{
  "id": "1610.06374",
  "version": "v1",
  "published": "2016-10-20T12:14:40.000Z",
  "updated": "2016-10-20T12:14:40.000Z",
  "title": "Hausdorff dimension and uniform exponents in dimension two",
  "authors": [
    "Yann Bugeaud",
    "Yitwah Cheung",
    "Nicolas Chevallier"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $\\mu$ $\\in$ (1/2, 1) is 2(1 -- $\\mu$) when $\\mu$ $\\ge$ $\\sqrt$ 2/2, whereas for $\\mu$ \\textless{} $\\sqrt$ 2/2 it is greater than 2(1 -- $\\mu$) and at most (3 -- 2$\\mu$)(1 -- $\\mu$)/(1 + $\\mu$ + $\\mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $\\mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $\\mu$ $\\ge$ 0.565. .. .",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-20T12:14:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "uniform exponent",
      "singular two-dimensional vectors",
      "lower bound",
      "dimension tends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}