Glasner property for unipotently generated group actions on tori

A theorem of Glasner from 1979 shows that if $A \subset $\mathbb{T}$ = $\mathbb{R}$/$\mathbb{Z}$$isinfinitethenforeach$\epsilon > 0$ 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 $f(x)\in \mathbb{Z}[x]$. 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 $ϵ$-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 $\mathrm{\Gamma }\le {\mathrm{SL}}_d(\mathbb{Z})$ 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.