Riesz sequences and arithmetic progressions

Abstract

Given a set $\mathcal{S}$ of positive measure on the circle and a set of integers $Λ$, one may consider the family of exponentials $E\left(Λ\right):=\left\{ e^{iλt}\right\}_{λ\inΛ}$ and ask whether it is a Riesz sequence in the space $L^{2}\left(\mathcal{S}\right)$. We focus on this question in connection with some arithmetic properties of the set of frequencies. Improving a result of Bownik and Speegle, we construct a set $\mathcal{S}$ such that $E\left(Λ\right)$ is never a Riesz sequence if $Λ$ contains arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N^{1-\varepsilon}\right)$. On the other hand, we prove that every set $\mathcal{S}$ admits a Riesz sequence $E\left(Λ\right)$ such that $Λ$ does contain arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N\right)$.