Additive number theory and the Dyson transform

TL;DR

This paper explores Dyson's generalization of Mann's theorem on the lower bounds of Shnirel'man density, introduces the Dyson transform, and demonstrates its applications in additive number theory, including Kneser's inequality.

Contribution

It provides a detailed explanation of Dyson's proof, the Dyson transform, and new applications in additive number theory, expanding understanding of sumset properties.

Findings

Dyson's theorem extends Mann's results to rank r sumsets.

The Dyson transform is a versatile tool in additive number theory.

Applications include a generalized form of Kneser's inequality.

Abstract

In 1942 Mann solved a famous problem, the $\alpha+\beta$ conjecture, about the lower bound of the Shnirel'man density of sums of sets of positive integers. In 1945, Dyson generalized Mann's theorem and obtained a lower bound for the Shnirel'man density of rank $r$ sumsets. His proof introduced the Dyson transform, an important tool in additive number theory. This paper explains the background of Dyson's work, gives Dyson's proof of his theorem, and includes several applications of the Dyson transform, such as Kneser's inequality for sums of finite subsets of an arbitrary additive abelian group.