Remarks on a generalization of the Davenport constant

TL;DR

This paper explores a generalized version of the Davenport constant for finite abelian groups, establishing bounds, asymptotic behavior, and exact values for specific groups, advancing understanding of zero-sum sequences.

Contribution

It introduces bounds and asymptotic properties of the generalized Davenport constant and determines exact values for certain elementary 2-groups, extending prior results.

Findings

The sequence (D_k(G)) is eventually an arithmetic progression with difference exp(G).

Exact values of D_k(G) are determined for elementary 2-groups of rank four and five.

Bounds on D_k(G) are established for general finite abelian groups.

Abstract

A generalization of the Davenport constant is investigated. For a finite abelian group $G$ and a positive integer $k$, let $D_k(G)$ denote the smallest $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint non-empty zero-sum subsequences. For general $G$, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence $(D_k(G))_{k\in\mathbb{N}}$ is eventually an arithmetic progression with difference $\exp(G)$, and several questions arising from this fact are investigated. For elementary 2-groups, $D_k(G)$ is investigated in detail; in particular, the exact values are determined for groups of rank four and five (for rank at most three they were already known).