{
  "id": "2302.04017",
  "version": "v1",
  "published": "2023-02-08T12:12:17.000Z",
  "updated": "2023-02-08T12:12:17.000Z",
  "title": "Heights and transcendence of $p$--adic continued fractions",
  "authors": [
    "Ignazio Longhi",
    "Nadir Murru",
    "Francesco Maria Saettone"
  ],
  "comment": "12 pages; comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-08T12:12:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "11J87"
    ],
    "keywords": [
      "transcendence",
      "transcendental real numbers",
      "adic euclidean algorithm",
      "adic continued fraction expansion",
      "adic problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}