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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Multiple lattice tiles and Riesz bases of exponentials
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by Mihail N. Kolountzakis PDF
Proc. Amer. Math. Soc. 143 (2015), 741-747 Request permission

Abstract:

Suppose $\Omega \subseteq \mathbb {R}^d$ is a bounded and measurable set and $\Lambda \subseteq \mathbb {R}^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the $\Lambda$-translates of $\Omega$ cover almost every point of $\mathbb {R}^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp (2\pi i t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\mathbb {R}^d$, which forms a Riesz basis of $L^2(\Omega )$. This result was recently proved by Grepstad and Lev under the extra assumption that $\Omega$ has boundary of measure $0$, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies $T$ turns out to be a finite union of shifted copies of the dual lattice $\Lambda ^*$. It can be chosen knowing only $\Lambda$ and $k$ and is the same for all $\Omega$ that tile multiply with $\Lambda$.
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Additional Information
  • Mihail N. Kolountzakis
  • Affiliation: Department of Mathematics and Applied Mathematics, University of Crete, GR-700 13, Heraklion, Crete, Greece
  • Email: kolount@math.uoc.gr
  • Received by editor(s): May 12, 2013
  • Published electronically: October 8, 2014
  • Additional Notes: The author was supported in part by grant No 3803 from the University of Crete.
  • Communicated by: Alexander Iosevich
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 741-747
  • MSC (2010): Primary 42B99
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12310-0
  • MathSciNet review: 3283660