Three-way tiling sets in two dimensions

TL;DR

This paper demonstrates the existence of measurable sets in the plane that tile the plane under complex group actions, advancing the understanding of tiling theory and wavelet sets with new solutions to open problems.

Contribution

It introduces measurable tiling sets in two dimensions that tile under an expansive matrix, affine Weyl group, and lattice, solving part of an open problem and extending previous work on wavelet and fractal analysis.

Findings

Existence of measurable tiling sets in the plane under complex group actions

Positive solution to a previously open conjecture in tiling theory

Extension of wavelet set theory to new group actions

Abstract

In this article we show that there exist measurable sets W in the plane with finite measure that tile the plane in a measurable way under the action of a expansive matrix A, an affine Weyl group W, and a full rank lattice G. This note is follow-up research to the earlier article "Coxeter groups and wavelet sets" by the first and second authors, and is also relevant to the earlier article "Coxeter groups, wavelets, multiresolution and sampling" by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on "Operator methods in fractal analysis, wavelets and dynamical systems," December 2 -- 7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. Olafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.