Name
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, C1,...,Cr of an Abelian group so that there exists b ∈ B such that |A + (B \ {b})| ≥ |A| + |B| - 1 and |A + (B \ {b}) + Σi=1r Ci| ≥ |A| |B| + Σi=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. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1016/j.ejc.2004.06.011 | Eur. J. Comb. |
Keywords | Field | DocType |
finite subsets,finite subset,quasi-periodic decomposition,abelian group,finite set,nonrecursive description,finite subsets b,recursive description,finite subset b,finite subsets a,kemperman structure theorem,satisfiability | Structured program theorem,Abelian group,Discrete mathematics,Combinatorics,Finite set,Periodic graph (geometry),Mathematics | Journal |
Volume | Issue | ISSN |
26 | 5 | 0195-6698 |
Citations | PageRank | References |
4 | 0.80 | 5 |
Authors | ||
1 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| David J. Grynkiewicz | 1 | 42 | 10.33 |