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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

DOI: https://doi.org/10.1090/S0002-9939-2014-12310-0
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