Disproof of a Conjectured Upper Bound for the Davenport Constant

Source abstract

Let $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $r(G)=r$ denote its rank. The Davenport constant $\mathsf D(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\mathsf D^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture (\cite[Conjecture 3.7]{GG06}) on the general upper bound of $\mathsf D(G)$ asserts that $\mathsf D(G)\le\mathsf D^*(G)+r(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{r(G)=r}\bigl(\mathsf D(G)-\mathsf D^*(G)\bigr)=\infty \qquad\text{for every fixed }r\ge8.$ Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank, contrary to what has long been believed in the past some decades.