Hausdorff dimension of the set of nonergodic directions (with an Appendix by M. Boshernitzan)

Abstract

It is known that nonergodic directions in a rational billiard form a subset of the unit circle with Hausdorff dimension at most $1/2$. Explicit examples realizing the dimension $1/2$ are constructed using Diophantine numbers and continued fractions. A lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction.