Improved lower bound on generalized Erdos-Ginzburg-Ziv constants

TL;DR

This paper improves the lower bounds on the generalized Erdős-Ginzburg-Ziv constants for finite Abelian groups, providing sharper estimates for the minimal sequence length ensuring zero-sum subsequences.

Contribution

The authors refine existing bounds on $s_{kn}(C_n^r)$, offering a more precise lower bound that enhances previous results for these zero-sum constants.

Findings

New lower bound for $s_{kn}(C_n^r)$ with sharper constants

Improved estimates extend previous bounds by Bitz et al.

Results applicable to finite Abelian groups with exponent n.

Abstract

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then $s_{2n}(C_{n}^{r}) > \frac{n}{2}[\frac{5}{4}-O(n^{-\frac{3}{2}})]^{r}$ and $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{1}{e k}-O(\frac{1}{n})]^{r}$ for $k > 2$. In this note, we sharpen their general bound by showing that $s_{k n}(C_{n}^{r}) > \frac{k n}{4} [1+\frac{(k-1)^{(k-1)}}{k^k}-O(\frac{1}{n})]^{r}$ for $k > 2$.