On the lower bounds of Davenport constant

Abstract

Let $G=C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $1<n_1|\cdots |n_r$ be a finite abelian group. The Davenport constant $\mathsf{D}(G)$ is the smallest integer $t$ such that every sequence $S$ over $G$ of length $|S|\ge t$ has a non-empty zero-sum subsequence. It is a starting point of zero-sum theory but only has a trivial lower bound ${\mathsf{D}}^{\ast }(G)=n_1+\cdots +n_r-r+1$, which equals $\mathsf{D}(G)$ over $p$-groups. We investigate the non-dispersive sequences over group $C_n^r$, thereby revealing the growth of $\mathsf{D}(G)-{\mathsf{D}}^{\ast }(G)$ over non-$p$-groups $G=C_n^r\oplus C_{kn}$ with $n,k\ne 1$. We give a general lower bound of $\mathsf{D}(G)$ over non-$p$-groups and show that, let $G$ be abelian groups with $\exp (G)=m$ and rank $r$, fix $m>0$ a non-prime-power, then for each $N>0$ there exists an $\epsilon >0$ such that if $|G|/m^r<\epsilon$, then $\mathsf{D}(G)-{\mathsf{D}}^{\ast }(G)>N$.