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


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.


Yitwah Cheung

Department of Mathematics, Northwestern University, Evanston, IL 60208, United States

Michael Boshernitzan

Department of Mathematics, Rice University, Houston, TX 77251, United States