Modified Erdős-Ginzburg-Ziv Constants for $(\mathbb{Z}/n\mathbb{Z}{)}^2$

Abstract

For an abelian group $G$ and an integer $t>0$, the modified Erdős-Ginzburg-Ziv constant $s_t^{\prime }(G)$ 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 $s_t^{\prime }(G)$ 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 Erdős-Ginzburg-Ziv constant for $G={\left(\mathbb{Z}/n\mathbb{Z}\right)}^2$ and subsequences of length $tn$.

11 pages. arXiv admin note: substantial text overlap with arXiv:1808.08486 by other authors