Discrete Math Seminar: Rational Distances with Rational Angles

Given n points in the real plane, the unit distance problem asks for an asymptotic upper bound on the number of unit distances between pairs of the points. We consider this problem under the restriction that the line segments between the points make a rational angle (in degrees) with the x-axis. In the complex plane, that allows us to think of such segments of length 1 as roots of unity. Given a point set with many such segments, we deduce simple linear equations with many solutions in roots of unity. Using an algebraic theorem of Mann from 1965, we can give a uniform bound on the number of such solutions, which will give us a tight asymptotic bound on the number of unit distances with rational angles. These results can then be extended to rational distances. This is joint work with Jozsef Solymosi.