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Riesz sequences and arithmetic progressions

Itay Londner; Alexander Olevskiĭ — 2014

Studia Mathematica

Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E (Λ) : = e_{λ \in Λ}^{i λ t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $ℓ = O (N^{1 - ε})$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...

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