Representation of finite abelian group elements by subsequence sums

Let G ≅ C n 1 ⊕ … ⊕ C n r be a finite and nontrivial abelian group withn₁|n₂| . . . |nr A conjecture of Hamidoune says that ifW=w₁ · . . . ·wn is a sequence of integers, all but at most one relatively prime to |G|, andSis a sequence overGwith |S| ≥ |W| + |G| − 1 ≥ |G| + 1, the maximum multiplicity ofSat most |W|, andσ(W)≡ 0 mod |G|, then there exists a nontrivial subgroupHsuch that every elementg∈Hcan be represented as a weighted subsequence sum of the form g = ∑ i = 1 n w i s i , withs₁ · . . . ·sn a subsequence ofS. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W| ≥ ½|G|. A theorem of Gao, generalizing an older result of Olson, says that ifGis a finite abelian group, andSis a sequence overGwith |S| ≥ |G| + D(G) − 1, then either every element ofGcan be represented as a |G|-term subsequence sum fromS, or there exists a cosetg+Hsuch that all but at most |G/H| − 2 terms ofSare fromg+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S| ≥ |G| + D(G) − 1 can be relaxed to |S| ≥ |G| + d*(G), where d * ( G ) = ∑ i = 1 r ( n i − 1 ) . We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d*(G) of thewi are relatively prime to |G|.