An analogue of the Erdős–Ginzburg–Ziv Theorem over [formula omitted

Abstract

Let S be a multiset of integers. We say S is a 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 lcm(2,3,4,…,max(2,2k−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+k2−k contains a zero-sum subsequence of length t, and that this is the minimal length for which this property holds.