The structure of maximal zero-sum free Sequences

TL;DR

This paper investigates the structure of maximal zero-sum free sequences in the group Z/nZ x Z/nZ, providing new conditions under which such sequences must contain a zero-sum, especially focusing on the highest element multiplicities.

Contribution

It introduces a systematic approach to property B by analyzing the highest multiplicities in zero-sum free sequences for large prime n.

Findings

Sequences with second highest multiplicity at least 2/3 n contain a zero-sum.

Sequences with highest multiplicity greater than (1-c)n contain a zero-sum.

Sequences with second highest multiplicity less than c n also contain a zero-sum.

Abstract

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.