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 , C 1 , … , C r of an Abelian group so that there exists b ∈ B such that | A + ( B ∖ { b } ) | ≥ | A | + | B | − 1 and | A + ( B ∖ { b } ) + ∑ i = 1 r C i | ≥ | A | + | B | + ∑ i = 1 r | C i | − ( 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 .