Davenport's constant for groups with large exponent (Q2869123)

This paper yields a new upper bound for the Davenport constant of finite abelian groups \(G\). The Davenport constant \(D(G)\) is the least integer \(k\) such that every sequence \(g_1, \dots, g_k\) of elements of \(G\) contains a subsequence \(g_{i_1}, \dots, g_{i_\ell}\) with \(g_{i_1}+ \dots+ g_{i_\ell} = 0\). Write \(G = \mathbb{Z}_{n_1} \oplus \dots \oplus \mathbb{Z}_{n_r}\), where \(n_1 \mid \dots \mid n_r\), and where \(\mathbb Z_n\) denotes the cyclic group with \(n\) elements. Set \(\exp(G) := n_r\). A well-known (and easy to prove) lower bound for \(D(G)\) is \(M(G) := \sum_i n_i - r + 1\), and it is known that if \(r = 2\), then one has equality: \(D(G) = M(G)\). In that case, we can also write \(M(G) = \exp(G) + \frac{|G|}{\exp(G)} - 1\). The authors prove that NEWLINE\[ D(G) \leq \exp(G) + \frac{|G|}{\exp(G)} - 1\] NEWLINEholds in general.NEWLINENEWLINEAs written here, this bound would be a bit unsatisfactory, since for \(\exp(G) < \sqrt{|G|}\), it becomes worse when \(\exp(G)\) becomes smaller (whereas one would expect better bounds for smaller \(\exp(G)\)). However, the authors also prove that if \(\exp(G) < \sqrt{|G|}\), then NEWLINE\[ D(G) \leq \sqrt{|G|} + \frac{|G|}{\sqrt{|G|}} - 1 = 2\sqrt{|G|} - 1. \]NEWLINENEWLINEFor the entire collection see [Zbl 1253.00023].