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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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
  • MR Author ID: 1047019
  • Email:
  • Nir Lev
  • Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
  • MR Author ID: 760715
  • Email:
  • 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:
  • MathSciNet review: 3811528