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

TL;DR

This paper proves that large subsets of integer lattices contain twisted patterns related to group actions, with applications showing that certain sets can represent all integers as sums and differences of squares within those sets.

Contribution

It establishes the existence of scaled and shifted patterns within large subsets of  lattices under group invariance, extending combinatorial and number-theoretic results.

Findings

Large subsets contain scaled patterns invariant under group actions.

Any finite set can be embedded into sum-of-squares structures within these subsets.

Results apply to sets with positive density, including Bohr-sets, enabling representation of all integers.

Abstract

Let $E \subset \mathbb{Z}^N$ be a set of positive upper Banach density and let $\Gamma < \operatorname{GL}_N(\mathbb{Z})$ be a finitely generated, strongly irreducible subgroup whose Zariski closure in $\operatorname{GL}_N(\mathbb{R})$ is a Zariski connected semisimple group with no compact factors. Let $Y$ be any set and suppose that $\Psi : \mathbb{Z}^N \rightarrow Y$ 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 \[ \Psi(kF) \subset \Psi(E-b) \quad \textrm{for some $b \in E$}. \] 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.