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Transactions of the American Mathematical Society

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Riesz bases, Meyer's quasicrystals, and bounded remainder sets


Authors: Sigrid Grepstad and Nir Lev
Journal: Trans. Amer. Math. Soc. 370 (2018), 4273-4298
MSC (2010): Primary 42C15, 52C23, 11K38
DOI: https://doi.org/10.1090/tran/7157
Published electronically: February 28, 2018
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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.


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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
Email: sigrid.grepstad@ntnu.no

Nir Lev
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: levnir@math.biu.ac.il

DOI: https://doi.org/10.1090/tran/7157
Keywords: Riesz basis, quasicrystal, cut-and-project set, bounded remainder set
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
Article copyright: © Copyright 2018 American Mathematical Society

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