Quasi-periodic decompositions and the Kemperman structure theorem

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 b∈B such that |A+(B∖{b})|≥|A|+|B|−1, and to (b) give conditions on finite sets A,B,C1,…,Cr of an Abelian group so that there exists b∈B such that |A+(B∖{b})|≥|A|+|B|−1 and |A+(B∖{b})+∑i=1rCi|≥|A|+|B|+∑i=1r|Ci|−(r+2)+1. 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 A⊆G with |A|≥2, and (b) giving, for a finite subset B⊆G for which |A+B|≥min{|G|,|A|+|B|−1} holds for every finite subset A⊆G, a nonrecursive description of the structure of those finite subsets A⊆G such that |A+B|=|A|+|B|−1.