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

Abstract

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 $|S|\ge \ell$ has a zero-sum subsequence with length $k$. The celebrated Erdős-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 $k\ge d$, where $d=\lceil \frac{D(G)}{q}\rceil$ and $p$ is a prime satisfying $p\ge 2d+3\lceil \frac{D(G)}{2q}\rceil -3$, 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 $k\ge p+d$ and $2p\ge 7d-3$, 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 $k\exp (G)\ge D(G)$. We also obtain a general bound $s_{kn}(C_n^d)\le 9kn$ for $n$ with large prime factors and $k$ sufficiently large. Our methods are inspired by the algebraic method of Kubertin, who proved that $s_{kq}(C_q^d)\le (k+C{d}^2)q-d$ whenever $k\ge d$ and $q$ is a prime power.

References in corpus (2)

• On Kemnitz' Conjecture Concerning Lattice Points in the Plane
• Long $n$-zero-free sequences in finite cyclic groups

Cited by in corpus (5)

• On generalized Erdős-Ginzburg-Ziv constants for ${\mathbb{Z}}_2^d$
• Extremal problems on the hypercube and the codegree Turán density of complete $r$-graphs
• Exponential Lower Bounds on the Generalized Erdős-Ginzburg-Ziv Constant
• Modified Erdős-Ginzburg-Ziv constants for ${\mathbb{Z}}_2^d$
• On generalized Erdős-Ginzburg-Ziv constants of $C_n^r$