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 Erd\H{o}s-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 $$\mathsf s_{kp^tn}(C_n^r)\le(kp^t+r)n+c_r\frac{n}{\ln n},$$ 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.