Representation of finite abelian group elements by subsequence sums

Grynkiewicz, David J., Marchan, Luz E., and Ordaz, Oscar. "Representation of finite abelian group elements by subsequence sums." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 559-587. <http://eudml.org/doc/10899>.

@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 -