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

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 $\ge 1$.