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

Sigrid Grepstad, Nir Lev

Research output: Contribution to journalArticlepeer-review


We consider systems of exponentials with frequencies belonging to simple quasicrystals in ℝd. We ask if there exist domains S in ℝd which admit such a system as a Riesz basis for the space L2 (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.

Original languageEnglish
Pages (from-to)4273-4298
Number of pages26
JournalTransactions of the American Mathematical Society
Issue number6
StatePublished - 2018


  • Bounded remainder set
  • Cut-and-project set
  • Quasicrystal
  • Riesz basis

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics


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