On n-sums in an Abelian group


Let G be an additive abelian group, let n ≥1 be an integer, let S be a sequence over G of length |S| ≥n + 1, and let (S) denote the maximum multiplicity of a term in S. Let Σ n (S) denote the set consisting of all elements in G which can be expressed as the sum of terms from a subsequence of S having length n. In this paper, we prove that either ng ϵ Σ n(S) for every term g in S whose multiplicity is at least (S) -1 or |Σ n (S)| ≥min{n + 1, |S| -n + | supp (S)| -1}, where |supp(S)| denotes the number of distinct terms that occur in S. When G is finite cyclic and n = |G|, this confirms a conjecture of Y. O. Hamidoune from 2003.

Publication Title

Combinatorics Probability and Computing