Simultaneous dilation and translation tilings of $\mathbb R^n$

TL;DR

This paper characterizes when certain sets can tile 2^n through dilation and translation, solving the wavelet set existence problem for higher dimensions and providing conditions on matrix dilations.

Contribution

It generalizes the wavelet set existence characterization to all dimensions and establishes new conditions on matrix dilations for wavelet set existence.

Findings

Characterization of lattices and matrices for wavelet set tilings in 2^n

Existence of wavelet sets for matrices with eigenvalues 2 1

Strengthening of previous wavelet set existence results

Abstract

We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $\Gamma\subset \mathbb R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + \gamma: \gamma \in \Gamma\}$ and $\{A^j(W): j\in \mathbb Z\}$ are tilings of $\mathbb R^n$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $\lambda$ satisfy $|\lambda| \ge 1$. As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$.