Remarks on a generalization of the Davenport constant

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).

Various expository changes, updated and slightly expanded bibliography

References in corpus (5)

• On the existence of zero-sum subsequences of distinct lengths
• Remarks to Arsovski's proof of Snevily's conjecture
• An application of coding theory to estimating Davenport constants
• 1-Saturating Sets, Caps and Round Sets in Binary Spaces
• Exterior algebras and two conjectures on finite abelian groups

Cited by in corpus (2)

• An application of coding theory to estimating Davenport constants
• Zero-sum problems with congruence conditions