Zero-sum subsequences of length over finite abelian -groups

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• 作者：Xiaoyu He ^{xiaoyuhe@college.harvard.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Additive combinatorics ; Zero-sum subsequences
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：399-407
• 全文大小：392 K

文摘

For a finite abelian group G$G$ and a positive integer k$k$, let $s_k\left(G\right)$ denote the smallest integer ℓ∈N$\ell \in \mathbb{N}$ such that any sequence S$S$ of elements of G$G$ of length |S|≥ℓ$\left|S\right|\ge \ell$ has a zero-sum subsequence with length k$k$. The celebrated Erdős–Ginzburg–Ziv theorem determines $s_n\left(C_n\right)=2n-1$ for cyclic groups C_n$C_n$, while Reiher showed in 2007 that $s_n\left(C_n^2\right)=4n-3$. In this paper we prove for a p$p$-group G$G$ with exponent exp(G)=q$exp\left(G\right)=q$ the upper bound $s_{kq}\left(G\right)\le (k+2d-2)q+3D\left(G\right)-3$ whenever k≥d$k\ge d$, where $d=\lceil \frac{D\left(G\right)}{q}\rceil$ and p$p$ is a prime satisfying $p\ge 2d+3\lceil \frac{D\left(G\right)}{2q}\rceil -3$, where $D\left(G\right)$ is the Davenport constant of the finite abelian group G$G$. This is the correct order of growth in both k$k$ and d$d$. Subject to the same assumptions, we show exact equality $s_{kq}\left(G\right)=kq+D\left(G\right)-1$ if k≥p+d$k\ge p+d$ and p≥4d−2$p\ge 4d-2$, resolving a case of the conjecture of Gao, Han, Peng, and Sun that $s_{kexp\left(G\right)}\left(G\right)=kexp\left(G\right)+D\left(G\right)-1$ whenever $kexp\left(G\right)\ge D\left(G\right)$. We also obtain a general bound $s_{kn}\left(C_n^d\right)\le 9kn$ for n$n$ with large prime factors and k$k$ sufficiently large. Our methods extend the algebraic method of Kubertin, who proved that $s_{kq}\left(C_q^d\right)\le (k+C{d}^2)q-d$ if k≥d$k\ge d$ and q$q$ is a prime power.