Avoiding zero-sum subsequences of prescribed length over the integers

Abstract

Let $t$ and $k$ be a positive integers, and let $I_k=\{i\in \mathbb{Z}:-k\le i\le k\}$. Let ${\mathsf{s}}_t^{\prime }(I_k)$ 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 ${\mathsf{s}}_t^{\prime }(I_k)=\infty$. In this paper, we prove that ${\mathsf{s}}_t^{\prime }(I_k)$ 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 ${\mathsf{s}}_t^{\prime }(I_k)$ is finite, then $t+k(k-1)\le {\mathsf{s}}_t^{\prime }(I_k)\le t+(2k-2)(2k-3)$. We also show that ${\mathsf{s}}_t^{\prime }(I_k)=t+k(k-1)$ holds for $k\le 3$ and conjecture that this equality holds for any $k\ge 1$.

13 pages. Added a new reference, corrected typos in the proof of Lemma 13, and added more details to the proof of Theorem 5