Abstract

The Kemperman structure theorem (KST) yields a recursive description of the structure of a pair of finite subsets A and B of an Abelian group satisfying |A+B|≤|A|+|B|−1. In this paper, we introduce a notion of quasi-periodic decompositions and develop their basic properties in relation to KST. This yields a fuller understanding of KST, and gives a way to more effectively use KST in practice. As an illustration, we first use these methods to (a) give conditions on finite sets A and B of an Abelian group so that there exists bB such that |A+(B b )|≥|A|+|B|−1, and to (b) give conditions on finite sets A,B,C₁,…,C_r of an Abelian group so that there exists bB such that |A+(B b )|≥|A|+|B|−1 and . Additionally, we simplify two results of Hamidoune, by (a) giving a new and simple proof of a characterization of those finite subsets B of an Abelian group G for which |A+B|≥min |G|−1,|A|+|B| holds for every finite subset AG with |A|≥2, and (b) giving, for a finite subset BG for which |A+B|≥min |G|,|A|+|B|−1 holds for every finite subset AG, a nonrecursive description of the structure of those finite subsets AG such that |A+B|=|A|+|B|−1.

Article Outline

1. Introduction

2. Quasi-periodic decompositions and KST

3. Some illustrative examples

Acknowledgements

References