Representation of finite abelian group elements by subsequence sums
Let G ≅ Cn1 ⊕ ⋯ ⊕ Cnr be a finite and nontrivial abelian group with n1|n2| ⋯ |nr. A conjecture of Hamidoune says that if W = w1 · ⋯ · wn is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S| > |W| + |G| - 1 ≥ |G| + 1, the maximum multiplicity of S at most |W|, and σ(W) ≡ 0 mod |G|, then there exists a nontrivial subgroup H such that every element g ∈ H can be represented as a weighted subsequence sum of the form g = (Formula presented) wisi, with s1 ·⋯ ·sn a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W| ≥ 1/2 |G|. A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S| ≥ |G| + D(G) - 1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g + H such that all but at most |G/H| - 2 terms of S are from g + 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) = (Formula presented) (ni - 1). We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d*(G) of the wi are relatively prime to |G|.
Journal de Theorie des Nombres de Bordeaux
Grynkiewicz, D., Marchan, L., & Ordaz, O. (2009). Representation of finite abelian group elements by subsequence sums. Journal de Theorie des Nombres de Bordeaux, 21 (3), 559-587. https://doi.org/10.5802/jtnb.689