Representation of finite abelian group elements by subsequence sums

@article{Grynkiewicz2009,
abstract = {Let $G\cong C_\{n_1\}\oplus \ldots \oplus C_\{n_r\}$ be a finite and nontrivial abelian group with $n_1|n_2|\ldots |n_r$. A conjecture of Hamidoune says that if $W=w_1\cdot \ldots \cdot w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\ge |W|+|G|-1\ge |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma (W)\equiv 0~\@mod \;|G|$, 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 $g=\underset\{i=1\}\{\overset\{n\}\{\sum \}\}w_is_i$, with $s_1\cdot \ldots \cdot s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\ge \frac\{1\}\{2\}|G|$.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 $|S|\ge |G|+\mathbb\{D\}(G)-1$, 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 $|S|\ge |G|+\mathbb\{D\}(G)-1$ can be relaxed to $|S|\ge |G|+\mathsf \{d\}^*(G)$, where $\mathsf \{d\}^*(G)=\underset\{i=1\}\{\overset\{r\}\{\sum \}\}(n_i-1)$. We also use this method to derive a variation on Hamidoune’s conjecture valid when at least $\mathsf \{d\}^*(G)$ of the $w_i$ are relatively prime to $|G|$.},
affiliation = {Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria.; Departamento de Matemáticas Decanato de Ciencias y Tecnologías Universidad Centroccidental Lisandro Alvarado Barquisimeto, Venezuela.; Departamento de Matemáticas y Centro ISYS Facultad de Ciencias Universidad Central de Venezuela Ap. 47567 Caracas 1041-A, Venezuela.},
author = {Grynkiewicz, David J., Marchan, Luz E., Ordaz, Oscar},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {zero-sum problem; Davenport constant; weighted subsequence sums; setpartition; $\mathsf \{d\}^*(G)$},
language = {eng},
number = {3},
pages = {559-587},
publisher = {Université Bordeaux 1},
title = {Representation of finite abelian group elements by subsequence sums},
url = {http://eudml.org/doc/10899},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Grynkiewicz, David J.
AU - Marchan, Luz E.
AU - Ordaz, Oscar
TI - Representation of finite abelian group elements by subsequence sums
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 559
EP - 587
AB - Let $G\cong C_{n_1}\oplus \ldots \oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|\ldots |n_r$. A conjecture of Hamidoune says that if $W=w_1\cdot \ldots \cdot w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\ge |W|+|G|-1\ge |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma (W)\equiv 0~\@mod \;|G|$, 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 $g=\underset{i=1}{\overset{n}{\sum }}w_is_i$, with $s_1\cdot \ldots \cdot s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\ge \frac{1}{2}|G|$.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 $|S|\ge |G|+\mathbb{D}(G)-1$, 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 $|S|\ge |G|+\mathbb{D}(G)-1$ can be relaxed to $|S|\ge |G|+\mathsf {d}^*(G)$, where $\mathsf {d}^*(G)=\underset{i=1}{\overset{r}{\sum }}(n_i-1)$. We also use this method to derive a variation on Hamidoune’s conjecture valid when at least $\mathsf {d}^*(G)$ of the $w_i$ are relatively prime to $|G|$.
LA - eng
KW - zero-sum problem; Davenport constant; weighted subsequence sums; setpartition; $\mathsf {d}^*(G)$
UR - http://eudml.org/doc/10899
ER -