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Acta Applicandae Mathematicae

, 108:529 | Cite as

Three-Way Tiling Sets in Two Dimensions

  • David R. Larson
  • Peter MassopustEmail author
  • Gestur Ólafsson
Article

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.

Keywords

Affine Weyl groups Tilings Wavelet sets 

Mathematics Subject Classification (2000)

20F55 28A80 42C40 51F15 46E25 65T60 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • David R. Larson
    • 1
  • Peter Massopust
    • 2
    Email author
  • Gestur Ólafsson
    • 3
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Helmholtz Zentrum München—National Research Center for Environmental Health, Institute of Biomathematics and Biometry, and Centre of Mathematics M6Technische Universität MünchenMünchenGermany
  3. 3.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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