Onn-Sums in an Abelian Group

Abstract

LetGbe an additive abelian group, letn⩾ 1 be an integer, letSbe a sequence overGof length |S| ⩾n+ 1, and let${\mathsf h}$(S) denote the maximum multiplicity of a term inS. Let Σn(S) denote the set consisting of all elements inGwhich can be expressed as the sum of terms from a subsequence ofShaving lengthn. In this paper, we prove that eitherng∈ Σn(S) for every termginSwhose multiplicity is at least${\mathsf h}$(S) − 1 or |Σn(S)| ⩾ min{n+ 1, |S| −n+ | supp (S)| − 1}, where |supp(S)| denotes the number of distinct terms that occur inS. WhenGis finite cyclic andn= |G|, this confirms a conjecture of Y. O. Hamidoune from 2003.