Almost Periodic Functions in terms of Bohr's Equivalence Relation

TL;DR

This paper introduces an equivalence relation for almost periodic functions to refine their characterization, showing that limit points of translates are equivalent functions and applying this to approximate the Riemann zeta function.

Contribution

It defines a new equivalence relation on almost periodic functions and uses it to refine Bochner's characterization, with applications to the Riemann zeta function.

Findings

Limit points of translates are exactly the equivalent functions.

Exponential sums equivalent to ζ(s) can be approximated by vertical translates.

Provides a new perspective on the structure of almost periodic functions.

Abstract

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner's result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, $\zeta(s)$, can be uniformly approximated in $\{s=\sigma+it:\sigma>1\}$ by certain vertical translates of $\zeta(s)$.