## Discrete Math Seminar: Rational Distances with Rational Angles

- Date: 03/23/2010

Frank De Zeeuw (UBC)

Ryan Schwartz (UBC)

University of British Columbia

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.

4:00 - 5:00pm, WMAX 216.