Incidences and pairs of dot products

Abstract

Let $\mathbb{F}$ be a field, let $P\subseteq {\mathbb{F}}^d$ be a finite set of points, and let $\alpha ,\beta \in \mathbb{F}\setminus \{0\}$. 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 $|{\Pi }_{\alpha ,\beta }|$ 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 $|{\Pi }_{\alpha ,\beta }|$ in a variety of settings.

arXiv comment9 pages; This version corrects mistakes in Theorems 2 and 7 from the previous version

References in corpus (3)

• Further improvements to incidence and Beck-type bounds over prime finite fields
• Upper bounds on pairs of dot products
• Bisector energy and few distinct distances