## 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 $\Lambda$.## References

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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