Representation of finite abelian group elements by subsequence sums. (English) Zbl 1214.11034

Let \(G\) be a finite abelian group. Hamidoune conjectures that if \(W=w_1\cdot\dots\cdot w_n\) is a sequence of integers, all but at most one relatively prime to \(|G|\), and \(S\) is a sequence over \(G\) with \(|S|\geq |W|+|G|-1\geq |G|+1\), the maximum multiplicity of \(S\) at most \(|W|\), and \(\sigma(W)\equiv 0 \pmod {|G|}\), then there exists a nontrivial subgroup \(H\) such that every element \(g\in H\) can be represented as a weighted subsequence sum of the form \(g=\sum_{i=1}^nw_is_i\), with \(s_1\cdot\dots\cdot s_n\) a subsequence of \(S\). The authors give two examples showing this does not hold in general, and characterize the counterexamples for large \(|W|\geq \frac{1}{2}|G|\).
A theorem of Gao states that if \(S\) is a sequence over \(G\) with \(|S|\geq |G|+D(G)-1\), 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 \(g+H\). The authors 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 they use to also improve the previously mentioned result of Gao by showing that the hypothesis \(|S|\geq |G|+D(G)-1\)can be relaxed to \(|S|\geq |G|+d^*(G)\), where \(d^*(G)=\sum_{i=1}^r (n_i-1)\). They also use this method to derive a variation of Hamidoune’s conjecture valid when at least \(d^*(G)\) of the \(w_i\) are relatively prime to \(|G|\).


11B75 Other combinatorial number theory
20K01 Finite abelian groups
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