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arXiv · 1608.04125

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

Aaron Berger
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Abstract

Let $\mathcal S$ be a multiset of integers. We say $\mathcal S$ is a $\textit{zero-sum sequence}$ 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 $\mathrm{LCM}(2,3,4,\dots,2k-1)$. 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.

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Aaron Berger. 2016-08-14. An Analogue of the Erd\H{o}s-Ginzburg-Ziv Theorem over $\mathbb Z$. https://arxiv.org/abs/1608.04125

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