On the lower bounds of Davenport constant

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 $\mathsf D^*(G) = n_1 + \cdots + n_r - r + 1$, which equals $\mathsf D(G)$ over $p$-groups. We investigate the non-dispersive sequences over group $C_n^r$, thereby revealing the growth of $\mathsf D(G)-\mathsf D^*(G)$ 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 $\varepsilon>0$ such that if $|G|/m^r<\varepsilon$, then $\mathsf D(G)-\mathsf D^*(G)>N$.