An inverse theorem for sumsets of sets of positive density in the integers

TL;DR

This paper characterizes sets of positive density in integers where the sumset's density equals the sum of individual densities, providing a new inverse theorem related to Kneser's inequality and addressing a longstanding open problem.

Contribution

It offers a novel inverse theorem for sumsets of positive density sets in integers, extending Kneser's inequality and partially answering Erdős and Graham's open question.

Findings

Characterization of sets with sumset density equal to sum of densities

Provides a new inverse theorem for Kneser's inequality

Addresses a long-standing open problem in additive number theory

Abstract

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erd\H{o}s and Graham.