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Proceedings of the American Mathematical Society

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Multiple lattice tiles and Riesz bases of exponentials


Author: Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 143 (2015), 741-747
MSC (2010): Primary 42B99
DOI: https://doi.org/10.1090/S0002-9939-2014-12310-0
Published electronically: October 8, 2014
MathSciNet review: 3283660
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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

Keywords: Riesz bases of exponentials, tiling
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
Article copyright: © Copyright 2014 American Mathematical Society