Davenport Constants of Finite Abelian Groups

Abstract

A central problem in zero-sum theory is to determine the Davenport constant of a finite abelian group. For a finite abelian group $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$, with $1<n_1\mid\cdots\mid n_r$, let $\mathsf D(G)$ be the least integer $t$ such that every sequence of $t$ elements of $G$ contains a nonempty zero-sum subsequence, and denote $\mathsf D^*(G):= 1+\sum_{i=1}^r(n_i-1)$. The standard construction shows that $\mathsf D(G)\ge\mathsf D^*(G)$. Equality holds for several important classes of groups but fails in general, and even the question whether $\mathsf D(G)/\mathsf D^*(G)$ is bounded by an absolute constant remains open. In this paper, we prove the upper bound \[ \mathsf D(G)<4.86\,\exp(G)\frac{\log|G|}{\log\exp(G)} \] for every finite abelian group. When $|G|$ is large compared with $\exp(G)$, it improves the previous classical upper bound by a factor of order $\log\exp(G)$. We also prove that, for almost all finite abelian groups (counting isomorphism classes by order), \[ \mathsf D(G)\sim\exp(G)\frac{\log|G|}{\log\exp(G)}\sim\mathsf D^*(G). \] We also study the difference $\mathsf D(G)-\mathsf D^*(G)$: for every function $L$ with $L(x)\to\infty$, almost all groups satisfy $\mathsf D(G)-\mathsf D^*(G)\le L(\mathsf D^*(G))$. However, for every fixed $C\ge0$, the groups with $\mathsf D(G)-\mathsf D^*(G)>C$ have positive lower density.