On generalized Erdős-Ginzburg-Ziv constants of $C_n^r$

Abstract

Let $G$ be an additive finite abelian group with exponent $\exp (G)=m$. For any positive integer $k$, the $k$-th generalized Erdős-Ginzburg-Ziv constant ${\mathsf{s}}_{km}(G)$ 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 $km$. It is easy to see that ${\mathsf{s}}_{kn}(C_n^r)\ge (k+r)n-r$ where $n,r\in \mathbb{N}$. Kubertin conjectured that the equality holds for any $k\ge r$. In this paper, we mainly prove the following results: (1) For every positive integer $k\ge 6$, we have $${\mathsf{s}}_{kn}(C_n^3)=(k+3)n+O(\frac{n}{\ln n}).$$ (2) For every positive integer $k\ge 18$, we have $${\mathsf{s}}_{kn}(C_n^4)=(k+4)n+O(\frac{n}{\ln n}).$$ (3) For $n\in \mathbb{N}$, assume that the largest prime power divisor of $n$ is $p^a$ for some $a\in \mathbb{N}$. For any fixed $r\ge 5$, if $p^t\ge r$ for some $t\in \mathbb{N}$, then for any $k\in \mathbb{N}$ we have $${\mathsf{s}}_{k{p}^{t}n}(C_n^r)\le (k{p}^t+r)n+c_r\frac{n}{\ln n},$$ where $c_r$ is a constant depends on $r$. Note that the main terms in our results are consistent with the conjectural values proposed by Kubertin.

14 pages