Analogues of Fourier quasicrystals for a strip

TL;DR

This paper explores a new class of discrete measures on a strip, establishing a correspondence with zero sets of exponential polynomials, extending the theory of Fourier quasicrystals to a complex setting.

Contribution

It introduces a novel analogue of Fourier quasicrystals for measures on a strip and links their supports to zero sets of exponential polynomials.

Findings

Supports of measures correspond to zero sets of exponential polynomials.

Results extend to measures related to Dirichlet series with bounded spectrum.

Abstract

We study a certain family of discrete measures with unit masses on a horizontal strip as an analogue of Fourier quasicrystals on the real line. We prove a one-to-one correspondence between supports of measures from this family and zero sets of exponential polynomials with imaginary frequencies. This result is the special case of a general result on measures whose supports correspond to zero sets of absolutely convergent Dirichlet series with bounded spectrum.