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. Let (G, +, 0) be an abelian group. If A, B ⊆ G, then their sumset, A + B, is the set of all possible pairwise sums, i.e. {a + b | a ∈ A, b ∈ B}. A set A ⊆ G is Ha-periodic, if it is the union of Ha-cosets for some subgroup Ha of G (note this definition of periodic differs slightly from the usual by allowing Ha to be trivial). A set which is maximally Ha-periodic, with Ha the trivial group, is aperiodic, and otherwise we refer to A as nontrivially periodic. For notational convenience, we use φa : G → G/Ha to denote the natural homomorphism. If S is a sequence of terms from G, then an n-set partition of S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct. Thus such subsequences can be considered as sets. Let A = A1, . . . , An be an n-set partition of a sequence S of terms from G whose sumset (i.e. the sumset of whose terms) is Ha-periodic. Let y ∈ G/Ha. If y ∈ φa(Ai) for all i, then y is an Ha-nonexception, and otherwise y is an Ha-exception. The number of y ∈ G/Ha that are Ha-nonexceptions of A is denoted by N(A,Ha). The number of terms x of S such that φa(x) is an Ha-exception of A is denoted by E(A,Ha). Note N(A,Ha) = 1 |Ha| | n ⋂ i=1 (Ai+Ha)| and E(A,Ha) = n ∑