Representation of finite abelian group elements by subsequence sums

Grynkiewicz, David J.; Marchan, Luz E.; Ordaz, Oscar

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 559-587.

Soit $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ un groupe abélien fini non trivial avec $n_1|n_2|...|n_r$. Une conjecture d’Hamidoune dit que si $W=w_1·...·w_n$ est une suite d’entiers, tous, sauf au plus un, premiers à $\left|G\right|$, et $S$ une suite d’éléments de $G$ avec $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, la multiplicité maximale de $S$ au plus $\left|W\right|$, et $\sigma \left(W\right)\equiv 0mod\left|G\right|$, alors il existe un sous-groupe non trivial $H$ tel que tout élément $g\in H$ peut être représenté par une somme pondérée de la forme $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, avec $s_1·...·s_n$ une sous-suite de $S$. Nous donnons deux exemples qui montrent que cela n’est pas vrai en général, et nous caractérisons les contre-exemples pour les grands $\left|W\right|\ge \frac{1}{2}\left|G\right|$.

Un théorème de Gao, généralisant un résultat plus ancien d’Olson, dit que si $G$ est un groupe abélien fini, et $S$ une suite d’éléments de $G$ avec $\left|S\right|\ge \left|G\right|+𝔻\left(G\right)-1$, alors, soit tout élément de $G$ peut être représenté par une sous-somme de $S$ à $\left|G\right|$ termes, soit il existe une classe $g+H$ telle que tous sauf au plus $|G/H|-2$ termes de $S$ sont dans $g+H$. Nous établissons quelques cas très spéciaux d’un analogue pondéré de ce théorème, conjecturé par Ordaz et Quiroz, et quelques conclusions partielles dans les autres cas, qui impliquent un résultat récent d’Ordaz et Quiroz. Cela est fait, en partie, en étendant un théorème de Grynkiewicz sur les partitions pondérées, que nous utilisons également pour améliorer le résultat de Gao cité précédemment en montrant que l’hypothèse $\left|S\right|\ge \left|G\right|+𝔻\left(G\right)-1$ peut être affaiblie en $\left|S\right|\ge \left|G\right|+{\mathsf{d}}^{*}\left(G\right)$, où ${\mathsf{d}}^{*}\left(G\right)=\underset{i=1}{\sum ^r}(n_i-1)$. Nous utilisons aussi cette méthode pour déduire une variante de la conjecture d’Hamidoune valide si au moins ${\mathsf{d}}^{*}\left(G\right)$ des $w_i$ sont premiers à $\left|G\right|$.

Let $G\cong C_{n_1}\oplus ...\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1·...·w_n$ is a sequence of integers, all but at most one relatively prime to $\left|G\right|$, and $S$ is a sequence over $G$ with $\left|S\right|\ge \left|W\right|+\left|G\right|-1\ge \left|G\right|+1$, the maximum multiplicity of $S$ at most $\left|W\right|$, and $\sigma \left(W\right)\equiv 0mod\left|G\right|$, 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 $g=\underset{i=1}{\sum ^n}{w}_i{s}_i$, with $s_1·...·s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $\left|W\right|\ge \frac{1}{2}\left|G\right|$.

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 $\left|S\right|\ge \left|G\right|+𝔻\left(G\right)-1$, then either every element of $G$ can be represented as a $\left|G\right|$-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 $\left|S\right|\ge \left|G\right|+𝔻\left(G\right)-1$ can be relaxed to $\left|S\right|\ge \left|G\right|+{\mathsf{d}}^{*}\left(G\right)$, where ${\mathsf{d}}^{*}\left(G\right)=\underset{i=1}{\sum ^r}(n_i-1)$. We also use this method to derive a variation on Hamidoune’s conjecture valid when at least ${\mathsf{d}}^{*}\left(G\right)$ of the $w_i$ are relatively prime to $\left|G\right|$.

Classification : 11B75, 20K01
Mots clés : zero-sum problem, Davenport constant, weighted subsequence sums, setpartition, $\mathsf {d}^*(G)$

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