2025/04/04 by S. Favorov, Favorov, Sergii Yu.
Materials Science · Mathematics · #Quasicrystal Structures and Properties #Analytic and geometric function theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2504.03365
Let f be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let μ be the measure on the set of zeros of f whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for μ and prove that it is a pure point measure on \R whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of f. Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for f to be a finite product of sines.