Optimal Arithmetic Structure in Exponential Riesz Sequences

Itay Londner*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider exponential systems E(Λ)={eiλt}λ∈Λ for Λ ⊂ Z. It has been previously shown by Londner and Olevskii (Stud Math 255(2):183–191, 2014) that there exists a subset of the circle, of positive Lebesgue measure, so that every set Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step ℓ= O(Nα) , α< 1 , cannot be a Riesz sequence in the L2 space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and stepℓ= O(N). In this paper we show that every set S⊂ T of positive measure belongs to a unique class, defined through the optimal growth rate of the step of arithmetic progressions with respect to the length that can be found in Riesz sequences in the space L2(S). We also give a partial geometric description of each class.

Original languageEnglish
Article number1
JournalJournal of Fourier Analysis and Applications
Issue number1
StatePublished - 1 Feb 2020
Externally publishedYes


  • Arithmetic progressions
  • Interpolation sets
  • Riesz sequences


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