Glasner property for unipotently generated group actions on tori

Abstract

A theorem of Glasner from 1979 shows that if $A\subset \mathbb{T}=\mathbb{R}/\mathbb{Z}$ is infinite then for each $\epsilon >0$ there exists an integer $n$ such that $nA$ is $\epsilon$-dense and Berend-Peres later showed that in fact one can take $n$ to be of the form $f(m)$ for any non-constant $f(x)\in \mathbb{Z}[x]$. Alon and Peres provided a general framework for this problem that has been used by Kelly-Lê and Dong to show that the same property holds for various linear actions on ${\mathbb{T}}^d$. We complement the result of Kelly-Lê on the $\epsilon$-dense images of integer polynomial matrices in some subtorus of ${\mathbb{T}}^d$ by classifying those integer polynomial matrices that have the Glasner property in the full torus ${\mathbb{T}}^d$. We also extend a recent result of Dong by showing that if $\Gamma \le {\mathrm{SL}}_d(\mathbb{Z})$ is generated by finitely many unipotents and acts irreducibly on ${\mathbb{R}}^d$ then the action $\Gamma \curvearrowright {\mathbb{T}}^d$ has a uniform Glasner property.

10 pages, 0 figures