Behaving sequences

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.