An additive property of almost periodic sets

TL;DR

This paper establishes a characterization of almost periodic sets through exponential sums and demonstrates how this property enables solving binary additive problems using the circle method.

Contribution

It provides a new criterion linking almost periodicity to exponential sum concentration, facilitating additive problem analysis.

Findings

Almost periodic sets correspond to exponential sums concentrated in minor arcs.

The circle method can be effectively applied to binary additive problems involving almost periodic sets.

A new characterization of almost periodicity in terms of exponential sums is proposed.

Abstract

We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method.