## Riesz bases, Meyer’s quasicrystals, and bounded remainder sets

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- by Sigrid Grepstad and Nir Lev PDF
- Trans. Amer. Math. Soc.
**370**(2018), 4273-4298 Request permission

## Abstract:

We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb {R}^d$. We ask if there exist domains $S$ in $\mathbb {R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several dimensions (and to the non-periodic setting) the results obtained earlier in dimension one.## References

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## Additional Information

**Sigrid Grepstad**- Affiliation: Institute of Financial Mathematics and Applied Number Theory, Johannes Kepler University, 4040 Linz, Austria
- Address at time of publication: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
- MR Author ID: 1047019
- Email: sigrid.grepstad@ntnu.no
**Nir Lev**- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
- MR Author ID: 760715
- Email: levnir@math.biu.ac.il
- Received by editor(s): May 27, 2016
- Received by editor(s) in revised form: November 21, 2016
- Published electronically: February 28, 2018
- Additional Notes: The first author was supported by the Austrian Science Fund (FWF), Project F5505-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”

The second author was partially supported by the Israel Science Foundation grant No. 225/13 - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 4273-4298 - MSC (2010): Primary 42C15, 52C23, 11K38
- DOI: https://doi.org/10.1090/tran/7157
- MathSciNet review: 3811528