{
  "id": "1602.05463",
  "version": "v1",
  "published": "2016-02-17T16:11:55.000Z",
  "updated": "2016-02-17T16:11:55.000Z",
  "title": "Effective irrationality measures for real and $p$-adic roots of rational numbers close to $1$, with an application to parametric families of Thue-Mahler equations",
  "authors": [
    "Yann Bugeaud"
  ],
  "comment": "12 pages, comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \\over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under the assumption that $a$ is $p$-adically very close to $b$, that is, that a large power of $p$ divides $a-b$. As an application, we solve completely certain families of Thue-Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-17T16:11:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J86",
      "11J61",
      "11J68",
      "11D61"
    ],
    "keywords": [
      "effective irrationality measures",
      "rational numbers close",
      "thue-mahler equations",
      "adic roots",
      "parametric families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}