Representation of Finite Abelian Group Elements by Subsequence Sums
Representation of Finite Abelian Group Elements by Subsequence Sums
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DOI：
10.5802/jtnb.689
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发表时间：
2008-06
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通讯作者：
D. Grynkiewicz;L. Marchan;Oscar Ordaz
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作者：
D. Grynkiewicz;L. Marchan;Oscar Ordaz
设$G\cong C_{n_1}\oplus. \o + C_{n_r}$是一个有限非平凡的阿贝尔群，其中n_1|氮气|...| n_r$。Hamidoune的一个猜想说，如果$W=w_1. w_n$是一个整数序列，除了最多一个与$互质|G| $S$是一个在$G$上的序列，其中$|S|格克|W| +| G|-1\geq| G| +1$，$S$的最大多重性至多为$|W| $，和$\sigma（W）\equiv 0\mod| G| $，则存在一个非平凡子群$H$使得H$中的每个元素$g\都可以表示为形式为$g=\sum_{i=1}^{n}w_is_i$的加权子序列和，其中$s_1. s_n$是$S$的子序列。我们给出了两个例子，表明这并不成立，并描述了反例大$|W|\geq {1/2}| G| $.高的一个定理推广了奥尔森的一个较早的结果，称如果$G$是有限阿贝尔群，并且$S$是$G$上的序列，其中$|S|格克|G| +D（G）-1$，则要么G$的每个元素都可以表示为一个$|G| $-term subsequence sum from $S$，或者存在陪集$g+H$使得除了至多$|G/H|-2$条件$S$是从$g+H$。我们在Ordaz和Quiroz给出的这个定理的一个加权类比中建立了一些非常特殊的情形，并在其余情形中建立了一些部分结论，这些结论暗示了Ordaz和Quiroz的一个最近结果.这部分是通过推广Grynkiewicz的一个加权集划分定理来实现的，然后我们用它来改进Gao前面提到的结果，证明假设|S|格克|G| +D（G）-1 $可放宽为$|S|格克|G| +d^*（G）$，其中d^*（G）=\Sum_{i=1}^{r}（n_i-1）$.我们还利用这种方法导出了Hamidoune猜想的一个变形，当至少$w_i$中的$d^*（G）$与$互素时，该猜想成立|G| $.
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 $|G|$, 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 $\sigma(W)\equiv 0\mod |G|$, 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 $g=\sum_{i=1}^{n}w_is_i$, 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 $|W|\geq {1/2}|G|$. 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 $|S|\geq |G|+D(G)-1$, 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 $g+H$. 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 $|S|\geq |G|+D(G)-1$ can be relaxed to $|S|\geq |G|+d^*(G)$, where $d^*(G)=\Sum_{i=1}^{r}(n_i-1)$. We also use this method to derive a variation on Hamidoune's conjecture valid when at least $d^*(G)$ of the $w_i$ are relatively prime to $|G|$.