Simultaneous dilation and translation tilings of 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 (2006), 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 extend the Ionascu-Wang characterization to higher dimensions for dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$. Finally, we show the existence of wavelet sets for all dilations $A$ with integer entries satisfying $|\det A|\ne 1$.