Exponential Lower Bounds on the Generalized Erd\H{o}s-Ginzburg-Ziv Constant

TL;DR

This paper establishes exponential lower bounds for the generalized Erdős-Ginzburg-Ziv constant in finite abelian groups, improving understanding of its growth and providing new probabilistic methods for such bounds.

Contribution

It introduces probabilistic techniques to derive exponential lower bounds for the generalized Erdős-Ginzburg-Ziv constant, surpassing previous linear bounds.

Findings

Exponential lower bound for _{2n}(C_n^r)

Exponential lower bound for _{kn}(C_n^r)

Improved understanding of the growth rate of the constant

Abstract

For a finite abelian group $G$, the generalized Erd\H{o}s--Ginzburg--Ziv constant $\mathsf s_{k}(G)$ is the smallest $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. If $n = \exp(G)$ is the exponent of $G$, the previously best known bounds for $\mathsf s_{kn}(C_n^r)$ were linear in $n$ and $r$ when $k\ge 2$. Via a probabilistic argument, we produce the exponential lower bound \[ \mathsf s_{2n}(C_n^r) > \frac{n}{2}[1.25 - O(n^{-3/2})]^r \] for $n > 0$. For the general case, we show \[ \mathsf s_{kn}(C_n^r) > \frac{kn}{4}\Big(1+\frac{1}{ek} + O\Big(\frac{1}{n}\Big)\Big)^r. \]