On the real projections of zeros of almost periodic functions

TL;DR

This paper investigates the structure of the real projections of zeros of almost periodic functions, providing criteria to determine their closure and showing that with linearly independent Fourier exponents, these sets lack isolated points.

Contribution

It offers new practical methods to analyze the closure of real zero projections and proves that such sets are dense without isolated points when Fourier exponents are rationally independent.

Findings

The set of real projections has no isolated points under certain conditions.

Provides criteria to determine if a real number is in the closure of the zero projections.

Shows the structure of zero projections depends on the linear independence of Fourier exponents.

Abstract

This paper deals with the set of the real projections of the zeros of an arbitrary almost periodic function defined in a vertical strip $U$. It provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents $\{\lambda_1,\lambda_2,\lambda_3,\ldots\}$ of an almost periodic function are linearly independent over the rational numbers, such a set has no isolated points in $U$.