#### Title

Quasi-periodic decompositions and the Kemperman structure theorem

#### 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 b ∈ B such that A + (B \ {b}) ≥ A + B - 1, and to (b) give conditions on finite sets A, B, Cl..., Cr of an Abelian group so that there exists b ∈ B such that A + (B \ {b}) ≥ A + B - 1, and A + (B \ {b}) + ∑i=1r Ci ≥ 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. © 2004 Elsevier Ltd. All rights reserved.

#### Publication Title

European Journal of Combinatorics

#### Recommended Citation

Grynkiewicz, D.
(2005). Quasi-periodic decompositions and the Kemperman structure theorem.* European Journal of Combinatorics**, 26* (5), 559-575.
https://doi.org/10.1016/j.ejc.2004.06.011