[Paper Review] Representation of Finite Abelian Group Elements by Subsequence Sums

David J. Grynkiewicz, E. Marchan+1 more|ArXiv.org|Jun 2, 2008

Limits and Structures in Graph Theory13 references3 citations

TL;DR

This paper investigates weighted subsequence sums in finite abelian groups, disproving a conjecture by Hamidoune on subgroup representation via subsequence sums and characterizing counterexamples for large weight sequences. It establishes a weighted analog of Gao's theorem with a relaxed condition using the invariant $\mathsf{d}^*(G)$, and proves that $\Sigma_{|W|}(W,S) = \Sigma(W,S)$ under certain multiplicity and gcd conditions, improving prior bounds in zero-sum combinatorics.

ABSTRACT

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 $σ(W)\equiv 0\mod |G|$, 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 $|G|$.

Motivation & Objective

• To investigate the representation of group elements as weighted subsequence sums in finite abelian groups.
• To test and disprove a conjecture by Hamidoune on the existence of nontrivial subgroups generated by such sums under specific conditions.
• To extend Gao's classical zero-sum theorem to a weighted setting, improving the required sequence length condition.
• To characterize the structure of sequences that fail the conjectured representation property, especially for large weight sequences.
• To refine the invariant $\mathsf{d}^*(G)$ as a replacement for $\mathsf{D}(G)$ in weighted zero-sum theorems.

Proposed method

• Uses a weighted setpartition theorem of Grynkiewicz to analyze subsequence sums under integer weights.
• Applies the invariant $\mathsf{d}^*(G) = \sum_{i=1}^r (n_i - 1)$ as a refined measure of group complexity in place of $\mathsf{D}(G)$.
• Employs the notation $\Sigma_n(W,S)$ for the set of all $n$-term weighted subsequence sums from sequences $W$ and $S$.
• Introduces the condition $\sigma(W) \equiv 0 \pmod{|G|}$ to ensure global zero-sum behavior in weighted sums.
• Uses the stabilizer and periodicity structure of sumsets to analyze subgroup structure in the image of weighted sums.
• Applies a modified version of a result from [7] to show $\Sigma(W,S) = \Sigma_{|W|}(W,S)$ when the maximum multiplicity of $S$ is at least $\mathsf{D}(G) - 1$.

Experimental results

Research questions

• RQ1 Does every element of a nontrivial subgroup $H$ of a finite abelian group $G$ arise as a weighted subsequence sum $\sum w_i s_i$ for a subsequence $s_i$ of $S$, under Hamidoune's conditions?
• RQ2 What are the structural properties of sequences $S$ that violate the conjectured representation property in Hamidoune's conjecture?
• RQ3 Can the classical zero-sum theorem of Gao be extended to a weighted setting with a relaxed length condition?
• RQ4 Under what conditions does $\Sigma(W,S) = \Sigma_{|W|}(W,S)$ hold, particularly when $\mathsf{h}(S) \geq \mathsf{D}(G) - 1$?
• RQ5 How does the invariant $\mathsf{d}^*(G)$ compare to $\mathsf{D}(G)$ in determining the threshold length for weighted subsequence sum coverage?

Key findings

• The paper provides two counterexamples to Hamidoune's conjecture, showing it does not hold in general.
• For $|W| \geq \frac{1}{2}|G|$, the paper fully characterizes the counterexamples to Hamidoune's conjecture.
• The authors prove that $\Sigma(W,S) = \Sigma_{|W|}(W,S)$ holds when $\mathsf{h}(S) \geq \mathsf{D}(G) - 1$, under the condition $\sigma(W) \equiv 0 \pmod{|G|}$.
• The hypothesis $|S| \geq |G| + \mathsf{D}(G) - 1$ in Gao's theorem is improved to $|S| \geq |G| + \mathsf{d}^*(G)$, where $\mathsf{d}^*(G) = \sum_{i=1}^r (n_i - 1)$.
• A weighted analog of Gao's theorem is established under the condition that at least $\mathsf{d}^*(G)$ of the weights $w_i$ are coprime to $|G|$.
• The paper derives a variation of Hamidoune's conjecture that holds when at least $\mathsf{d}^*(G)$ of the $w_i$ are relatively prime to $|G|$.