论文标题
凸几何和Erdős-Ginzburg-Ziv问题
Convex geometry and the Erdős-Ginzburg-Ziv problem
论文作者
论文摘要
用$ {\ Mathfrak s}（{\ Mathbb f} _p^d）$ eRD {\ h o} s - ginzburg- ziv常数$ {\ mathbb f} _p^d $，也就是说，即$ s $ s $ $ s $ $ n $ $ s $ p^$ p^$ p^$ p^p^p f \ p f \ \ f \ \ p f。总和为零的向量。 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 any integers $α_1, \ldots, α_s \ge 0$ with sum $p$ we have $α_1 v_1 + \ ldots +α_sv_s \ neq 0 $，除非$α_i= p $ for Some $ i $。 在1995年，Alon-- dubiner证明了$ {\ mathfrak s}（\ Mathbb f_p^d）$在修复$ d $时在$ p $中生长。在这项工作中，我们确定线性的常数：对于固定的$ d $和$ p $，我们表明$ {\ mathfrak s}}（{\ Mathbb f} _p^d）\ sim {\ sim {\ mathfrak w}（{\ Mathbb f} _p f} _p^d）p $。此外，对于任何$ p $和$ d $，我们表明$ {\ mathfrak w}（{\ mathbb f} _p^d）\ le {2dd-1 \ select ofect d}+1 $。特别是，$ {\ Mathfrak s}（{\ Mathbb f} _p^d）\ le 4^d p $用于所有足够大的$ p $和修复$ d $。
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 any 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 any 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 ${\mathfrak s}({\mathbb F}_p^d) \sim {\mathfrak w}({\mathbb F}_p^d) p$. Furthermore, for any $p$ and $d$ 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 $d$.
