Optimal Arithmetic Structure in Exponential Riesz Sequences

摘要

We consider exponential systems Eei lambda t lambda is an element of subset of 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 which contains, for arbitrarily large N, an arithmetic progressions of length N and step l=ON alpha alpha 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 In this paper we show that every set S subset of 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. We also give a partial geometric description of each class.