A Step Beyond Kemperman's Structure Theorem

TL;DR

This paper extends Kemperman's structure theorem by providing a complete description of subset pairs in abelian groups where the sumset size equals the sum of the subset sizes, focusing on the case of equality.

Contribution

It offers a new comprehensive characterization of subset pairs in abelian groups when the sumset size exactly equals the sum of the subset sizes, complementing the classical inequality case.

Findings

Complete description of sumsets where |A+B|=|A|+|B|

Extension of Kemperman's theorem to the equality case

New structural insights into abelian group subsets

Abstract

A classical result of Kemperman gives a complete recursive description of the structure of those subsets $A$ and $B$ of an abelian group that fail to satisfy the triangle inequality, i.e., $|A+B|<|A|+|B|$. In this paper, we achieve the complete description in the case when equality holds: $|A+B|=|A|+|B|$.