A zero-sum theorem over Z

TL;DR

This paper proves that for any zero-sum sequence of integers from a bounded set, if the sequence is longer than 2k-1, it must contain a proper zero-sum subsequence, establishing a precise threshold.

Contribution

The paper generalizes zero-sum sequence results by proving that the minimal length threshold is exactly 2k-1 for sets of integers from [-k,k].

Findings

Established that (k)=2k-1 for all k>1.

Proved a more general zero-sum sequence theorem.

Confirmed the minimal length for guaranteed zero-sum subsequences.

Abstract

A zero-sum sequence of integers is a sequence of nonzero terms that sum to 0. Let $k>0$ be an integer and let $[-k,k]$ denote the set of all nonzero integers between $-k$ and $k$. Let $\ell(k)$ be the smallest integer $\ell$ such that any zero-sum sequence with elements from $[-k,k]$ and length greater than $\ell$ contains a proper nonempty zero-sum subsequence. In this paper, we prove a more general result which implies that $\ell(k)=2k-1$ for $k>1$.