{
  "id": "1410.2927",
  "version": "v1",
  "published": "2014-10-10T23:29:41.000Z",
  "updated": "2014-10-10T23:29:41.000Z",
  "title": "The sequence of fractional parts of roots",
  "authors": [
    "Kevin O'Bryant"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the function M(t,n) = Floor[ 1 / {t^(1/n)} ], where t is a positive real number, Floor[.] and {.} are the floor and fractional part functions, respectively. In a recent article in the Monthly, Nathanson proved that if log(t) is rational, then for all but finitely many positive integers n one has M(t,n) = Floor[ n / log(t) - 1/2 ]. We extend this by showing that, without condition on t, all but a zero-density set of integers n satisfy M(t,n) = Floor[ n / log(t) - 1/2 ]. Using a metric result of Schmidt, we show that almost all t have asymptotically log(t) log(x)/12 exceptional n<x. Using continued fractions, we produce uncountably many t that have only finitely many exceptional n, and also give uncountably many explicit t that have infinitely many exceptional n.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-10T23:29:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B68",
      "11J25",
      "11J71",
      "11A25",
      "11B83",
      "11J99",
      "11J70"
    ],
    "keywords": [
      "fractional part functions",
      "exceptional",
      "positive real number",
      "zero-density set",
      "metric result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.2927O"
    }
  }
}