On the lower bounds of Davenport constant

Abstract

Let G=Cn1⊕…⊕Cnr with 1<n1|…|nr be a finite abelian group. The Davenport constant D(G) is the smallest integer t such that every sequence S over G of length |S|≥t has a non-empty zero-sum subsequence. It is a starting point of zero-sum theory. It has a trivial lower bound D⁎(G)=n1+…+nr−r+1, which equals D(G) over p-groups. We investigate the non-dispersive sequences over groups Cnr, thereby revealing the growth of D(G)−D⁎(G) over non-p-groups G=Cnr⊕Ckn with n,k≠1. We give a general lower bound of D(G) over non-p-groups and show that if G is an abelian group with exp⁡(G)=m and rank r, fix m>0 a non-prime-power, then for each N>0 there exists an ε>0 such that if |G|/mr<ε, then D(G)−D⁎(G)>N.