Incidences and pairs of dot products

TL;DR

This paper investigates the number of point triples with specified dot product conditions in finite fields, establishing new bounds by linking the problem to point-hyperplane incidence geometry.

Contribution

It introduces a novel connection between dot product incidences and point-hyperplane incidences, leading to improved bounds on the number of such triples.

Findings

Established new upper bounds on the number of dot product triples.

Linked dot product incidence problems to well-studied incidence geometry.

Strengthened existing bounds in various finite field settings.

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 \[|\Pi_{\alpha, \beta}| = \{(p,q,r) \in P \times P \times P \mid p \cdot q = \alpha, p \cdot r = \beta \}.\] 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.