arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1509.01072math.CO
收藏

点积的发生率和对

Incidences and pairs of dot products

Ben Lund

备注：
9 pages; This version corrects mistakes in Theorems 2 and 7 from the previous version
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$\mathbb{F}$是一个域，$P \subseteq \mathbb{F}^d$是一个有限点集，$α,β\in \mathbb{F} \setminus \{0\}$。我们研究的数量\[|Π_{α, β}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = α, p \cdot r = β\}.\]我们观察到的问题之间的连接，把一个上限$|Π_{α,β}|$和一个很好的研究问题之间的发病率点和超平面，并使用这种连接证明新的和加强的上界$|Π_{α,β}|$在各种设置。

英文摘要：

Let $\mathbb{F}$ be a field, let $P \subseteq \mathbb{F}^d$ be a finite set of points, and let $α,β\in \mathbb{F} \setminus \{0\}$. We study the quantity \[|Π_{α, β}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = α, p \cdot r = β\}.\] We observe a connection between the question of placing an upper bound on $|Π_{α,β}|$ and a well-studied question on the number of incidences betwen points and hyperplanes, and use this connection to prove new and strengthened upper bounds on $|Π_{α,β}|$ in a variety of settings.

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。

↑