Glasner property for unipotently generated group actions on tori

TL;DR

This paper investigates the Glasner property for unipotently generated group actions on tori, classifying matrices with this property and extending results to certain linear group actions, revealing new density and irreducibility phenomena.

Contribution

It classifies integer polynomial matrices with the Glasner property on the full torus and extends Glasner's results to actions of unipotent-generated groups on tori.

Findings

Classified matrices with Glasner property in full tori.

Extended Glasner property to certain linear group actions.

Proved irreducible unipotent actions have uniform Glasner property.

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-L\^{e} and Dong to show that the same property holds for various linear actions on $\mathbb{T}^d$. We complement the result of Kelly-L\^{e} 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 $\Gamma \leq \operatorname{SL}_d(\mathbb{Z})$ 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.