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

### Michael Björklund

Chalmers University of Technology, Gothenburg, Sweden### Kamil Bulinski

University of Sydney, Australia

## 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

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

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.

## Cite this article

Michael Björklund, Kamil Bulinski, Twisted patterns in large subsets of $\mathbb Z^N$. Comment. Math. Helv. 92 (2017), no. 3, pp. 621–640

DOI 10.4171/CMH/420