Zero-sum Subsequences of Length kq over Finite Abelian p-Groups

For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The celebrated Erd\H{o}s-Ginzburg-Ziv theorem determines $s_{n}(C_{n})=2n-1$ for cyclic groups $C_{n}$, while Reiher showed in 2007 that $s_{n}(C_{n}^{2})=4n-3$. In this paper we prove for a $p$-group $G$ with exponent $\exp(G)=q$ the upper bound $s_{kq}(G)\le(k+2d-2)q+3D(G)-3$ whenever $k\geq d$, where $d=\Big\lceil\frac{D(G)}{q}\Big\rceil$ and $p$ is a prime satisfying $p\ge2d+3\Big\lceil\frac{D(G)}{2q}\Big\rceil-3$, where $D(G)$ is the Davenport constant of the finite abelian group $G$. This is the correct order of growth in both $k$ and $d$. As a corollary, we show $s_{kq}(C_{q}^{d})=(k+d)q-d$ whenever $k\geq p+d$ and $2p\geq7d-3$, resolving a case of the conjecture of Gao, Han, Peng, and Sun that $s_{k\exp(G)}(G)=k\exp(G)+D(G)-1$ whenever $k\exp(G)\geq D(G)$. We also obtain a general bound $s_{kn}(C_{n}^{d})\leq9kn$ for $n$ with large prime factors and $k$ sufficiently large. Our methods are inspired by the algebraic method of Kubertin, who proved that $s_{kq}(C_{q}^{d})\leq(k+Cd^{2})q-d$ whenever $k\geq d$ and $q$ is a prime power.