Abstract
In this article we show that there exist measurable sets W⊂ℝ2 with finite measure that tile ℝ2 in a measurable way under the action of a expansive matrix A, an affine Weyl group \(\widetilde{W}\) , and a full rank lattice \(\widetilde{\varGamma}\subset\mathbb{R}^{2}\) . 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. Ólafsson, 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.
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The research of the first author was partially supported by NSF grant DMS-0139386 and of the third author by NSF grant DMS-0402068. The research of the second author was partially supported by the grant MEXT-CT-2004-013477, Acronym MAMEBIA, of the European Commission. The essential part of this research was accomplished while all three authors participated in the Banff workshop on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006
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Larson, D.R., Massopust, P. & Ólafsson, G. Three-Way Tiling Sets in Two Dimensions. Acta Appl Math 108, 529–546 (2009). https://doi.org/10.1007/s10440-008-9424-y
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DOI: https://doi.org/10.1007/s10440-008-9424-y