Periodicity of almost periodic zero sets and almost periodic functions

TL;DR

This paper investigates the structure of zeros of almost periodic functions, showing that under certain conditions, these functions can be expressed as products of periodic functions with related periods, extending to some Dirichlet series.

Contribution

It establishes a new link between the zero sets of almost periodic functions and their representation as products of periodic functions with commensurable periods.

Findings

Zeros of certain almost periodic functions form discrete difference sets.

Such functions can be represented as infinite products of periodic functions.

Results apply to specific classes of Dirichlet series.

Abstract

Whenever all differences between zeros of two holomorphic almost periodic functions in a strip form a discrete set, then both functions are infinite products of periodic functions with commensurable periods. In particular, the result is valid for some classes of Dirichlet series.