{
  "id": "2208.01680",
  "version": "v1",
  "published": "2022-08-02T18:15:31.000Z",
  "updated": "2022-08-02T18:15:31.000Z",
  "title": "Symmetries of the Three Gap Theorem",
  "authors": [
    "Aneesh Dasgupta",
    "Roland Roeder"
  ],
  "comment": "To appear in The American Math Monthly. This is the author's original manuscript (AOM), and this version does not include several improvements that took place due to the refereeing process. Comments welcome!",
  "categories": [
    "math.NT",
    "math.CO",
    "math.DS"
  ],
  "abstract": "The Three Gap Theorem states that for any $\\alpha \\in \\mathbb{R}$ and $N \\in \\mathbb{N}$, the fractional parts of $\\{ 0\\alpha, 1\\alpha, \\dots, (N - 1)\\alpha \\}$ partition the unit circle into gaps of at most three distinct lengths. We prove a result about symmetries in the order with which the sizes of gaps appear on the circle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-02T18:15:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K06",
      "11J71",
      "37A44"
    ],
    "keywords": [
      "symmetries",
      "gap theorem states",
      "gaps appear",
      "unit circle",
      "distinct lengths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}