Avoiding zero-sum subsequences of prescribed length over the integers

Abstract

Let $t$ and $k$ be a positive integers, and let $I_k=\{i\in \mathbb{Z}:\; -k\leq i\leq k\}$. Let $\mathsf{s}'_t(I_k)$ be the smallest positive integer $\ell$ such that every zero-sum sequence $S$ over $I_k$ of length $|S|\ge \ell$ contains a zero-sum subsequence of length $t$. If no such $\ell$ exists, then let $\mathsf{s}'_t(I_k)=\infty$. In this paper, we prove that $\mathsf{s}'_t(I_k)$ is finite if and only if every integer in $[1,D(I_k)]$ divides $t$, where $D(I_k)=\max\{2,2k-1\}$ is the Davenport constant of $I_k$. Moreover, we prove that if $\mathsf{s}'_t(I_k)$ is finite, then $t+k(k-1)\leq \mathsf{s}'_t(I_k)\leq t+(2k-2)(2k-3)$. We also show that $\mathsf{s}'_t(I_k)=t+k(k-1)$ holds for $k\leq 3$ and conjecture that this equality holds for any $k\geq1$.