{
  "id": "math/0403522",
  "version": "v1",
  "published": "2004-03-30T16:24:16.000Z",
  "updated": "2004-03-30T16:24:16.000Z",
  "title": "On the rational approximations to the powers of an algebraic number",
  "authors": [
    "Pietro Corvaja",
    "Umberto Zannier"
  ],
  "comment": "12 pages, plain Tex",
  "categories": [
    "math.NT"
  ],
  "abstract": "About fifty years ago Mahler proved that if $\\alpha>1$ is rational but not an integer and if $0<l<1$ then the fractional part of $\\alpha^n$ is $>l^n$ apart from a finite set of integers $n$ depending on $\\alpha$ and $l$. Answering completely a question of Mahler we show that the same conclusion holds for all algebraic numbers which are not $d$-th roots of Pisot numbers. By related methods, we also answer a question by Mendes France, characterizing completely the quadratic irrationals $\\alpha$ such that the continued fraction of $\\alpha^n$ has period length tending to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-03-30T16:24:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic number",
      "rational approximations",
      "fractional part",
      "period length",
      "conclusion holds"
    ],
    "note": {
      "typesetting": "Plain TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3522C"
    }
  }
}