On generalized Erd\H{o}s-Ginzburg-Ziv constants of $C_n^r$

TL;DR

This paper investigates the generalized Erdős-Ginzburg-Ziv constants for groups of the form $C_n^r$, providing asymptotic formulas and bounds that support Kubertin's conjecture for various parameters.

Contribution

The paper proves asymptotic formulas and bounds for the generalized Erdős-Ginzburg-Ziv constants of $C_n^r$, extending previous conjectures and results in additive combinatorics.

Findings

For $k extgreater 6$, $ extsf{s}_{kn}(C_n^3)=(k+3)n+O(n/\ln n)$.

For $k extgreater 18$, $ extsf{s}_{kn}(C_n^4)=(k+4)n+O(n/\ln n)$.

Bounds for $ extsf{s}_{kp^tn}(C_n^r)$ when the largest prime power divisor of $n$ is $p^a$, supporting conjectural values.

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.