多面体中的台球：一种将二维均匀性转化为三维均匀性的方法
Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity
AI总结：
本文通过将正则八边形曲面与单位环面作笛卡尔积，利用算术级数转移定理，把二维均匀性推广到三维均匀性。
摘要：
二维不可积平坦动力系统这一类问题已有相当丰富的文献和许多深刻的结果，但针对这类问题所发展的方法，无论是通过Teichmüller几何的传统途径，还是我们近期提出的短线祖先方法，似乎都专门适用于平面。因此，对于三维系统，我们几乎不了解任何具有真正意义的内容。本文旨在描述二维均匀性向三维均匀性的一些非常有限的推广。我们考虑一个三维流形，它是正则八边形曲面与单位环面的笛卡尔积。这是一个受限系统，因为其中一个方向是可积的。然而，这一限制也使我们能够利用Beck、Donders和Yang先前建立的关于算术级数的转移定理。
英文摘要：
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 Teichmüller 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
