Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Simultaneous dilation and translation tilings of $\mathbb R^n$ (2109.10323v1)

Published 21 Sep 2021 in math.CA

Abstract: We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $\Gamma\subset \mathbb Rn$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that ${W + \gamma: \gamma \in \Gamma}$ and ${Aj(W): j\in \mathbb Z}$ are tilings of $\mathbb Rn$. 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 $\ge 1$.

Summary

We haven't generated a summary for this paper yet.