Davenport 常数的一个猜想上界之反例
Disproof of a Conjectured Upper Bound for the Davenport Constant
- School of Mathematical Sciences Tiangong University（天津工业大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文否证了关于 Davenport 常数上界的长期猜想，证明对任意固定秩 r≥8，Davenport 常数与经典下界之差可任意大，推翻了过去数十年的普遍信念。
AI中文摘要：
设 $G\cong C_{n_1}\oplus\cdots\oplus C_{n_r}$ 为有限交换群，其中 $1<n_1\mid\cdots\mid n_r$，并令 $r(G)=r$ 表示其秩。Davenport 常数 $\mathsf D(G)$ 是使得 $G$ 中任意长度为 $\ell$ 的序列都包含非空零和子序列的最小整数 $\ell$，而 $\mathsf D^*(G)=1+\sum_{i=1}^r(n_i-1)$ 是其经典下界。一个长期存在的猜想（\cite[猜想3.7]{GG06}）关于 $\mathsf D(G)$ 的一般上界断言 $\mathsf D(G)\le\mathsf D^*(G)+r(G)-1$。在本文中，我们否证了这一猜想。更强地，我们证明对每个固定的 $r\ge8$，有 $\sup_{r(G)=r}\bigl(\mathsf D(G)-\mathsf D^*(G)\bigr)=\infty$。因此，经典下界并不能在仅依赖于秩的加性误差范围内逼近 Davenport 常数，这与过去几十年长期以来的信念相反。
英文摘要：
Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture \cite[Conjecture 3.7]{GG06} asserts that $\DD(G)\le\DD^*(G)+\rr(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{\rr(G)=r}\bigl(\DD(G)-\DD^*(G)\bigr)=\infty$ for every fixed $r\ge8$. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. Our result also disproves the Narkiewicz--Śliwa conjecture of 1982 \cite{NS82} on the Narkiewicz constant, arising in algebraic number theory from the quantitative study of algebraic integers with unique factorization. The same amplification of the Davenport excess yields counterexamples to Girard's conjecture \cite[Conjecture 1.2]{Girard08} on the cross numbers of long zero-sum-free sequences. As a further main result, we establish the uniform upper bound $$\DD(G)\le\frac{16}{5}\rr(G)\exp(G)$$ for every nontrivial finite abelian group $G$. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives $\DD(G)\le\exp(G)\left(1+\log\frac{|G|}{\exp(G)}\right) \le\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$. Our bound removes the logarithmic factor $\log\exp(G)$ from this classical estimate, and replaces it with the absolute constant $16/5$.
