{
  "id": "math/0409232",
  "version": "v2",
  "published": "2004-09-14T19:15:41.000Z",
  "updated": "2004-10-05T21:40:24.000Z",
  "title": "On two exponents of approximation related to a real number and its square",
  "authors": [
    "Damien Roy"
  ],
  "comment": "13 pages, minor corrections",
  "journal": "Canad. J. Math, vol. 59 (2007), 211-224",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any irrational real number xi, let lambda(xi) denote the supremum of all real numbers lambda such that, for each sufficiently large X, the inequalities |x_0| < X, |x_0*xi-x_1| < X^{-lambda} and |x_0*xi^2-x_2| < X^{-lambda} admit a solution in integers x_0, x_1 and x_2 not all zero, and let omega(xi) denote the supremum of all real numbers omega such that, for each sufficiently large X, the dual inequalities |x_0+x_1*xi+x_2*xi^2| < X^{-omega}, |x_1| < X and |x_2| < X admit a solution in integers x_0, x_1 and x_2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents lambda(xi) where xi ranges through all irrational non-quadratic real numbers form a dense subset of the interval [1/2, (sqrt{5}-1)/2] while, for the same values of xi, the dual exponents omega(xi) form a dense subset of [2, (sqrt{5}+3)/2]. Part of the proof rests on a result of V. Jarnik showing that lambda(xi) = 1-1/omega(xi) for these real numbers xi.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-10-05T21:40:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J13",
      "11J82"
    ],
    "keywords": [
      "approximation",
      "irrational non-quadratic real numbers form",
      "dense subset",
      "irrational real number xi",
      "dual exponents omega"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9232R"
    }
  }
}