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On syndetic Riesz sequences

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Abstract

Applying the solution to the Kadison–Singer problem, we show that every subset S of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials {eiλx}λ∈Λ such that Λ ⊂ ℤ is a set with gaps between consecutive elements bounded by \(\frac{C}{|\mathcal{S}|}\). In the case when \(\mathcal{S}\) is an open set we demonstrate, using quasicrystals, how such Λ can be deterministically constructed.

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Author information

Author notes

  1. Marcin Bownik

    Present address: Current address: Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Marcin Bownik

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Itay Londner

Authors
  1. Marcin Bownik
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  2. Itay Londner
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Corresponding author

Correspondence to Marcin Bownik.

Additional information

The authors are grateful to Prof. Olevskii for discussions on the subject of this paper.

The first author was supported in part by the NSF grant DMS-1665056.

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Bownik, M., Londner, I. On syndetic Riesz sequences. Isr. J. Math. 233, 113–131 (2019). https://doi.org/10.1007/s11856-019-1903-5

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  • DOI: https://doi.org/10.1007/s11856-019-1903-5

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