# 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.