Some remarks on Davenport constant

Direct zero-sum problems for certain groups of rank three

“…In the present paper, we obtain these values for the η-constant and, assuming a now well-supported conjecture, for the Erdős-Ginzburg-Ziv constant as well. Our results confirm Gao's conjecture (Conjecture 2.2) for this type of groups, and generalize previous results obtained in the case n 1 = n 2 = 2 (see [5, Theorem 1.2 (1)] and [4,Theorem 1.3]). Moreover, they show that recent results of Luo [16] are essentially optimal.…”

“…Proof of Theorem 3.4. When n ≥ 2, it follows from Theorems 2.7, 2.3 and 3.1 that D(G) = D(C 2 ⊕ C 2m ) + (2mn − 1) and η(G) ≤ D(G) + 2mn, so that [12, Theorem 6.1.5 (1)] (see also [15] and [1]) yields the desired result. Now, assume that n = 1.…”

“…Therefore, we have (the first inequality by [1,Proposition 2.6], which is can be proved using the inductive method)…”

The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics

“…The concept of the kth Davenport constants D k (G) has been introduced by Halter-Koch [50] for abelian groups in order to study the asymptotic behavior of arithmetical counting functions in rings of integers of algebraic number fields (see [40, Theorem 9.1.8], [67,Theorem 1]). They have been further studied in [15,30]. In the last years the third author and Grynkiewicz [39,48] studied the (small and the large) Davenport constant of non-abelian groups, and among others determined their precise values for groups having a cyclic subgroup of index two.…”

A stronger connection between the Erdős-Burgess and Davenport constants

“…The study of this group invariant traces back to a 1963 paper of Rogers [16] and has appeared more recently in a variety of contexts. (See, e.g., [1,2,6,9,19]. )…”