abstract: We solve the wavelet set existence problem. That is, we characterize the full-rank lattices ⁿ and invertible n n matrices A for which there exists a measurable set W such that \W+: \ and \Aʲ (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 satisfy || 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.
Bownik et al. (Tue,) studied this question.
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