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 ℓ such that each sequence over G of length at least ℓ 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∈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).

An application of coding theory to estimating Davenport constants

“…For example, η(C 6 3 ) was determined only recently [27], despite the fact that the problem of determining η(C r 3 ) is fairly popular (see [10] for a detailed outline of several problems, and their respective history, that are equivalent to determining η(C It is known, in particular by the work of Delorme, Ordaz, and Quiroz [8], that the invariants s ≤x (G) can be used to derive upper bounds for D j (G). More specifically, we have (this is Lemma 2.4 in [14])…”

“…For results in the converse scenario, that is fixed but arbitrary r, and j goes to infinity, see [14].…”

“…But in general computing (even bounding) D j (G) is quite more complicated than for D(G), in particular for (elementary) p-groups. For example, D j (G) for all j, is only known for the following elementary p-groups of rank greater than two: C [8,14,3]). The main difference is that it seems that the types of arguments used to determine D(G) for p-groups cannot be applied.…”

On the Erdős–Ginzburg–Ziv constant of finite abelian groups of high rank

“…More information can be found in the surveys [9,13]. Both the Davenport constant and the Erdős-Ginzburg-Ziv constant have found far reaching generalizations, and for these generalized versions, the precise values have been determined for groups with rank at most two (see [15,Section 6.1], [7], [14,Theorem 5.2], [17]). …”

New upper bounds for the Davenport and for the Erdős–Ginzburg–Ziv constants

“…We shall follow the notations as given in [8] (One may also refer to [11,22]). For a non-empty subset I of natural numbers and a finite abelian group G, let s I (G) denote the smallest l ∈ N ∪ {∞} such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence of length in I.…”