Optimal Arithmetic Structure in Exponential Riesz Sequences

We consider exponential systems E(Λ)={eiλt}λ∈Λ for Λ ⊂ 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 Λ which contains, for arbitrarily large N, an arithmetic progressions of length N and step ℓ= O(N^α) , α< 1 , 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(N). In this paper we show that every set S⊂ 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(S). We also give a partial geometric description of each class.