Hausdorff dimension of the set of nonergodic directions (with an appendix by M. Boshernitzan)
classification
🧮 math.DS
math.NT
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dimensiondirectionshausdorffnonergodicappendixbilliardboshernitzancertain
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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.
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