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 ϵ > 0 there exists an integer n such that nA is ϵ-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) ∈ 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 ϵ-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 Γ ≤ SLd(ℤ) is generated by finitely many unipotents and acts irreducibly on ℝd, then the action \(\Gamma \curvearrowright {\mathbb{T}^d}\) has a uniform Glasner property.