On a Conjecture of Hamidoune for Subsequence Sums

Let G be an abelian group of order m, let S be a sequence of terms from G with k distinct terms, let m ∧ S denote the set of all elements that are a sum of some m-term subsequence of S, and let |S| be the length of S. We show that if |S| ≥ m + 1, and if the multiplicity of each term of S is at most m − k + 2, then either |m ∧ S| ≥ min{m, |S| − m + k − 1}, or there exists a proper, nontrivial subgroup Ha of index a, such that m ∧ S is a union of Ha-cosets, Ha ⊆ m ∧ S, and all but e terms of S are from the same Ha-coset, where e ≤ min{|S|−m+k−2 |Ha| − 1, a − 2} and |m ∧ S| ≥ (e + 1)|Ha|. This confirms a conjecture of Y. O. Hamidoune.