Generalized Fourier quasicrystals and almost periodic sets

TL;DR

This paper characterizes positive measures with discrete Fourier transforms on the real line, constructs entire almost periodic functions with prescribed zeros, and provides conditions for their exponential growth, advancing understanding of Fourier quasicrystals.

Contribution

It introduces explicit forms of almost periodic functions with given zero sets and characterizes measures with purely point Fourier transforms, based on properties of almost periodic sets.

Findings

Explicit form of entire almost periodic functions with specified zeros

Necessary and sufficient conditions for exponential growth of these functions

A simple representation of almost periodic sets on the line

Abstract

Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire almost periodic function with a set of zeros $\Lambda$, taking multiplicities into account. A necessary and sufficient condition for the exponential growth of this function is also found. Our constructions are based on the properties of almost periodic sets on the line. In particular, in the article we find a simple representation of such sets.