v_m}-1\\}^{\\ell} \\cap W^{p_m^{v_m}}(Y)}e^{2\\pi i g(\\bss)a_m\/p_m^{v_m}}, \\end{equation*} where $a_1,\\dots,a_r$ are the unique residues satisfying $a\/q \\equiv a_1\/p_1^{v_1}+\\cdots+a_{r}\/p_r^{v_r} \\ \\text{mod }1. $\n\n\\noindent Suppose $p^v=p_m^{v_m}$ with $\\gamma(p)>1$ and $v<2\\gamma(p)$. By definition of $\\gamma$, $\\grad g$ identically vanishes modulo $p^{\\gamma(p)-1}$. Since $p^{2\\gamma(p)-1}\\leq p^{3(\\gamma(p)-1)}$, we can bound $p^{v}$ by the cube of a modulus at which $\\grad g$ identically vanishes, trivially bound the corresponding sum, and absorb it into the constant $C_1$ in the conclusion of the lemma. \n\n\\noindent Next suppose $p^v=p_m^{v_m}$ with $p\\leq Y$ and $v=\\gamma(p)=1$. Recalling that $j(p)$ is the number of zeros of $\\grad g$ modulo $p$  and applying Theorem \\ref{delmain}, we have for $p\\nmid b$ that  $$\\left|\\sum_{\\bss \\in \\{0,\\dots,p-1\\}^{\\ell}\\cap W^{p}(Y)}e^{2\\pi i g(\\bss)b\/p}\\right| \\leq (k-1)^{\\l}p^{\\ell\/2}+j(p), $$ provided $g$ is Deligne modulo $p$, and the remaining such primes are absorbed into $C_1$. \n\n\\noindent Now suppose that $p^v=p_m^{v_m}$ with $p\\leq Y$ and $v\\geq 2\\gamma(p)$, and let $w=2\\gamma(p)-1$. If $\\bss \\in \\{0,\\dots, p^v-1\\}^{\\ell}$ and $\\tilde{\\bss}$ is the reduced residue class of $\\bss$ modulo $p^{w}$, then we have that $g(\\bss)\\equiv p^{w}t+g(\\tilde{\\bss}) \\ (\\text{mod }p^v)$ for some $0\\leq t\\leq p^{v-w}-1$. Conversely,  if $\\tilde{\\bss}\\in \\{0,\\dots, p^w-1\\}^{\\ell}$ with  $\\grad g(\\tilde{\\bss})\\not\\equiv \\bszero \\  (\\text{mod }p^{\\gamma(p)})$, then for every $0\\leq t \\leq p^{v-w}-1$, Lemma \\ref{hensel} applied to the polynomial $g(\\bsx)-(p^{w}t+g(\\tilde{\\bss}))$ yields $\\bss \\in \\{0,\\dots,p^v-1\\}^{\\ell}$ with $g(\\bss)\\equiv p^{w}t+g(\\tilde{\\bss}) \\ (\\text{mod }p^v)$.\n \n\\noindent In other words, the map $F$ on $\\Z\/p^{v-w}\\Z$ defined by $g(p^{w}t+\\tilde{\\bss})\\equiv p^{w}F(t)+g(\\tilde{\\bss}) \\ (\\text{mod }p^v)$ is a bijection.  In particular, if $p\\nmid b$, then \\begin{align*}\\sum_{\\bss\\in \\{0,\\dots, p^v-1\\}^{\\ell} \\cap W^{p^v}(Y)}e^{2\\pi i g(\\bss)b\/p^v} &= \\sum_{\\substack{\\tilde{\\bss}\\in \\{0,\\dots,p^w-1\\}^{\\ell} \\\\ \\grad g(\\tilde{\\bss})\\not\\equiv \\bszero \\ (\\text{mod }p^{\\gamma(p)})}} \\sum_{t=0}^{p^{v-w}-1}e^{2\\pi \\i g(p^{w}t+\\tilde{\\bss})b\/p^v}\\\\  &=\\sum_{\\substack{\\tilde{\\bss}\\in \\{0,\\dots,p^w-1\\}^{\\ell} \\\\ \\grad g(\\tilde{\\bss})\\not\\equiv \\bszero \\ (\\text{mod }p^{\\gamma(p)})}} \\sum_{t=0}^{p^{v-w}-1}e^{2\\pi \\i \\left(p^{w}t+g(\\tilde{\\bss})\\right)b\/p^v} \\\\ &=0,\\end{align*} where the last equality is the fact that the sum in $t$ runs over the full collection of $p^{v-w}$-th roots of unity. \n\n\\noindent Finally, suppose $p^v=p_m^{v_m}$ with $p> Y$. We note that $W^{p^v}(Y)=\\N$ and we only exploit cancellation in a single variable. To this end, for each $\\tilde{\\bss}=(s_2,\\dots,s_{\\ell})\\in \\{0,\\dots,p^v-1\\}^{\\ell-1}$, we define $\\tilde{g}$ by $\\tilde{g}(x)=g(x,\\tilde{\\bss})$.  Utilizing the  standard single-variable complete sum estimate (see \\cite{Chen} for example), we have for $b\\nmid p$ that  \\begin{align*}\\left|\\sum_{\\bss\\in\\{0,\\dots,p^v-1\\}^{\\ell}}e^{2\\pi i g(\\bss)b\/p^v}\\right| &\\leq  \\sum_{\\tilde{\\bss}\\in \\{0,\\dots,p^v-1\\}^{\\ell-1}} \\left| \\sum_{s=0}^{p^v-1} e^{2\\pi i \\tilde{g}(s)b\/p^v} \\right| \\\\ &\\ll_k  p^{v(1-1\/k)}\\sum_{\\tilde{\\bss}\\in \\{0,\\dots,p^v-1\\}^{\\ell-1}}\\gcd(\\text{cont}(\\tilde{g}),p^v)^{1\/k}. \\end{align*} \nTo analyze the remaining sum, we note that at the expense of the term $\\gcd(\\text{cont}(g),p^v)^{1\/k}$ in our final estimate, we can cancel factors of $p$ from the coefficients of $g$ and assume that $p\\nmid \\text{cont}(g)$. In this case, suppose $a_{\\bsi}=a_{i_1,\\dots,i_{\\ell}}$ with $0<|\\bsi|\\leq k$ is a coefficient of $g$, corresponding to $x_1^{i_1}\\cdots x_{\\l}^{i_{\\l}}$, that is not divisible by $p$. Further, assume that $i_1>0$, as if $i_1=0$ then we could just relabel our coordinates. In this case, for each $0\\leq w \\leq v$, $\\gcd(\\text{cont}(\\tilde{g}),p^v)=p^w$ only if $p^w\\mid s_2^{i_2}\\cdots s_{\\ell}^{i_{\\l}}$, so in particular $p^{\\lceil w\/k \\rceil}\\mid s_2\\cdots s_{\\ell}$, which occurs for fewer than $(w+1)^{\\ell-1}p^{v(\\ell-1)-w\/k}$ choices of $\\tilde{\\bss}$. In particular, \\begin{align*}\\sum_{\\tilde{s}\\in \\{0,\\dots,p^v-1\\}^{\\ell-1}}\\gcd(\\text{cont}(\\tilde{g}),p^v)^{1\/k}&\\leq \\gcd(\\text{cont}(g),p^v)^{1\/k} \\sum_{w=0}^{v} (w+1)^{\\ell-1}p^{v(\\ell-1)-w\/k}p^{w\/k} \\\\ &\\leq (v+1)^{\\ell}\\gcd(\\text{cont}(g),p^v)^{1\/k}p^{v(\\ell-1)}. \\end{align*}  The $\\gcd(\\text{cont}(g),p^v)^{1\/k}$ term can be absorbed into $C_1$, and the remaining bound on the exponential sum modulo $p^v$ is a constant depending on $k$ times $p^{v(1-1\/k)}(v+1)^{\\ell} p^{\\ell(v-1)}=(v+1)^{\\ell}p^{v(\\ell-1\/k)}$, as required. Having accounted for all prime divisors of $q$, the proof is complete.\n \\end{proof} \n\nLemma \\ref{locgen} combines with Lemma \\ref{gradconst} as well as the estimates $\\prod_{p\\mid q}\\left(1+\\frac{C}{p} \\right)\\leq (q\/\\phi(q))^C$ and $\\prod_{p\\mid q}\\left(1+\\frac{C}{p^{3\/2}} \\right)\\ll_C 1$  to yield item (\\ref{locitem}) of Theorem \\ref{standalonethm}, restated below. \n\n\\begin{corollary}\\label{qcor} If $\\ell\\geq 2$, $g\\in \\Z[x_1,\\dots, x_{\\l}]$ with $\\deg(g)=k\\geq 2$, and $(a,q)=1$, then $$\\left| \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q} \\right| \\leq C_1\\begin{cases} (k-1)^{\\l \\omega(q)}\\Phi(q,\\l) q^{\\ell\/2} &\\text{if }q\\leq Y \\\\ C_2^{\\omega(q)}\\tau(q)^{\\l}q^{\\ell-1\/k} &\\text{for all }q \\end{cases},$$ where $C_2=C_2(k)$, $\\Phi(q,2)=(q\/\\phi(q))^{C_2}$, $\\Phi(q,\\l)\\ll_{k,\\l} 1$ for $\\l\\geq 3$, and $C_1$ depends only on the moduli at which $\\grad g$ identically vanishes and the primes $p\\leq Y$ dividing $q$ modulo which $g$ is not Deligne.\n\n\\end{corollary}\n\n\\subsection{Oscillatory integral estimate} In order to establish (\\ref{SminII}) in the case that $\\alpha$ is close, but not too close, to a rational with very small denominator, we need to control the oscillatory integral in the asymptotic formula given by Lemma \\ref{Sasym}. To achieve this, we invoke the following standard estimate, given for example in Lemma 2.8 of \\cite{vaughan}. \n\n\\begin{lemma}[Van der Corput's Lemma] \\label{vdcl} If  $X>0$, $\\beta\\neq 0$, $k\\in \\N$, and $g\\in \\Z[x]$ with $\\deg(g)=k$, then $$\\left|\\int_0^X e^{2\\pi i g(x)\\beta} dx \\right| \\ll |\\beta|^{-1\/k}. $$\n\n\\end{lemma}\n\nUtilizing Lemma \\ref{vdcl} to exploit cancellation in a single variable, then trivially bounding the integral in the remaining variables, we have the following bound for the integral in the conclusion of Lemma \\ref{Sasym}.\n\n\\begin{corollary} \\label{vdccor} If $X>0$, $\\beta\\neq 0$, $k,\\ell\\in \\N$, and $g\\in \\Z[x_1,\\dots,x_{\\l}]$ with $\\deg(g)=k$, then $$\\left|\\int_{[0,X]^{\\ell}}e^{2\\pi i g(\\bsx)\\beta}d\\bsx\\right| \\ll \\min\\{ X^{\\ell}, X^{\\ell-1}|\\beta|^{-1\/k}\\}.$$\n\n\\end{corollary}\n\n\\subsection{Minor arc estimates}\nIn an effort to establish item (\\ref{minitem}) of Theorem \\ref{standalonethm}, we begin by invoking a variation of the most traditional minor arc estimate, Weyl's Inequality.\n\n\\begin{lemma}[Lemma 3, \\cite{CLR}] \\label{weyl2}  Suppose $k\\in \\N$, $g(x)=a_0+a_1x+\\cdots+a_{k}x^{k}$ with $a_0\\dots,a_k \\in \\R$ and $a_{k} \\in \\N$. If $X>0$, $a,q\\in \\N$ with $(a,q)=1$, and $|\\alpha-a\/q|<q^{-2}$, then \n$$ \\left|\\sum_{n=1}^X e^{2\\pi \\i g(n)\\alpha} \\right| \\ll_{k} X \\left(a_{k}\\log^{k^2}(a_{k}qX)\\left(q^{-1}+X^{-1}+\\frac{q}{a_kX^k}\\right) \\right)^{2^{-k}}.$$\n\\end{lemma}\n\nWe now carefully adapt Lemma \\ref{weyl2} to our particular sieve, and to the multivariate setting, though as in Corollary \\ref{vdccor}, we ultimately only exploit cancellation in a single variable. \n\n\\begin{lemma}\\label{weyl3} Suppose $k,\\ell\\in \\N$ and $g(\\bsx)=\\sum_{|\\bsi|\\leq k} a_{\\bsi} \\bsx^{\\bsi} \\in \\Z[x_1,\\dots,x_{\\ell}]$ with $\\deg(g)=k$. Suppose further that $X,Y,Z\\geq 2$, $YZ\\leq X$, and $a,q\\in \\N$ with $(a,q)=1$, and let $J=\\sum_{|\\bsi|\\leq k} |a_{\\bsi}|$. If $|\\alpha-a\/q|<q^{-2}$, then $$\\left|\\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)} e^{2\\pi \\i g(\\bsn)\\alpha} \\right| \\ll_{k,\\ell} \\textnormal{cont}(g)^6(\\log Y)^{ek} X^{\\ell}\\left(e^{-\\frac{\\log Z}{\\log Y}}+\\left(J\\log^{k^2}(JqX)\\left(q^{-1}+\\frac{Z}{X}+\\frac{qZ^k}{X^k}\\right) \\right)^{2^{-k}} \\right). $$\n\\end{lemma}\n\n\\begin{proof} Suppose $k,\\ell\\in \\N$ and $g(\\bsx)=\\sum_{|\\bsi|\\leq k} a_{\\bsi} \\bsx^{\\bsi} \\in \\Z[x_1,\\dots,x_{\\ell}]$ with $\\deg(g)=k$. We begin by conducting an invertible (over $\\Z$) change of variables to reduce to the case where the $x_1^k$ coefficient $a_{(k,0,\\dots,0)}$ is nonzero. To this end, consider the polynomial $\\tilde{g}\\in \\Z[x_2,\\dots,x_{\\ell}]$ defined by $\\tilde{g}(x_2,\\dots,x_{\\ell})=g^k(1,x_2,\\dots,x_{\\ell})$, where $g^k$ denotes the top degree homogeneous part of $g$, noting that $\\tilde{g}$ is not identically zero. Let $(c_2,\\dots,c_{\\ell})\\in \\{0,1,\\dots,k\\}^{\\ell-1}$ be such that $\\tilde{g}(c_2,\\dots,c_{\\ell})\\neq 0$. \n\n\\noindent As an aside, the existence of such a ``small integer non-root'' of a general nonzero multivariate polynomial $F\\in \\Z[x_1,\\dots,x_j]$ can be shown via induction, which we sketch here. The base case $j=1$ corresponds to nonzero univariate polynomials, which have at most $k$ roots, hence at least one non-root in $\\{0,1,\\dots,k\\}$. Then, for higher degrees, fix one variable that appears at least once in $F$ (without loss of generality, assume $x_1$ appears at least once), let $d$ be the degree of $F$ as a polynomial in $x_1$ only, and let $\\tilde{F}(x_2,\\dots,x_j)$ be the polynomial of degree at most $k-d$ that forms the $x_1^d$ coefficient. By the inductive hypothesis, we can choose $(m_2,\\dots,m_j)\\in \\{0,\\dots,k\\}^{j-1}$ such that $\\tilde{F}(m_2,\\dots,m_j)\\neq 0$. Then, $F(x_1,m_2,\\dots,m_j)$ is a nonzero degree-$d$ polyomial in $x_1$, which has a non-root in $\\{0,\\dots,k\\}$, completing the induction. \n\n \n\\noindent Back to the proof at hand, we see that the change of variables $x_1=y_1$ and $x_j=y_j+c_jy_1$ for $2\\leq j \\leq \\ell$ yields a $y_1^k$ coefficient of $\\tilde{g}(c_2,\\dots,c_{\\ell})\\neq 0$. Let $M$ denote the $\\ell\\times \\ell$ matrix satisfying $M\\bsx =  \\boldsymbol{y}$ corresponding to the described change of variables, and let $f(y_1,\\dots,y_{\\l})=\\sum_{|\\bsi|\\leq k} b_{\\bsi} y^{\\bsi}$ be the polynomial satisfying $f(\\boldsymbol{y})=g(M^{-1}\\boldsymbol{y})$. By taking the complex conjugate of the relevant exponential sum if necessary, we can assume that $b=b_{(k,0,\\dots,0)}>0$. Further, the effect of the transformation on the size of this coefficient is well-controlled, in that $b\\ll_{k,l} J$.  \n\n\\noindent Let $T=M([1,X]^{\\ell})$, so $$ \\sum_{\\bsn \\in [1,X]^{\\ell} \\cap W(Y)}e^{2\\pi \\i g(\\bsn)\\alpha}=\\sum_{\\bsn \\in T \\cap W(Y)}e^{2\\pi \\i f(\\bsn)\\alpha},$$ where $W(Y)$ is defined on each side in terms of the corresponding polynomial. \n\n\\noindent Let $\\tilde{T}$ denote the projection of $T$ onto the last $\\ell-1$ coordinates, noting that $|\\tilde{T}|\\leq (2kX)^{\\ell-1}$ due to the details of our change of variables. For each fixed $\\tilde{\\bsn}=(n_2,\\dots,n_{\\ell})\\in \\N^{\\ell-1}$, we let $I=\\{n\\in \\N: (n,\\tilde{\\bsn})\\in T\\}$, which is an interval of integers of length at most $X$, we let $\\tilde{W}(Y)=\\{n\\in \\N: (n,\\tilde{\\bsn})\\in W(Y)\\}$, and we let $\\tilde{f}(x)=f(x,\\tilde{\\bsn})$. We see trivially that \\begin{equation}\\label{T} \\left|\\sum_{\\bsn \\in T \\cap W(Y)} e^{2\\pi \\i f(\\bsn)\\alpha} \\right| \\leq (2kX)^{\\ell-1}\\max_{\\tilde{\\bsn}\\in \\tilde{T}} \\left|\\sum_{n \\in I \\cap \\tilde{W}(Y)} e^{2\\pi \\i \\tilde{f}(n)\\alpha} \\right|. \\end{equation} \n\n\\noindent We now proceed with $\\tilde{\\bsn}=(n_2,\\dots,n_{\\ell})$ fixed, and we define $L$ and $m$ so that $I=[m,L+m]$, so in particular $L\\leq X$. All subsequent conclusions will be independent of $\\tilde{\\bsn}$. Let  $P$ be the set of products $p_1^{\\gamma(p_1)}\\cdots p_s^{\\gamma(p_s)}$ for primes $p_1<\\cdots<p_s\\leq Y$, let $P_1$ denote the set of elements of $P$ that are at most $Z$, and let $P_2$ denote the set of elements of $P$ that are greater than $Z$. \n\n\\noindent By inclusion-exclusion, we have \\begin{equation}\\label{P1P2} \\left|\\sum_{n \\in I \\cap \\tilde{W}(Y)} e^{2\\pi \\i \\tilde{f}(n)\\alpha}\\right| =\\left|\\sum_{D\\in P} (-1)^{\\omega(D)} \\sum_{\\substack{n\\in I \\\\ \\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)}} e^{2\\pi \\i \\tilde{f}(n)\\alpha}\\right|,\n\\end{equation}\nwhere $\\omega(D)$ is the number of distinct prime factors of $D$. For $D\\in P_1$, we use the fact that the set of $n$ for which $\\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)$ is contained in the set of $n$ for which $\\tilde{f}'(n)\\equiv 0 \\ (\\text{mod }D)$. Noting that $\\tilde{f}'$ can have at most $k$ roots modulo any prime at which it does not identically vanish, we have \n\\begin{align*}\\left|\\sum_{D\\in P_1} (-1)^{\\omega(D)} \\sum_{\\substack{n\\in I \\\\ \\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)}} e^{2\\pi \\i \\tilde{f}(n)\\alpha}\\right| \\ll_k (\\text{cont}(g))^2 \\sum_{D \\in P_1} k^{\\omega(D)} \\max_{0\\leq c \\leq D}\\left| \\sum_{n=0}^{L\/D} e^{2\\pi \\i \\tilde{f}(Dn+m+c)\\alpha}\\right|,\n\\end{align*} where the $\\text{cont}(g)^2$ term accounts for the primes $p$ for which $\\gamma(p)>1$ by Proposition \\ref{idzero}. Further, we see from Lemma \\ref{weyl2} and the estimate $1\\leq b\\ll_{k,l} J$ that \\begin{align*}\\sum_{D \\in P_1} k^{\\omega(D)} \\max_{0\\leq c \\leq D}\\left| \\sum_{n=0}^{L\/D} e^{2\\pi \\i \\tilde{f}(Dn+m+c)\\alpha}\\right| &\\ll_{k,l} \\sum_{D\\in P_1} k^{\\omega(D)} \\frac{L}{D} \\left(b\\log^{k^2}(bqL)\\left(q^{-1}+\\frac{D}{L}+\\frac{qD^k}{bL^k}\\right) \\right)^{2^{-k}} \\\\ &\\ll_{k,l} X \\left(J\\log^{k^2}(JqX)\\left(q^{-1}+\\frac{Z}{X}+\\frac{qZ^k}{X^k}\\right) \\right)^{2^{-k}}\\sum_{D\\in P_1} \\frac{k^{\\omega(D)}}{D} \\\\ &\\ll_{k,l} X(\\log Y)^k\\left(J\\log^{k^2}(JqX)\\left(q^{-1}+\\frac{Z}{X}+\\frac{qZ^k}{X^k}\\right) \\right)^{2^{-k}},\\end{align*} where the last inequality uses that if $C>0$, then \\begin{equation}\\label{Z}\\sum_{D\\in P} \\frac{C^{\\omega(D)}}{D} = \\prod_{p\\leq Y} \\left(1+\\frac{C}{p^{\\gamma(p)}}\\right) \\leq   \\prod_{p\\leq Y} \\left(1+\\frac{C}{p}\\right) \\ll (\\log Y)^C.\\end{equation} This combines with (\\ref{T}) to close the book on the contributions to (\\ref{P1P2}) from $P_1$. It remains to account for the contribution to (\\ref{P1P2}) from $P_2$. Because $P_2$ has so many elements, it is crucial for us to exploit the cancellation provided by the term $(-1)^{\\omega(D)}$. \n\n\\noindent To this end, for a  fixed $n\\in I$, let $P^n=\\{D\\in P:  \\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)\\}$, and let $P^n_2=P^n\\cap P_2$. The only issue is the possibility that way more elements of $P^2_n$ have an even number of prime factors than odd, or vice versa, which we show below does not happen.\n\n\\noindent Let $q$ be the largest prime power of the form $p^{\\gamma(p)}$ with $p\\leq Y$, and let $q_n$ be the largest such prime power lying in $P^n$,  noting that $q_n\\leq q\\ll_k \\cont(g)Y$ by Proposition \\ref{idzero}. \\noindent Let $A$ denote the set of elements of $P^n$ that have an even number of prime factors, let $B$ denote the set of elements of $P^n$ that have odd number of prime factors, and let $A'$ and $B'$, respectively, denote the same for elements of $P^n_2$. The quantity we need control of is $\\left||A'|-|B'|\\right|$. \n\n\\noindent  Let $A_1$ be the elements of $A$ that are greater than $Z$ and not divisible by $q_n$, and let $A_2$ be the elements of $A$  that are greater than $q_nZ$ and divisible by $q_n$. Likewise define $B_1$ and $B_2$. The map $D\\to q_nD$ defines an injection from $A_1$ to $B_2$, while the map $D\\to D\/q_n$ defines an injection from $A_2$ to $B_1$. Letting $A_3$ denote all the elements of $A$ greater than $q_nZ$, we have $$|A_3|\\leq |A_1|+|A_2| \\leq |B_1|+|B_2| \\leq |B'|. $$ Symmetrically, we have $|B_3|\\leq |A'|$. Finally, letting $A_4$ and $B_4$ denote the elements of $A'$ and $B'$ satisfying $Z<D\\leq q_nZ$, we have $|A'|=|A_3|+|A_4|\\leq |B'|+|A_4|$ and similarly $|B'|\\leq |A'|+|B_4|$, so the magnitude of $|A'|-|B'|$ is bounded above by $|A_4|+|B_4|$, which is the size of the set $\\bar{P^n}$ of elements of $P^n$ satisfying $Z < D \\le q_nZ$.\n\n\\noindent We now see\n\\begin{align*} \\left|\\sum_{D\\in P_2} (-1)^{\\omega(D)} \\sum_{\\substack{n\\in I \\\\ \\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)}} e^{2\\pi i \\tilde{f}(n)\\alpha} \\right|& =\\left|\\sum_{n\\in I} e^{2\\pi i \\tilde{f}(n)\\alpha}\\sum_{D\\in P^n_2} (-1)^{\\omega(D)}\\right|  \\\\ & \\leq \\sum_{n\\in I} |\\bar{P^n}| \\\\ &= \\sum_{\\substack{D\\in P \\\\ Z< D \\leq qZ}} |\\{n\\in I : \\grad f(n,\\tilde{\\bsn})\\equiv \\bszero \\ (\\text{mod }D)\\}| \\\\ &\\ll_k (\\text{cont}(g))^2 \\sum_{\\substack{D\\in P \\\\ Z< D \\leq qZ}} k^{\\omega(D)}\\left( \\frac{L}{D}+1 \\right) \\\\ & \\ll (\\text{cont}(g))^3 X  \\sum_{\\substack{D\\in P \\\\ D> Z}} \\frac{k^{\\omega(D)}}{D},\n\\end{align*}\nprovided $YZ \\leq X$.  If $D\\in P$ with $D>Z$, then, since $D\\ll_k \\text{cont}(g)^2Y^{\\omega(D)}$ and $Y\\geq 2$, we know that  \\begin{equation}\\label{logQY} \\text{cont}(g)^3e^{\\omega(D)-\\frac{\\log Z}{\\log Y}} \\gg_k 1. \\end{equation} \nFinally, (\\ref{Z}) and (\\ref{logQY}) imply \\begin{align*} \\sum_{\\substack{D \\in P \\\\ D> Z}}\\frac{k^{\\omega(D)}}{D} &\\ll_k \\text{cont}(g)^3 e^{-\\frac{\\log Z}{\\log Y}} \\sum_{D\\in P} \\frac{(ek)^{\\omega(D)}}{D} \\\\& \\ll \\text{cont}(g)^3 e^{-\\frac{\\log Z}{\\log Y}} (\\log Y)^{ek}, \\end{align*} and the lemma follows.\n\\end{proof} \n\nWe now conclude our discussion by combining the tools developed in this section to establish (\\ref{SmajII}) and (\\ref{SminII}), thus completing the proof of Theorem \\ref{more}.\n\n\\subsection{Proof of (\\ref{SmajII}) and (\\ref{SminII})} We return to the  proof of Lemma \\ref{L2I} in Section \\ref{massproof}, recalling all assumptions,  notation, and fixed parameters. We let $Z=N^{c_0}$, and we let $J$ denote the sum of the absolute value of the coefficients of $h_d$, noting that \\begin{equation}\\label{Jb} J\\ll_h d^k \\leq Z^k. \\end{equation} Fixing $\\alpha\\in \\T$, the pigeonhole principle guarantees the existence of $1\\leq q \\leq M^k\/Z^{3k}$ and $(a,q)=1$ such that $$\\left|\\alpha-\\frac{a}{q} \\right|<\\frac{Z^{3k}}{qM^k}. $$ Letting $\\beta=\\alpha-a\/q$, we have by Lemma \\ref{Sasym}, as well as Lemma \\ref{gradconst}, Proposition \\ref{idzero}, and  Lemma \\ref{content}, that \\begin{equation} \\label{Sproofmaj} S(\\alpha)=\\frac{w}{w_qq^{\\l}} \\sum_{\\boldsymbol{s}\\in \\{0,\\dots,q-1\\}^{\\ell} \\cap W^{q}(Y)}e^{2\\pi i g(\\boldsymbol{s})a\/q}\\int_{[0,M]^{\\ell}}e^{2\\pi i g(\\bsx)\\beta}d\\bsx + O_h\\left(qM^{\\ell-1}\\log^C(Y) Z^{4k\\l}\\right)  ,\\end{equation} where $$w_q=\\prod_{\\substack{ p\\leq Y \\\\ p^{\\gamma(p)}\\mid q}}\\left(1-\\frac{j_d(p)}{p^{\\gamma_d(p)\\l}}\\right)\\gg_h 1. $$ Combining (\\ref{Sproofmaj}) with Corollary \\ref{qcor}, Lemma \\ref{content}, and Corollary \\ref{vdccor} yields (\\ref{SmajII}) if $$q\\leq Q\\text{ and }|\\beta|<\\gamma,$$ as well as (\\ref{SminII}) if $$q\\leq Q \\text{ and }|\\beta|\\geq \\gamma \\quad \\text{or} \\quad Q\\leq q\\leq Z^{3k}.$$ For this latter conclusion, when applying Corollary \\ref{qcor} we use standard estimates that assure $$C^{\\omega(q)}\\tau(q)^{\\l}\\ll_{k,\\l,\\epsilon} q^{\\epsilon} $$ for all $\\epsilon>0$. Finally, it follows from Lemma \\ref{weyl3} and Proposition \\ref{content} that (\\ref{SminII}) holds whenever $Z^{3k}\\leq q \\leq M^k\/Z^{3k}$.  \\qed \n\n\\section*{Acknowledgments}\nThe authors would like to thank Neil Lyall, \\'Akos Magyar, Steve Gonek, and Paul Pollack for their helpful conversations and references. The authors would also like to thank the anonymous referee for their comments and suggestions. The second author would like to thank Gouquan Li for alerting him to an oversight in the proof of Lemma 4.5 in \\cite{ricemax}, which is rectified in the proof of Lemma \\ref{weyl3} in this paper.\n\n\n\\setlength{\\parskip}{0pt}\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nNearly all Globular Clusters (GCs) with quality multi-wavelength\nphotometry appear to harbor multiple stellar populations \\citep[e.\\,g.\\,][and references therein]{2015AJ....149...91P}.\nNGC\\,2808 is one of the most-studied clusters in this context.\nMultiple populations of this GCs have been identified among stars at different evolutionary stages, including the main-sequence \\citep[][]{2005ApJ...631..868D, 2007ApJ...661L..53P, 2012arXiv1211.0685M}, the RGB \\citep[][]{2009Natur.462..480L, 2009A&A...505..117C, 2013MNRAS.431.2126M} the HB \\citep[][]{2008MNRAS.390..693D, 2011MNRAS.410..694D, 2014MNRAS.437.1609M} and even the AGB \\citep[][]{2017ApJ...843...66M}.  \n\n\\cite{2015ApJ...808...51M} found at least five distinct populations within the MS and RGB of NGC 2808.\n\nTheir work is based on photometric coordinates designed to be sensitive to light-element abundances, which vary across stellar subpopulations. The resulting plot is referred to as a `chromosome map' plane in the following, where we focus on\nRGB stars only. Chromosome maps are now widely used to identify and characterize multiple populations in about sixty GCs \\citep[e.g.][and references therein]{2019MNRAS.tmp.1350M}.\n\n\nIn \\cite{2015ApJ...808...51M}, RGB stars on the chromosome map are clustered into populations by hand, so the exact boundaries between populations and even the number of populations found is somewhat affected by subjective factors.\nIn principle, an automated clustering method would be desirable, especially in view of applications to a larger number of GCs, with the goal of extracting reliable statistical information on stellar populations.\nIn this paper we use NGC $2808$ as a benchmark to compare the results of clustering algorithms with the expectations of an expert human judge.\nAll the algorithms we consider are non-parametric, i.e. they make no explicit assumptions on the underlying statistical distribution of the data, as opposed to parametric clustering methods such as e.g. multivariate Gaussian mixture modeling. Since the first self-enrichment models \\citep[e.g.][]{2002A&A...395...69D}, considerable theoretical effort was devoted to understanding the mechanism of multiple population formation, but the issue is far from settled \\citep[see][for a recent review]{2018ARA&A..56...83B}. This is our reason for focusing on non-parametric models, so not to bias the results of our clustering by relying on theoretical assumptions that may later prove wrong.\nOurs is the first systematic comparison of this kind, giving us guidance on which clustering method to use to automatically extract information (such as the number of groups and their characteristics) from a large sample of GCs observed in the chromosome-map filter combinations.\n\n\\section{Data}\nThis work is based on the $\\Delta_{F275W,F336W,F438W}$ and $\\Delta_{F275W,F814W}$ pseudo-colors of NGC\\,2808 of RGB stars from \\cite{2017MNRAS.464.3636M} \\citep[but see also][]{2015ApJ...808...51M}. We show the relevant plot for reference in Fig.~\\ref{fig:jp}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{justplot-eps-converted-to.pdf}\n    \\caption{Plot of NGC\\,2808 RGB stars in the $\\Delta_{F275W,F336W,F438W}$ and $\\Delta_{F275W,F814W}$ pseudo-colors.}\n    \\label{fig:jp}\n\\end{figure}\n\n\n\\section{Methods}\nMost of the clustering algorithms we considered are described by \\cite{1990fgda.book.....K}. For all algorithms we used implementations from the R \\citep[][]{R_itself} libraries \\emph{stats}, \\emph{cluster}, and \\emph{dbscan}. In the following we provide references for both the theoretical description of the algorithm (and any subsequent improvements) and the implementation, on an algorithm-by-algorithm basis. Additionally, we briefly explain the workings of each algorithm.\n\n\\subsection{Partitioning methods}\nWe considered two partitioning methods -i.e. methods that divide the dataset into non-overlapping groups, whose number is specified in advance: \\emph{k-means} and Partitioning Around Medoids (PAM). These are usually the starting point when looking for groups in data but we will see that they do not perform well on our dataset.\n\n\\subsubsection{k-means}\nThe k-means algorithm \\citep[][]{forgey1965cluster, hartigan1979algorithm, lloyd1982least, macqueen1967some} partitions a dataset into a given number $k$ of non-overlapping groups. Its goal is to assign objects to groups so that the sum of their distances from their group mean is minimized. While in principle one could consider all possible partitions and choose the optimal one, this is computationally unfeasible. The algorithm solves this by an iterative procedure which is not guaranteed to converge to the global optimum but scales linearly with the number of groups sought and with the number of datapoints to be clustered.\nThe iterative procedure starts with a set of $k$ initially given means (possibly chosen at random) and assigns points to the nearest mean. Subsequently the means of the groups thus formed are recalculated, and this loop is iterated until the groups no longer change, i.e. convergence is reached.\nThis approach tends to produce approximately round groups. It is bound to obtain counter-intuitive results when groups are elongated, with nontrivial shapes. It also ignores changes in density, which are often used to trace group contours when clustering by eye. Finally, the use of group means leads to sensitivity to outliers, which may move the mean of a group far off from its real center. However, sensitivity to outliers is not the main issue with this method: we will actually see in Sect.~\\ref{DBPM} that the results of k-means applied to our dataset are still unsatisfactory, even when outliers are removed in a pre-processing step.\nWe used the implementation of k-means provided by the \\emph{stats} package in R \\citep[][]{statspackage}.\n\n\\subsubsection{PAM}\nThe PAM algorithm \\citep[][]{1990fgda.book.....K} is similar to k-means in that it clusters data around centroids using an iterative procedure, but the centroids in this case are actual datapoints (medoids) instead of means. This is useful if we intend to characterize groups based on a representative element \\citep[see e.g.][]{2019arXiv190105354P}. However in our case it is bound to suffer the same shortcomings of k-means, even though its results appear to be less affected by outliers.\nWe used the implementation of PAM provided by the \\emph{cluster} package in R \\citep[][]{clusterpackage}.\n\n\\subsection{Density-based methods}\n\\subsubsection{DBSCAN}\nDensity Based Spatial Clustering of Applications with Noise \\cite[DBSCAN; ][]{ester1996density} is the most popular density-based clustering algorithm. The implementation we use is from the R library \\emph{dbscan} by \\cite{dbscan}.\nThe idea of density-based clustering is more similar to our intuitive notion of grouping together datapoints that are connected by regions populated with high density. It allows for clusters to be elongated and of arbitrary shapes, even nested within larger clusters.\nDBSCAN relies on two parameters, \\emph{minPts} and \\emph{eps} (also written $\\epsilon$ in the following), to perform its grouping. A datapoint is a core point if at least minPts points are within distance eps of it. Core points essentially live in regions where density is at least \\emph{minPts}$\/\\epsilon^2$. A point $q$ is directly reachable from a core point $p$ if it is within distance eps from it. Point $q$ is reachable from a core point $p$ if there is a sequence $p_1$, ..., $p_n$ starting in $p$ and ending in $q$ where each point is directly reachable from the previous one.\nGroups are obtained by clustering together points (core or not) that are reachable from a given core point.\nSome points will be left out, as they are not reachable from any core point. These points are outliers or \\emph{noise points}.\nReachability corresponds to the intuitive notion that that points in the same group should be connected by high-denisty areas.\nConversely, noise points live in low density areas.\nIn the following we will use DBSCAN to find groups in our dataset, but also to remove outliers as a preprocessing step for other algorithms.\nA limitation of the DBSCAN algorithm is that \\emph{minPts} and \\emph{eps} are global, set once and for all for the whole dataset. So if groups have different intrinsic densities, DBSCAN may have difficulties in finding them all with a given \\emph{minPts} and \\emph{eps} setting. We will see that this is indeed an issue with our data.\n\n\\subsubsection{DBSCAN for outlier removal}\n\\label{DBSCANOutlierRemoval}\nIn addition to using DBSCAN directly for clustering we also used it in the pre-processing stage before applying other algorithms, to remove outliers. This was achieved by setting \\emph{MinPts} to a much lower value than in the previous case, which leads DBSCAN to consider a point as a member of a group even if it has only a few neighbors. The exact value chosen is $4$, which we associated with a relatively large value of \\emph{eps}$= 0.02$. In the following, whenever we discuss outlier removal the points we removed are those identified as noise points by DBSCAN with these settings. \n\n\\subsubsection{OPTICS}\nThe Ordering Points To Identify the Clustering Struture \\citep[OPTICS; ][]{ankerst1999optics} algorithm is a generalization of DBSCAN aimed at tackling the problem of identifying clusters of different density.\nThe OPTICS algorithm sorts points so that groups appear as stretches of adjacent points. Like DBSCAN, it takes as input two values, \\emph{minPts} and \\emph{eps}, but the latter only serves as an upper bound. A reachability distance is then defined and plotted for each point, sorted according to the relevant order. In the resulting reachability plot points that should be grouped together have small reachability distance from their neighbor and so groups appear as valleys in the reachability plot. It is then a matter of finding a rule to delimit these valleys and output definite groups. In the following we discuss a possible subdivision of the reachability plot done by hand. \n\n\n\\subsection{Hierarchical methods}\nUnlike partitioning methods, for which a value of the number of groups in which to split the dataset had to be specified in advance, hierarchical methods produce a tree-like structure (dendrogram) obtained by subsequent merging or splitting of groups. Agglomerative methods start with each datapoint in its own, separate group, and progressively merge nearby groups until all the data fits in a single group. Conversely, divisive methods start with the data all grouped together and proceed by splitting it into groups until each point is on its own. In both cases, the resulting dendrogram summarizes the clustering structure of the data set at different scales. Cutting the dendrogram at a given \\emph{height} returns a given number of groups, i.e. chooses the scale at which to look at the clustering structure.\n\n\\subsubsection{AGNES}\nAgglomerative methods progressively join datapoints to form groups. The AGlomerative NESting (AGNES) algorithm is described in \\cite{1990fgda.book.....K}.\nThe criterium for joining two points to form a group is based on their distance, with the two nearest points being joined first. Later on the algorithm needs to join either two groups into a new group or a lone point into a group. To do this, a notion of distance between groups is needed. There are many different variations on this, and our choice of a subset of them to test for the purposes of this paper is discussed and motivated later in the following Sect.~\\ref{linkage}.\nThe general update rule used to calculate the distance between groups based on the distances of the groups (and ultimately points) that were previously joined into them was introduced by \\cite{lance1966generalized}. We use the implementation of AGNES in the R library \\emph{cluster} by \\cite{cluster}.\n\n\\subsubsection{Linkages}\n\\label{linkage}\nTwo groups are joined in AGNES based on their distance. Different linkage choices correspond to different ways of defining the distance between groups based on the distance of the respective elements.\nIn the following we used\n\\begin{itemize}\n    \\item \\emph{single linkage}: $D(A, B) \\coloneqq min_{x \\in A, y \\in B} D(x, y)$ \\citep[][]{florek1951liaison, sneath1957application, johnson1967hierarchical}\n    \\item \\emph{average linkage}: $D(A, B) \\coloneqq mean_{x \\in A, y \\in B} D(x, y)$ \\citep[][]{michener1957quantitative, lance1966generalized}\n    \\item \\emph{complete linkage}: $D(A, B) \\coloneqq max_{x \\in A, y \\in B} D(x, y)$ \\citep[][]{mcquitty1960hierarchical, sokal1963principles}\n    \\item \\emph{Ward's method}: $D^2(A, B) \\coloneqq n_A n_B D^2(m_A, m_B) \/ (n_A + n_B)$ where $n_X$ is the number of objects in cluster $X$, and $m_X$ is its centroid, i.e. the point $m \\in X$ such that $\\sum_{x \\in X} D(x, m)$ is minimal \\citep[][]{ward1963hierarchical}.\n\\end{itemize}\nwhere $D(x,y)$ is the Euclidean distance between points $x$ and $y$ on the chromosome map plane.\n\nWith single linkage two groups are near to each other when they contain at least one point each which is near to the other cluster. Even a tiny `bridge' of points spanning the space between two groups may lead to a merger. We will see in Sect.~\\ref{resultsAGNES} that for our problem of clustering in the chromosome map space this leads to undesirable results. This is in line with the poor performance of single linkage in previous empirical studies \\citep{baker1974stability, milligan1980validation}.\nAverage and complete linkage are different takes on the same idea of using the distances between points in the two groups for defining the distance between the groups. Complete linkage uses the maximum distance, so two groups that are otherwise close to each other will not be merged if they have at least a couple of points that is very far from each other. Average linkage strikes a balance within these two extremes using the mean of all the pairwise distances of points to define the distance between groups. Being a mean, this can still be quite vulnerable to outliers because even a few points that are very far apart can delay the merging of two groups. As we will see in Sect.~\\ref{resultsAGNES} removing isolated points using DBSCAN (see Sect.\\ref{DBSCAN}) as pre-processing for AGNES improves its performance in the cases of average and complete linkage.\nThe definition of distance in Ward's method can be rewritten as follows:\n\\begin{equation}\n    D^2(A,B) = \\sum_{x \\in A \\cup B} D^2(x, m_{A \\cup B}) - \\left(\\sum_{x \\in A} D^2(x, m_A)  + \\sum_{x \\in B} D^2(x, m_B) \\right)\n\\end{equation}\nwhich is interpreted as the increase in the sum of distances to the respective centroids as groups are merged. This definition tends to produce somewhat round clusters and we will see in Sect.~\\ref{resultsAGNES} that it produces the most intuitive clustering even in the presence of outliers.\n\n\\subsubsection{DIANA}\nDivisive methods work in the opposite direction as agglomerative methods, in that they start with the whole dataset grouped together and progressively split it into smaller groups until individual points are reached.\nWe use the divisive algorithm DIvisive ANAlysis (DIANA) in its implementation in the R library \\emph{cluster} by \\cite{cluster}. The original algorithm is described by \\cite{1990fgda.book.....K} based on \\cite{mcnaughton1964dissimilarity}.\nThe algorithm initially considers all datapoints grouped together in a single cluster. It then finds the point that is most dissimilar to the others (i.e. further away from them) to initiate a splinter group. Points of the leftover group that are more similar to the splinter group than to the other members of that group get moved to the splinter group, until two groups are formed.\nThe group with the largest diameter (distance between its two members that are the furthest away from each other) among the resulting two groups is split in turn in the same fashion, producing three groups. The algorithm then proceeds iteratively until all points are assigned to their own separate subgroup.\nThe final output of the algorithm is a dendrogram with each node corresponding to a split and associated to the diameter of the group being split, which is the height of the respective node.\n\n\\section{Results}\n\\subsection{Partitioning methods}\n\\label{DBPM}\nFigures \\ref{fig:PART00} and \\ref{fig:PART01} show the results of applying the algorithms k-means and PAM respectively to our whole dataset, i.e. without removing outliers.\nWe explored four choices for the number of groups that the algorithms are required to find, namely $k=3$, $4$, $5$, and $6$.\nAs it can be easily seen, both k-means and PAM produce counterintuitive results for all values of $k$ we considered. For example, sharp linear boundaries between groups that cut through regions of high density or, vice versa, groups that extend across low-density regions.\nIn Fig.~\\ref{fig:PART02} and \\ref{fig:PART03} we check whether this result is affected by the presence of outliers, i.e. points located in low density regions far from the bulk of the dataset.\nTo eliminate outliers we used the DBSCAN algorithm, which we also use later for clustering (see Sect.~\\ref{DBSCAN}), with $\\epsilon = 0.02$ and \\emph{MinPts}$=4$ and labeled as outliers the resulting noise points (as described in Sect.~\\ref{DBSCANOutlierRemoval}). Visually, the outlier points that were removed can be identified by comparing e.g. Fig.~\\ref{fig:PART02} to Fig.~\\ref{fig:PART00}. As can be easily seen, the removal of outliers does not improve the results of clustering either with k-means or PAM, even though the latter seems more consistent across the two cases.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{partitioning_00-eps-converted-to.pdf}\n    \\caption{Results of applying the \\emph{k-means} algorithm to the whole dataset (including outliers). The number of groups is $k=3$, $4$, $5$, $6$ in the top left, top right, bottom left, and bottom right panel respectively.}\n    \\label{fig:PART00}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{partitioning_01-eps-converted-to.pdf}\n    \\caption{Results of applying the \\emph{PAM} algorithm to the whole dataset (including outliers). The number of groups is $k=3$, $4$, $5$, $6$ in the top left, top right, bottom left, and bottom right panel respectively.}\n    \\label{fig:PART01}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{partitioning_02-eps-converted-to.pdf}\n    \\caption{Results of applying the \\emph{k-means} algorithm to the dataset after outlier removal with DBSCAN. The number of groups is $k=3$, $4$, $5$, $6$ in the top left, top right, bottom left, and bottom right panel respectively.}\n    \\label{fig:PART02}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{partitioning_03-eps-converted-to.pdf}\n    \\caption{Results of applying the \\emph{PAM} algorithm to the dataset after outlier removal with DBSCAN. The number of groups is $k=3$, $4$, $5$, $6$ in the top left, top right, bottom left, and bottom right panel respectively.}\n    \\label{fig:PART03}\n\\end{figure}\n\n\\subsection{Density-based methods}\n\\label{DBSCAN}\n\\subsubsection{DBSCAN}\nIn Fig.~\\ref{fig:DBSCAN02} we present a selection of DBSCAN results for different combinations of the \\emph{eps} and \\emph{MinPts} parameters.\nThe effects of changes in these parameters can be interpreted by keeping in mind that \\emph{MinPts}$\/$\\emph{eps}$^2$ is essentially the density cutoff between what is considered a group and what is considered noise. Thus increasing \\emph{MinPts} for a given \\emph{eps} leads to smaller groups and more noise points.\n\nThe outcome of DBSCAN clustering depends heavily on the \\emph{eps} and \\emph{MinPts} parameters. The results in terms of the number of groups found and the fraction of points regarded as `noise' are presented in Fig.~\\ref{fig:DBSCAN01}. The behavior of the former is pretty erratic: as \\emph{MinPts} is changed by a few units for a given \\emph{eps} the number of groups can easily vary by a factor of two. If we were to use the raw output of DBSCAN for studying the properties of each stellar population (identified as a group in the chromosome map space) or the overall number of populations in view of e.g. a statistical study on a sample of globular clusters, this strong variability with the choice of the \\emph{eps} and \\emph{MinPts} parameters would be a major drawback.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{dbscan_01-eps-converted-to.pdf}\n    \\caption{Number of groups (top panel) and fraction of points considered outliers (bottom panel) by DBSCAN for $\\epsilon = 0.01$ (dotted line), $\\epsilon = 0.015$ (dashed line), and $\\epsilon = 0.02$ (solid line). The \\emph{MinPts} parameter (abscissae axis) varies from $4$ to $120$.}\n    \\label{fig:DBSCAN01}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{dbscan_02-eps-converted-to.pdf}\n    \\caption{Selection of DBSCAN results for $\\epsilon = 0.015$ (top row) and $\\epsilon = 0.02$ (bottom row). Each group is color coded based on the vertical position of its centroid, except for noise points that are gray. In each row the left and right panel correspond to different values of \\emph{MinPts}, illustrating how changing \\emph{MinPts} can either resolve the two populations in the top left corner of each plot (yellow in the left column, yellow$\/$green in the right column) or recover the small population in the bottom right (purple in the left column, gray - considered noise points - in the right column).}\n    \\label{fig:DBSCAN02}\n\\end{figure}\n\nAnother issue faced by DBSCAN is due to the varying density of the groups that we would like to identify. For example, the `bridge' between the two groups in the top left of each plot in Fig.~\\ref{fig:DBSCAN02} has a higher density than the group in the bottom right. So there is no choice of parameters for which DBSCAN will be able to both separate the two top left groups and identify the bottom right group. If it does separate the two top left groups then the bottom right group is considered noise.\nThis can be seen in Fig.~\\ref{fig:KDE} where we show the level curves of the Probability Density Function (PDF) of our datapoints obtained using kernel density estimation \\citep[through function \\emph{kde2d} of the \\emph{MASS} R package; ][]{MASS}. We used a bivariate Gaussian kernel with equal bandwidths in the $x$ and $y$ directions, equal to the minimum of the two bandwidths obtained separately for the two by using the \\emph{bandwidth.nrd} function in the MASS library.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{kdecontour-eps-converted-to.pdf}\n    \\caption{Kernel density estimation of the two-dimensional probability density function (green contours) of the datapoints (yellow dots). Contour levels were chosen to illustrate the issues faced by DBSCAN due to the range of density present in our dataset. Notice how the bottom right group that is identified in the left column of Fig.~\\ref{fig:DBSCAN02} has smaller density than the area between the two groups in the upper left, which get split only in the right column of Fig.~\\ref{fig:DBSCAN02} at the cost of making the former group disappear.}\n    \\label{fig:KDE}\n\\end{figure}\n\n\\subsubsection{OPTICS}\nAt the bottom of Fig.~\\ref{fig:optics00} we show the reachability plot obtained by OPTICS with \\emph{minPts}$=30$ and $\\epsilon$ set equal to the diameter of the whole dataset. The sorted points are arranged along the $x$ axis and their reachability distance is plotted in the $y$ axis. Groups in this plot correspond to `valleys', i.e. regions of low reachability distance. The intuitive explanation for this is that points grouped together are easy to reach from each other, because they are nearby. On the other hand, high reachability distance points corresponds to outliers.\nFrom the qualitative point of view, the results of OPTICS confirm our human-expert expectations: it is relatively easy to pick out, by hand, five valleys in the reachability plot that correspond to five `sensible' groups. This is shown in the top part of Fig.~\\ref{fig:optics00}, where each group is shown in the same color as the respective valley in the reachability plot.\nThe problem with this is that the valleys had to be selected by hand. Simple approaches, such as cutting the reachability plot at a given level, would run into the same issues that DBSCAN has: separating the blue from the teal group at the top makes the yellow group at the bottom disappear \\footnote{This is unsurprising as this operation is essentially what is used to extract a DBSCAN cluster from OPTICS output \\citep[see][]{ankerst1999optics}}. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{optics_00-eps-converted-to.pdf}\n    \\caption{Rechability plot obtained by OPTICS for our dataset (bottom) and corresponding groups (top) extracted by hand, identifying `valleys' in the reachability plots. The color coding and dashed lines associate each cluster with the corresponding valley.}\n    \\label{fig:optics00}\n\\end{figure}\n\n\\subsection{Hierarchical methods}\n\n\\subsubsection{AGNES}\n\\label{resultsAGNES}\nWe applied AGNES with different linkages to the NGC 2808 dataset. In Fig.~\\ref{fig:AGNES02}, \\ref{fig:AGNES00}, and \\ref{fig:AGNES04} we used the R function \\emph{cutree} to cut the tree and obtain $4$, $5$, and $6$ groups respectively. Similarly, Fig.~\\ref{fig:AGNES03}, \\ref{fig:AGNES01}, and \\ref{fig:AGNES05} represent the result of AGNES with the same settings but after outlier removal with DBSCAN as described in Sect.~\\ref{DBSCANOutlierRemoval}.\n\nIn each of these six figures the top left panel shows the groups obtained by applying AGNES with the single-linkage recipe for joining nearby groups: in all cases single linkage results in all points that are not outliers joining into a single group. This is clearly useless for distinguishing stellar populations within the chromosome map.\n\nThe top right panel of each figure shows the results of adopting average linkage: this always leads to a poor outcome in the presence of outliers, while the groups look more aligned with the expectations of an expert with outlier removal. The situation is similar for complete linkage (bottom left panel of each figure), which performs better after outlier removal. Still, even after outlier removal complete linkage leads to results that would be deemed unphysical: e.g. in Fig.~ \\ref{fig:AGNES01}.\n\nBy far the recipe that produces groups most in line with expert expectations is Ward's method (bottom right panel). Ward's method also gives pretty consistent results whether the outliers are removed or not. Average linkage instead produces comparable results only when outliers are removed.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_02-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). In all cases the hierarchical tree was cut at $k=4$ groups. Notice how in some cases (e.g. single-linkage) some groups contain only a few points.}\n    \\label{fig:AGNES02}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_00-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). In all cases the hierarchical tree was cut at $k=5$ groups.}\n    \\label{fig:AGNES00}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_04-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). In all cases the hierarchical tree was cut at $k=6$ groups.}\n    \\label{fig:AGNES04}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_03-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). Outliers were removed using DBSCAN (see discussion in text). In all cases the hierarchical tree was cut at $k=4$ groups.}\n    \\label{fig:AGNES03}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_01-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). Outliers were removed using DBSCAN (see discussion in text). In all cases the hierarchical tree was cut at $k=5$ groups.}\n    \\label{fig:AGNES01}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_05-eps-converted-to.pdf}\n    \\caption{Comparison of AGNES results with different linkages: single (top left), average (top right), complete (bottom left), and Ward's (bottom right). Outliers were removed using DBSCAN (see discussion in text). In all cases the hierarchical tree was cut at $k=6$ groups.}\n    \\label{fig:AGNES05}\n\\end{figure}\n\nHierarchical methods actually produce a dendrogram as groups are subsequently merged into larger groups, traversing the clustering structure of the dataset at different scales.\nFig.~\\ref{fig:AGNESdendrogram} shows the top part of the dendrogram produced by AGNES with average linkage and outliers removed. The dendrogram can be read from the bottom, so that moving towards the top merges different groups, until at the end we are left with the whole dataset grouped into one. The merging points of branches are plotted so that the vertical position of the merger is proportional to the distance (defined by the linkage, average distance in this case) between the groups that are being merged. As shown in Fig.~\\ref{fig:AGNESdendrogram} the dendrogram can be cut at different \\emph{heights}, obtaining a different number of groups. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{agglo_dendrogram-eps-converted-to.pdf}\n    \\caption{Different partitions are obtained by cutting the dendrogram (obtained by AGNES with average linkage in the absence of outliers, in this case) at different heights. Height corresponds to the distance between the clusters being merged at each branching point.}\n    \\label{fig:AGNESdendrogram}\n\\end{figure}\n\n\\subsubsection{DIANA}\n\nFig.~\\ref{fig:DIANA00} shows the groups obtained by DIANA with all points included, while Fig.~\\ref{fig:DIANA01} shows the same with outliers removed by DBSCAN. Despite sharing the hierarchical approach with AGNES, DIANA produces groups that are far from the expectations of an expert, as they appear sometimes concave. Additionally, by comparing the bottom left panels of Fig.~\\ref{fig:DIANA00} and Fig.~\\ref{fig:DIANA01} we see that outliers can heavily affect the outcome of DIANA's clustering.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{diana00-eps-converted-to.pdf}\n    \\caption{Comparison of DIANA results with all datapoints included. From top left to bottom right the number of groups ranges from $k=3$ to $k=6$.}\n    \\label{fig:DIANA00}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{diana01-eps-converted-to.pdf}\n    \\caption{Comparison of DIANA results with outliers removed using DBSCAN. From top left to bottom right the number of groups ranges from $k=3$ to $k=6$.}\n    \\label{fig:DIANA01}\n\\end{figure}\n\n\\section{Conclusions}\nWe applied a set of different non-parameteric clustering methods to a dataset of $2682$ RGB stars of NGC 2808 observed in the \\emph{chromosome map} photometric plane \\citep[][]{2015ApJ...808...51M}. Our ultimate goal is to identify groups that correspond to underlying distinct stellar populations, while avoiding strong assumptions on the underlying statistical distribution of our dataset. While a human expert can accomplish this with relative ease for NGC 2808, we still seek to compare the merits of different automatic clustering algorithms, in view of application to a larger sample of globular clusters. In that context, consistency and reproducibility are of paramount interest, especially if the results are to form the basis of a statistical study into the properties of multiple populations.\n\nWe considered three different approaches to clustering: partitioning methods (k-means, PAM), hierarchical methods (AGNES, DIANA), and density-based methods (DBSCAN, OPTICS). We also used DBSCAN to identify outliers and compared the results of all other algorithms with and without outliers.\n\nWe find that AGNES produces results most in line with human-expert expectations, as long as Ward's method \\citep[][]{ward1963hierarchical} is used to determine the distance between groups. Ward's method merges groups so that the increase in variance is minimal. This results in relatively round groups (as opposed, e.g. to single linkage that often produces elongated groups), in agreement with theoretical expectations that each stellar population is close to point-like except for broadening due to photometric errors. In Fig.~\\ref{fig:final} we show the final outcome with this method.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{final-eps-converted-to.pdf}\n    \\caption{Groups obtained by AGNES with Ward's method on the whole dataset (outliers included).}\n    \\label{fig:final}\n\\end{figure}\n\n\n\\section*{Acknowledgment}\nThis project has received funding from the European Union's Horizon $2020$\nresearch and innovation programme under the Marie Sk\\l{}odowska-Curie grant agreement No. $664931$\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}