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 "sufficiently large" subgroup. We show in this paper that for each positive integer $m$ there exists a positive integer $k$ with the following property: For every ${a\_1,\ldots,a\_m} \subset k \cdot \mathbb Z^N$, there are $\gamma\_1,\ldots,\gamma\_m \in \Gamma$ and $b \in E$ such that $$ \gamma\_i \cdot a\_i \in E - b, \quad \text{for all $i = 1,\ldots,m$}. $$ We use this „twisted" multiple recurrence result to study images of $E-b$ under various $\Gamma$-invariant maps. For instance, if $N \geq 3$ and $Q$ is an integer quadratic form on $\mathbb Z^N$ of signature $(p,q)$ with $p,q \geq 1$ and $p + q \geq 3$, then our twisted multiple recurrence theorem applied to the group $\Gamma = \mathrm {SO}(Q)(\mathbb Z)$ shows that $$ k^2 Q(F) \subset Q(E-b), $$ for every $F \subset k \cdot \mathbb Z^N$ with $m$ elements. In the case when $E$ is an aperiodic Bohr$\_o$ set, we can choose $b$ to be zero and $k = 1$, and thus $Q(\mathbb Z^N) \subset Q(E)$. Our result is derived from a non-conventional ergodic theorem which should be of independent interest. Important ingredients in our proofs are the recent breakthroughs by Benoist–Quint and Bourgain–Furman–Lindenstrauss–Mozes on equidistribution of random walks on automorphism groups of tori.