{
  "id": "math/0210182",
  "version": "v2",
  "published": "2002-10-11T21:41:06.000Z",
  "updated": "2002-10-24T21:43:55.000Z",
  "title": "Approximation to real numbers by cubic algebraic integers II",
  "authors": [
    "Damien Roy"
  ],
  "comment": "7 pages; major simplification of the original proof",
  "journal": "Annals of Mathematics 158 (2003), 1081-1087.",
  "categories": [
    "math.NT"
  ],
  "abstract": "It has been conjectured for some time that, for any integer n\\ge 2, any real number \\epsilon >0 and any transcendental real number \\xi, there would exist infinitely many algebraic integers \\alpha of degree at most n with the property that |\\xi-\\alpha| < H(\\alpha)^{-n+\\epsilon}, where H(\\alpha) denotes the height of \\alpha. Although this is true for n=2, we show here that, for n=3, the optimal exponent of approximation is not 3 but (3+\\sqrt{5})/2 = 2.618...",
  "revisions": [
    {
      "version": "v2",
      "updated": "2002-10-24T21:43:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J04",
      "11J82"
    ],
    "keywords": [
      "cubic algebraic integers",
      "approximation",
      "transcendental real number",
      "optimal exponent"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10182R"
    }
  }
}