Behaving sequences

Abstract

Let S be a sequence over an additively written abelian group. We denote by h(S) the maximum of the multiplicities of S, and by (S) the set of all subsums of S. In this paper, we prove that if S has no zero-sum subsequence of length in [1, h(S)], then either | (S)| 2|S| − 1, or S has a very special structure which implies in particular that (S) is an interval. As easy consequences of this result, we deduce several well-known results on zero-sum sequences.