On generalized Erdős–Ginzburg–Ziv constants of [formula omitted

Abstract

Let G be an additive finite abelian group with exponent exp(G)=n. For any positive integer k, the kth Erdős–Ginzburg–Ziv constant skn(G) is defined as the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. It is easy to see that skn(Cnr)≥(k+r)n−r where n,r∈N. Kubertin conjectured that the equality holds for any k≥r. In this paper, we prove the following results: •[(1)] For every positive integer k≥6, we have skn(Cn3)=(k+3)n+O(nlnn).•[(2)] For every positive integer k≥18, we have skn(Cn4)=(k+4)n+O(nlnn).•[(3)] For n∈N, assume that the largest prime power divisor of n is pa for some a∈N. Forany fixed r≥5, if pt≥r for some t∈N, then for any k∈N we have skptn(Cnr)≤(kpt+r)n+crnlnn,where cr is a constant that depends on r. Our results verify the conjecture of Kubertin asymptotically in the above cases.