{
  "id": "2007.14785",
  "version": "v1",
  "published": "2020-07-28T07:40:34.000Z",
  "updated": "2020-07-28T07:40:34.000Z",
  "title": "On simultaneous rational approximation to a $p$-adic number and its integral powers, II",
  "authors": [
    "Dzmitry Badziahin",
    "Yann Bugeaud",
    "Johannes Schleischitz"
  ],
  "comment": "17 pages. arXiv admin note: text overlap with arXiv:1906.05508",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime number. For a positive integer $n$ and a real number $\\xi$, let $\\lambda_n (\\xi)$ denote the supremum of the real numbers $\\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \\ldots , x_n)$ such that $| x_0 \\xi - x_1|_p, \\ldots , | x_0 \\xi^n - x_n|_p$ are all less than $X^{-\\lambda - 1}$, where $X$ is the maximum of $|x_0|, |x_1|, \\ldots , |x_n|$. We establish new results on the Hausdorff dimension of the set of real numbers $\\xi$ for which $\\lambda_n (\\xi)$ is equal to (or greater than or equal to) a given value.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-28T07:40:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J13"
    ],
    "keywords": [
      "simultaneous rational approximation",
      "integral powers",
      "adic number",
      "real number",
      "hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}