Optimal arithmetic structure in exponential Riesz sequences

2019/03/13 by Itay Londner, Londner, Itay
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1903.05570

To appear in the Journal of Fourier Analysis and Applications

arxiv created 2019/10/12 · arxiv updated 2019/10/15

Abstract

We consider exponential systems E(Λ)=\ e^iλt\ _λ∈Λ for Λ⊂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 Λwhich 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 step ℓ=O(N). In this paper we show that every set S⊂\mathbbT 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.

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