有限域上向量空间的零子和
Zero subsums in vector spaces over finite fields
摘要：
Olson常数$\mathcal{O}L(\mathbb{F}_{p}^{d})$表示最小正整数$t$，具有如下性质：基数$t$的每个子集$A\subset \mathbb{F}_{p}^{d}$包含具有零和的非空子集。估计$\mathcal{O}L(\mathbb{F}_{p}^{d})$的问题是加法组合学中最古老的问题之一，即使对于$d=1$的情况也有着悠久而有趣的历史。本文证明了对任何固定的$d \geq 2$和$ε> 0$，$\mathbb{F}_{p}^{d}$的Olson常数对所有充分大的素数$p$都满足不等式$$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+ε)p$$.这就解决了Hoi Nguyen和Van Vu的一个猜想。
英文摘要：
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. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
