Optimal arithmetic structure in exponential Riesz sequences

Abstract

We consider exponential systems $E\left(\Lambda\right)=\left\{ e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ for $\Lambda\subset\mathbb{Z}$. It has been shown by Londner and Olevskii in [9] that there exists a subset of the circle, of positive Lebesgue measure, so that every set \Lambda which contains, for arbitrarily large N, an arithmetic progressions of length N and step $\ell=O\left(N^{\alpha}\right)$, $\alpha<1$, cannot be a Riesz sequence in the $L^{2}$ space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and step $\ell=O\left(N\right)$. In this paper we show that every set $\mathcal{S}\subset\mathbb{T}$ 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 $L^{2}\left(\mathcal{S}\right)$. We also give a partial geometric description of each class.