Direct zero-sum problems for certain groups of rank three

TL;DR

This paper calculates exact zero-sum constants for specific rank three finite abelian groups, advancing understanding of their additive combinatorics properties.

Contribution

It provides the exact values of the η-constant and multiwise Davenport constants for certain rank three groups, and determines the Erdős-Ginzburg-Ziv constant under specific conditions.

Findings

Exact η-constant for groups of the form C_2 ⊕ C_{n_2} ⊕ C_{n_3}

Exact multiwise Davenport constants for these groups

Erdős-Ginzburg-Ziv constant determined under Property D or n_2 = n_3

Abstract

We determine the exact value of the $\eta$-constant and the multiwise Davenport constants for finite abelian groups of rank three having the form $G \simeq C_2 \oplus C_{n_2} \oplus C_{n_3}$ with $2 \mid n_2 \mid n_3$. Moreover, we determine the Erd\H{o}s-Ginzburg-Ziv constant of these groups under the assumption that $n_2/2$ has Property D or $n_2 = n_3$.