On the lower bounds of Davenport constant

摘要

Let G = C-n1 circle plus...circle plus C-nr with 1 < n(1)vertical bar...vertical bar n(r) a finite abelian group. The Davenport constant D(G) is the smallest integer t such that every sequence S over G of length vertical bar S vertical bar >= 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) = n(1) +...+ n(r) - r + 1, which equals D(G) over p-groups. We investigate the non-dispersive sequences over groups C-n(r), thereby revealing the growth of D(G) - D* (G) over non-p-groups G = C-n(r) circle plus C-kn with n, k not equal 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 epsilon > 0 such that if vertical bar G vertical bar/m(r) < epsilon, then D(G) - D* (G) > N. (C) 2019 Elsevier Inc. All rights reserved.