Some Results on Zero Sum Sequences in $Z_p^3$

TL;DR

This paper explores zero sum sequences in $Z_p^3$, providing bounds under certain conditions that extend the understanding of Kemnitz's conjecture from $Z_p^2$ to three dimensions.

Contribution

The paper introduces new bounds for zero sum sequences in $Z_p^3$ when additional conditions are applied, advancing the theory beyond existing results.

Findings

Established bounds for zero sum sequences in $Z_p^3$ with added conditions

Extended Kemnitz's conjecture to three dimensions under specific constraints

Provided new insights into the structure of zero sum sequences in finite abelian groups

Abstract

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a similar result we have to take a sequence of length atleast $9p-8$ . In this paper we will show that if we add a condition on the chosen sequence, then we can get a good upper and a lower bound for which similar results hold.