Generalized Erdős--Ginzburg--Ziv Constants for finite abelian groups of rank three

Let $G$ be a finite abelian group with exponent $\exp (G)$. For positive integer $k$, the generalized Erdős–Ginzburg–Ziv constant, denoted by ${\mathsf{s}}_{k\exp (G)}(G)$ is the least integer $\mathrm{\ell }$ such that every sequence of length at least $\mathrm{\ell }$ over $G$ has a zero-sum subsequence of length $k\exp (G)$. In this paper, we prove that, for any $n\ge 1$, $${\mathsf{s}}_{3\cdot 2^n}(C_{2^n}^3)=6\cdot 2^n-3\text{and}{\mathsf{s}}_{4\cdot 3^n}(C_{3^n}^3)=7\cdot 3^n-3.$$ This completes all the remaining cases in the study of ${\mathsf{s}}_{kq}(C_q^3)$, where $q$ is a prime power and $k\ge 3$. As a consequence, for every $k\ge 3$, we have $${\mathsf{s}}_{kn}(C_n^3)=(k+3)n+O(\frac{n}{\log n})\mathrm{.}$$ This verifies a conjecture of Kubertin asymptotically. Our approach combines the previous group ring techniques and a recent powerful generalization of the Chevalley-Warning and Ax-Katz theorems introduced by Grynkiewicz