Repeated Angles in the Plane for Angles with Algebraic Tangents

TL;DR

This paper constructs a large set of points in the plane that determine many triples forming a specific angle with algebraic tangent, matching known upper bounds and improving previous constructions.

Contribution

It introduces a new construction that achieves the maximum number of such angles for algebraic tangents, extending prior results to a broader class of algebraic numbers.

Findings

Constructed point sets with n^2 log n triples for algebraic tangent angles

Matched the upper bound established by Pach and Sharir

Improved upon previous constructions limited to specific algebraic forms

Abstract

We construct a set of points with $\Omega(n^2\log n)$ triples determining an angle $\theta$ whenever $\tan(\theta)$ is algebraic over $\mathbb{Q}$, matching the upper bound of Pach and Sharir. This improves upon the original construction, which was optimal only for $\tan(\theta)=a\sqrt{m}/b$ with $a,b,m$ positive integers.