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

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 $tn$.