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}) + r ∑ i=1 Ci| ≥ |A| + |B| + r ∑ i=1 |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.