Codes and Expansions (CodEx) Seminar

Darrin Speegle (Saint Louis University)
Simultaneous Dilation and Translation Tilings of \(R^n\)

We solve the wavelet set existence problem. That is, we characterize the 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 \(\vert\lambda\vert\geq 1\). As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is \(\geq 1\).

In this talk, I will focus on the connection between the wavelet set existence problem and computing the cardinality of \(A^{-j}(B) \cap \Gamma\) for \(j \geq 0\) (where \(B\) is the unit ball in \(R^n\)). In the case that one of the eigenvectors of \(A^{-1}\) associated with eigenvalue greater than \(1\) in modulus points in the direction of an element of the lattice, it is clear that this cardinality goes to infinity. However, there are other, “non-obvious” ways that the cardinality can go to infinity which have been deeply studied and which are related to the existence of wavelet sets.