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arXiv · 2403.19954

Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity

J. Beck·W. W. L. Chen·Y. Yang
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Abstract

The class of 2-dimensional non-integrable flat dynamical systems has a rather extensive literature with many deep results, but the methods developed for this type of problems, both the traditional approach via Teichm\"{u}ller geometry and our recent shortline-ancestor method, appear to be exclusively plane-specific. Thus we know very little of any real significance concerning 3-dimensional systems. Our purpose here is to describe some very limited extensions of uniformity in 2 dimensions to uniformity in 3 dimensions. We consider a 3-manifold which is the cartesian product of the regular octagonal surface with the unit torus. This is a restricted system, in the sense that one of the directions is integrable. However, this restriction also allows us to make use of a transference theorem for arithmetic progressions established earlier by Beck, Donders and Yang.

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J. Beck, W. W. L. Chen, Y. Yang. 2024-03-29. Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity. https://arxiv.org/abs/2403.19954

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