Rational Distances with Rational Angles

TL;DR

This paper investigates the maximum number of rational-angle unit distances among points in the plane, establishing new upper bounds and connecting algebraic number theory with combinatorial geometry.

Contribution

It introduces bounds on rational-angle distances, applying algebraic theorems to improve understanding of geometric configurations with rational constraints.

Findings

Upper bound of $n^{1+6/\sqrt{\log n}}$ for rational-angle unit distances

Bound on rational distances with no three collinear points

Connection between algebraic number theory and geometric distance problems

Abstract

In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is $u(n)<cn^{4/3}$, due to Spencer, Szemer\'edi and Trotter. We show that the upper bound $n^{1+6/\sqrt{\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two fixed vertices in the unit distance graph, giving a contradiction if there are too many unit distances with rational angle. This bound holds if we consider rational distances instead of unit distances as long as there are no three points on a line. A superlinear lower bound is given, due to Erd\H os and Purdy. If we have at most $n^{\alpha}$ points on a line then we get the bound $O(n^{1+\alpha})$ or $n^{1+\alpha+6/\sqrt{\log n}}$ for the number of rational distances with rational angle depending on whether $\alpha\ge 1/2$ or $\alpha < 1/2$ respectively.