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

Abstract: In this talk we present a solution of the wavelet set problem. That is, we characterize full-rank lattices $\Gamma\subset \R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + \gamma: \gamma \in \Gamma\}$ and $\{A^j(W): j\in \Z\}$ are tilings of $\R^n$. 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 $\ge 1$. > Based on a joint work with Darrin Speegle.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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