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

TL;DR

This paper extends the Erdős-Ginzburg-Ziv theorem to integer sequences within a bounded interval, establishing minimal lengths for zero-sum subsequences under specific divisibility conditions.

Contribution

It confirms a conjecture that minimal sequence length guarantees the existence of zero-sum subsequences of a given length when divisibility conditions are met.

Findings

Sequences longer than a certain minimal length always contain the zero-sum subsequence of length t.

The minimal length for such sequences is precisely t + k^2 - k.

The divisibility condition on t is necessary for the property to hold.

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.