# Modified Erd\H{o}s-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z})^2$

## TL;DR

This paper investigates bounds for the modified Erdős-Ginzburg-Ziv constants in specific finite abelian groups, extending known results and exploring the behavior of zero-sum subsequences of various lengths.

## Contribution

It provides new bounds for the modified Erdős-Ginzburg-Ziv constants in groups like  and , and analyzes the constants for subsequences of different lengths.

## Key findings

- Bounds for s'_{t}(G) in ()^2 and () groups.
- Bounds for s'_{t}(G) in ()^d with specified subsequence lengths.
- Analysis of the Erds-Ginzburg-Ziv constant for ()^2 with subsequences of length tn.

## Abstract

For an abelian group $G$ and an integer $t > 0$, the modified Erd\H{o}s-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 bounds for $s'_{t}(G)$ for $G = \left(\mathbb{Z}/n\mathbb{Z}\right)^2$ and $G = \left(\mathbb{Z}/n_1\mathbb{Z} \times \mathbb{Z}/n_2\mathbb{Z}\right)$. We also compute bounds for $G = \left(\mathbb{Z}/p\mathbb{Z}\right)^d$ where the subsequence can be any length in $\{p, \dots, (d-1)p\}$. Lastly, we investigate the Erd\H{o}s-Ginzburg-Ziv constant for $G = \left(\mathbb{Z}/n\mathbb{Z}\right)^2$ and subsequences of length $tn$.