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

Let $E\subset {\mathbb{Z}}^N$ be a set of positive upper Banach density and let $\mathrm{\Gamma }<{\mathrm{GL}}_N(\mathbb{Z})$ 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 \rightarrow Y$isa$\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 $\mathrm{\mid }F\mathrm{\mid }=m$, we have $$\mathrm{\Psi }(kF)\subset \mathrm{\Psi }(E-b)\text{for some }b\in E\mathrm{.}$$ 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}$$,wecanfind$x,y,z \in E_o$ such that $$k^{2}F\subset \{(u-x{)}^2+(v-y{)}^2-(w-z{)}^2:u,v,w\in E_o\}\mathrm{.}$$ 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.