{
  "id": "2403.19954",
  "version": "v1",
  "published": "2024-03-29T03:19:42.000Z",
  "updated": "2024-03-29T03:19:42.000Z",
  "title": "Billiards in polyhedra: a method to convert 2-dimensional uniformity to 3-dimensional uniformity",
  "authors": [
    "J. Beck",
    "W. W. L. Chen",
    "Y. Yang"
  ],
  "comment": "5 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-29T03:19:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E35",
      "11K38"
    ],
    "keywords": [
      "uniformity",
      "regular octagonal surface",
      "arithmetic progressions established earlier",
      "shortline-ancestor method",
      "real significance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}