{
  "id": "2301.05855",
  "version": "v1",
  "published": "2023-01-14T08:24:37.000Z",
  "updated": "2023-01-14T08:24:37.000Z",
  "title": "Uniform Diophantine approximation and run-length function in continued fractions",
  "authors": [
    "Bo Tan",
    "Qing-Long Zhou"
  ],
  "comment": "33 pages, any comments for improvements are appreciated",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\\hat{\\nu},$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\\mathcal{U}(y,\\hat{\\nu})=\\Big\\{x\\in[0,1)\\colon \\forall N\\gg1, \\exists~ n\\in[1,N], \\text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\\hat{\\nu}}\\Big\\}$$ for algebraic irrational points $y\\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-14T08:24:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80",
      "11J83"
    ],
    "keywords": [
      "uniform diophantine approximation",
      "continued fractions",
      "run-length function",
      "multifractal properties",
      "uniform approximation exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}