论文标题
有限字段的矢量空间中的零元素
Zero subsums in vector spaces over finite fields
论文作者
论文摘要
The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\subset \mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum.估计$ \ MATHCAL {O} L（\ Mathbb {f} _ {p}^{d}）$的问题是添加剂Compinatorics中最古老的问题之一，即使在情况下为$ d = 1 $，也有漫长而有趣的历史记录。 在本文中，我们证明，对于任何固定的$ d \ geq 2 $和$ε> 0 $，Olson常数为$ \ Mathbb {f} _ {p}^{d} $满足不平等$ \ MATHCAL {o} l（\ MATHBB {f} _ {p} p} p} p}^de for $ \ mathcal {o}足够大的素数$ p $。这解决了Hoi Nguyen和van Vu的猜想。
The Olson constant $\mathcal{O}L(\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\subset \mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{O}L(\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d \geq 2$ and $ε> 0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality $$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+ε)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.
