# On n-sums in an Abelian group

## Abstract

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

## Recommended Citation

Gao, W., Grynkiewicz, D., & Xia, X.
(2016). On n-sums in an Abelian group.* Combinatorics Probability and Computing**, 25* (3), 419-435.
https://doi.org/10.1017/S0963548315000255