Uniform Dilations in Higher Dimensions

Abstract

A theorem of Glasner says that if $X$ is an infinite subset of the torus $\mathbb{T}$, then for any $\hat{I}\mu >0$, there exists an integer $n$ such that the dilation $nX=\{nx:x\in \mathbb{T}\}$ is $\hat{I}\mu$-dense (i.e, it intersects any interval of length $2\hat{I}\mu$ 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 $\mathbf{A}(x)$ 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 $\hat{I}\mu$-dense in a translate of a subtorus of ${\mathbb{T}}^L$. We give a necessary and sufficient condition for matrices $\mathbf{A}$ for which this is true. We also prove an effective version of the result.