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.