2014/04/07 by Itay Londner, Londner, Itay, Alexander Olevskii +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA
paper · pdf · doi:10.48550/arxiv.1404.1796
arxiv created 2016/06/10 · arxiv updated 2016/06/13
Given a set S of positive measure on the circle and a set of integers Λ, one may consider the family of exponentials E(Λ):=\ eiλt\λ∈Λ and ask whether it is a Riesz sequence in the space L2(S). 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 S such that E(Λ) is never a Riesz sequence if Λ contains arbitrary long arithmetic progressions of length N and step ℓ=O(N1-ε). On the other hand, we prove that every set S admits a Riesz sequence E(Λ) such that Λ does contain arbitrary long arithmetic progressions of length N and step ℓ=O(N).