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.

Authors

Yitwah Cheung

Department of Mathematics
Northwestern University
Evanston, IL 60208
United States

Michael Boshernitzan

Department of Mathematics
Rice University
Houston, TX 77251
United States