Uniform Dilations in Higher Dimensions

A theorem of Glasner says that if $X$ is an infinite subset of the torus $\mathbb{T}$, then for any $\epsilon>0$, there exists an integer $n$ such that the dilation $nX=\{nx: x \in \mathbb{T} \}$ is $\epsilon$-dense (i.e, it intersects any interval of length $2\epsilon$ in $\mathbb{T}$). Alon and Peres provided a general framework for this problem, and showed quantitatively that one can restrict the dilation to be of the form $f(n)X$ where $f \in \mathbb{Z}[x]$ is not constant. Building upon the work of Alon and Peres, we study this phenomenon in higher dimensions. Let ${\bf A}(x)$ be an $L \times N$ matrix whose entries are in $\mathbb{Z}[x]$, and $X$ be an infinite subset of $\mathbb{T}^N$. Contrarily to the case $N=L=1$, it's not always true that there is an integer $n$ such that $\bA(n)X$ is $\epsilon$-dense in a translate of a subtorus of $\mathbb{T}^{L}$. We give a necessary and sufficient condition for matrices ${\bf A}$ for which this is true. We also prove an effective version of the result.