{
  "id": "0910.4448",
  "version": "v1",
  "published": "2009-10-23T06:39:34.000Z",
  "updated": "2009-10-23T06:39:34.000Z",
  "title": "Irrationality exponent and rational approximations with prescribed growth",
  "authors": [
    "Stéphane Fischler",
    "Tanguy Rivoal"
  ],
  "comment": "11 pages, to appear in Proc. Amer. Math. Soc",
  "journal": "Proc. Amer. Math. Soc. 138.8 (2010), 799-808",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\xi$ be a real irrational number. We are interested in sequences of linear forms in 1 and $\\xi$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $\\xi$. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin's new proof of Nesterenko's linear independence criterion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-23T06:39:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J82",
      "11J04",
      "11J13",
      "11J72"
    ],
    "keywords": [
      "irrationality exponent",
      "rational approximations",
      "prescribed growth",
      "linear forms",
      "nesterenkos linear independence criterion"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Proc. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.4448F"
    }
  }
}