Generalized Fourier quasicrystals, almost periodic sets, and zeros of Dirichlet series

2023/11/05 by Serhii Favorov, Favorov, Serhii · 1 citation
Mathematics · Materials Science · #Analytic and geometric function theory #Meromorphic and Entire Functions #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2311.02728

Abstract

Let S(z) be an absolutely convergent Dirichlet series with a bounded spectrum and only real zeros a_n, let μ be the sum of unit masses at points a_n. It is proven that the Fourier transform μ in the sense of distributions is a purely point measure. Conversely, in terms of the properties of μ, a sufficient condition was found when a_n are zeros of an absolutely convergent Dirichlet series with bounded spectrum. Also, in terms of the properties of μ, a criterion is established that a_n are zeros of an almost periodic entire function of exponential growth. In all cases, the multiplicity of zeros is taken into account. Almost periodic sets, introduced by B.Levin and M.Krein in 1948, play important role in our investigations. In particular, we show a new simple representation of such sets.

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