## Abstract

We consider systems of exponentials with frequencies belonging to simple quasicrystals in ℝ^{d}. We ask if there exist domains S in 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.

Original language | English |
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Pages (from-to) | 4273-4298 |

Number of pages | 26 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 6 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Publisher Copyright:© 2018 American Mathematical Society.

### Funding

Received by the editors May 27, 2016, and, in revised form, November 21, 2016. 2010 Mathematics Subject Classification. Primary 42C15, 52C23, 11K38. Key words and phrases. Riesz basis, quasicrystal, cut-and-project set, bounded remainder set. 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.

Funders | Funder number |
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Austrian Science Fund | F5505-N26 |

Israel Science Foundation | 225/13 |

## Keywords

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