On small sumsets in abelian groups. (English) Zbl 0956.11008

Vsevolod, Lev F.

Deshouillers, Jean-Marc (ed.) et al., Structure theory of set addition. Paris: Société Mathématique de France, Astérisque. 258, 317-321 (1999).

Let \(G\) be an Abelian group and \(A,B\) two of its finite subsets such that \(|A+B|<|A|+|B|-1\) and that \(\max\{|A|,|B|\}>1\). The main result of the paper says that then there exist a finite subgroup \(H\subseteq G\) and two finite subsets \(S_1,S_2\subseteq G\) such that \(A\subseteq S_1+H\), \(B\subseteq S_2+H\) and \(|A+B|\geq(|S_1|+|S_2|-2)|H|+1\) where if the maximum of \(\{|S_1|,|S_2|\}\) is \(1\) the first parenthesis should be replaced by \(1/2\). If the minimum of this set is \(>1\) then both \(S_1\) and \(S_2\) are true arithmetic progressions with common difference \(d\) of order at least \(|S_1|+|S_2|+1\).
For the entire collection see [Zbl 0919.00044].

Reviewer: Štefan Porubský (Praha)

MSC:

11B75	Other combinatorial number theory
20D60	Arithmetic and combinatorial problems involving abstract finite groups
11P99	Additive number theory; partitions
05E99	Algebraic combinatorics

Keywords:

sumsets; small doubling; abelian group; finite subsets