Upper bounds on pairs of dot products

TL;DR

This paper establishes upper bounds on the number of point triples in a finite set in the plane that realize a specific pair of dot products, contributing to geometric combinatorics and dot product problems.

Contribution

It provides new upper bounds on the number of triples of points with prescribed dot products in a finite planar set.

Findings

Derived explicit upper bounds for the number of such triples.

Extended understanding of dot product configurations in finite point sets.

Abstract

Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the size of the set $$\left\{(p,q,r)\in P \times P \times P : p \cdot q = \alpha, p \cdot r = \beta \right\}.$$