Twisted patterns in large subsets of ${\mathbb{Z}}^N$

Abstract

Let $E\subset {\mathbb{Z}}^N$ be a set of positive upper Banach density and let $\Gamma <{\mathrm{GL}}_N(\mathbb{Z})$ be a finitely generated, strongly irreducible subgroup whose Zariski closure in ${\mathrm{GL}}_N(\mathbb{R})$ is a Zariski connected semisimple group with no compact factors. Let $Y$ be any set and suppose that $\Psi :{\mathbb{Z}}^N\to Y$ is a $\Gamma$-invariant function. We prove that for every positive integer $m$, there exists a positive integer $k$ with the property that for every finite set $F\subset {\mathbb{Z}}^N$ with $|F|=m$, we have \[ Ψ(kF) \subset Ψ(E-b) \quad \textrm{for some $b\in E$}. \] Furthermore, if $E$ is an aperiodic Bohr${}_o$-set, we can choose $k=1$ and $b=0$. As one of many applications of this result, we show that if $E_o\subset \mathbb{Z}$ has positive upper Banach density, then, for any integer $m$, there exists an integer $k$ with the property for \emph{every} finite set $F\subset \mathbb{Z}$, we can find $x,y,z\in E_o$ such that \[ k^2 F \subset \big\{ (u-x)^2 + (v-y)^2 - (w-z)^2 \, : \, u,v,w \in E_o \big\}. \] In particular, if $E_o\subset \mathbb{Z}$ is an aperiodic Bohr${}_o$-set, then every integer can be written on the form $u^2+v^2-w^2$ for some $u,v,w\in E_o$. Our techniques use recent results by Benoist-Quint and Bourgain-Furman-Lindenstrauss-Mozes on equidistribution of random walks on automorphism groups of tori.

arXiv comment17 pages, no figures