arXiv 2002.098922020-02math.CO

凸几何与埃罗什-辛格-齐问题

Convex geometry and the Erdős-Ginzburg-Ziv problem

Dmitrii Zakharov

备注：: 55 pages, final version published in Discrete Analysis
DOI：: 10.19086/da.165216

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AI总结：

本文确定了埃罗什-辛格-齐常数的线性增长常数，并给出了其上界。

摘要：

令${\mathfrak s}({\mathbb F}_p^d)$表示${\mathbb F}_p^d$的埃罗什-辛格-齐常数，即满足任何$s$个向量序列中包含$p$个向量其和为零的最小$s$。令${\mathfrak w}({\mathbb F}_p^d)$表示最大长度的向量序列$v_1,\ldots,v_s\in{\mathbb F}_p^d$，使得对于所有整数$α_1,\ldots,α_s\ge0$其和为$p$，有$α_1v_1+\ldots+α_sv_s\ne0$除非存在某个$i$使得$α_i=p$。1995年，阿隆-杜宾证明当$d$固定时，$\mathfrak s(\mathbb F_p^d)$随$p$线性增长。在本工作中，我们确定了线性增长的常数：对于固定$d$和增长的$p$，我们证明${\mathfrak s}(\mathbb F_p^d)=(1+o(1))\mathfrak w(\mathbb F_p^d)p$。此外，对于每个素数$p$和每个$d$，我们证明${\mathfrak w}(\mathbb F_p^d)\le{2d-1\choose d}+1$。特别是，对于足够大的$p$和固定的$d$，有${\mathfrak s}(\mathbb F_p^d)\le4^dp$。

英文摘要：

Denote by ${\mathfrak s}({\mathbb F}_p^d)$ the Erd{\H o}s--Ginzburg--Ziv constant of $\mathbb F_p^d$, that is, the minimum $s$ such that every sequence of $s$ vectors in ${\mathbb F}_p^d$ contains $p$ vectors whose sum is zero. Let ${\mathfrak w}({\mathbb F}_p^d)$ be the maximum size of a sequence of vectors $v_1, \ldots, v_s \in {\mathbb F}_p^d$ such that, for all integers $α_1, \ldots, α_s \ge 0$ with sum $p$, we have $α_1 v_1 + \ldots + α_s v_s \neq 0$ unless $α_i = p$ for some $i$. In 1995, Alon--Dubiner proved that $\mathfrak s(\mathbb F_p^d)$ grows linearly in $p$ when $d$ is fixed. In this work, we determine the constant of linearity: for fixed $d$ and growing $p$, we show that ${\mathfrak s}({\mathbb F}_p^d) = (1+o(1)) {\mathfrak w}({\mathbb F}_p^d) p$. Furthermore, for every prime $p$ and every $d$, we show that ${\mathfrak w}({\mathbb F}_p^d) \le {2d-1 \choose d}+1$. In particular, ${\mathfrak s}({\mathbb F}_p^d) \le 4^d p$ for all sufficiently large $p$ and fixed $d$.

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摘要：

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