An Analogue of the Erd\H{o}s-Ginzburg-Ziv Theorem over $\mathbb{Z}$

Let $\mathcal{S}$ be a multiset of integers. We say $\mathcal{S}$ is a $\mathit{\text{zero-sum}}\mathit{\text{sequence}}$ if the sum of its elements is 0. We study zero-sum sequences whose elements lie in the interval $[-k,k]$ such that no subsequence of length $t$ is also zero-sum. Given these restrictions, Augspurger, Minter, Shoukry, Sissokho, Voss show that there are arbitrarily long $t$-avoiding, $k$-bounded zero-sum sequences unless $t$ is divisible by $\mathrm{L}\mathrm{C}\mathrm{M}(2,3,4,\dots ,2k-1)$. We confirm a conjecture of these authors that for $k$ and $t$ such that this divisibility condition holds, every zero-sum sequence of length at least $t+k^2-k$ contains a zero-sum subsequence of length $t$, and that this is the minimal length for which this property holds.