Simultaneous dilation and translation tilings of ℝ n

We solve the wavelet set existence problem. That is, we characterize the full-rank lattices Γⁿ and invertible n× n matrices A for which there exists a measurable set W such that +γ:γ∈Γ\ and ʲ(W):j\ are tilings of Rⁿ. 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 λ satisfy |λ|≥ 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 ≥ 1. Finally, we show the existence of wavelet sets for all dilations A with integer entries satisfying | A|≠ 1.