Modified Erd\"os--Ginzburg--Ziv Constants for $\mathbb Z/n\mathbb Z$ and $(\mathbb Z/n\mathbb Z)^2$

TL;DR

This paper determines the modified Erdős–Ginzburg–Ziv constants for cyclic groups and their squares, providing exact values for specific cases, which advances understanding of zero-sum sequences in finite abelian groups.

Contribution

The paper explicitly computes the modified Erdős–Ginzburg–Ziv constants for Z/n and (Z/n)^2 for certain parameters, filling gaps in the existing literature.

Findings

Computed s_t'(G) for G = Z/n and t = n.

Derived exact values for s_t'(G) when G = (Z/n)^2.

Extended the understanding of zero-sum sequences in finite abelian groups.

Abstract

For an abelian group $G$ and an integer $t > 0$, the \emph{modified Erd\"os--Ginzburg--Ziv constant} $s_t'(G)$ is the smallest integer $\ell$ such that any zero-sum sequence of length at least $\ell$ with elements in $G$ contains a zero-sum subsequence (not necessarily consecutive) of length $t$. We compute $s_t'(G)$ for $G = \mathbb Z/n\mathbb Z$ and for $t = n$, $G = (\mathbb Z/n\mathbb Z)^2$.