{
  "id": "1510.00282",
  "version": "v1",
  "published": "2015-10-01T15:27:29.000Z",
  "updated": "2015-10-01T15:27:29.000Z",
  "title": "On the $b$-ary expansions of $\\log (1 + \\frac{1}{a})$ and ${\\mathrm e}$",
  "authors": [
    "Yann Bugeaud",
    "Dong Han Kim"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $b \\ge 2$ be an integer and $\\xi$ an irrational real number. We prove that, if the irrationality exponent of $\\xi$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $\\xi$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of ${\\mathrm e}$, and $\\log (1 + \\frac{1}{a})$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its $b$-ary expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-01T15:27:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11J82",
      "68R15"
    ],
    "keywords": [
      "ary expansion",
      "irrationality exponent",
      "irrational real number",
      "non-zero rational powers",
      "result applies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151000282B"
    }
  }
}