Uniform Dilations in Higher Dimensions

TL;DR

This paper extends the study of uniform dilations of infinite sets on the torus to higher dimensions, providing necessary and sufficient conditions for when dilations produce dense subsets, and offering effective bounds.

Contribution

It characterizes when matrix-based dilations in higher dimensions lead to dense sets, generalizing previous one-dimensional results and providing effective criteria.

Findings

Identifies conditions for matrix dilations to produce dense subsets in higher dimensions.

Provides a necessary and sufficient condition for the density of dilated sets.

Offers an effective version of the density criterion.

Abstract

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.