Geodesic
Hausdorff dimension of the set of nonergodic directions (with an Appendix by M. Boshernitzan)
Yitwah Cheung1 ; Michael Boshernitzan2
1 Department of Mathematics, Northwestern University, Evanston, IL 60208, United States
2 Department of Mathematics, Rice University, Houston, TX 77251, United States
Annals of mathematics, Tome 158 (2003) no. 2, pp. 661-678.

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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.
DOI : 10.4007/annals.2003.158.661

Yitwah Cheung 1 ; Michael Boshernitzan 2

1 Department of Mathematics, Northwestern University, Evanston, IL 60208, United States
2 Department of Mathematics, Rice University, Houston, TX 77251, United States 
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Yitwah Cheung; Michael Boshernitzan. Hausdorff dimension of the set of nonergodic directions (with an Appendix by M. Boshernitzan). Annals of mathematics, Tome 158 (2003) no. 2, pp. 661-678. doi : 10.4007/annals.2003.158.661. http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.661/

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