{
  "id": "1805.10436",
  "version": "v1",
  "published": "2018-05-26T06:56:48.000Z",
  "updated": "2018-05-26T06:56:48.000Z",
  "title": "Hausdorff dimension in inhomogeneous Diophantine approximation",
  "authors": [
    "Yann Bugeaud",
    "Dong Han Kim",
    "Seonhee Lim",
    "Michał Rams"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $\\alpha$ be an irrational real number. We show that the set of $\\epsilon$-badly approximable numbers \\[ \\mathrm{Bad}^\\varepsilon (\\alpha) := \\{x\\in [0,1]\\, : \\, \\liminf_{|q| \\to \\infty} |q| \\cdot \\| q\\alpha -x \\| \\geq \\varepsilon \\} \\] has full Hausdorff dimension for some positive $\\epsilon$ if and only if $\\alpha$ is singular on average. The condition is equivalent to the average $\\frac{1}{k} \\sum_{i=1, \\cdots, k} \\log a_i$ of the logarithms of the partial quotients $a_i$ of $\\alpha$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-26T06:56:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K60",
      "28A80",
      "37E10"
    ],
    "keywords": [
      "inhomogeneous diophantine approximation",
      "irrational real number",
      "full hausdorff dimension",
      "stronger result",
      "partial quotients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}