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 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 $\Lambda$ 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.