A short proof of Kneser's addition theorem for abelian groups

TL;DR

This paper presents a concise proof of Kneser's addition theorem for abelian groups, establishing a lower bound on the size of the sumset using an intersection union argument.

Contribution

It provides a simplified and elegant proof of Kneser's theorem, improving understanding and accessibility of this fundamental result in additive combinatorics.

Findings

Short proof based on intersection union argument

Clarifies the structure of sumsets in abelian groups

Enhances pedagogical approach to additive theorems

Abstract

Martin Kneser proved the following addition theorem for every abelian group $G$. If $A,B \subseteq G$ are finite and nonempty, then $|A+B| \ge |A+K| + |B+K| - |K|$ where $K = \{g \in G \mid g+A+B = A+B \}$. Here we give a short proof of this based on a simple intersection union argument.