{
  "id": "2301.02391",
  "version": "v1",
  "published": "2023-01-06T05:55:26.000Z",
  "updated": "2023-01-06T05:55:26.000Z",
  "title": "On effective irrationality exponents of cubic irrationals",
  "authors": [
    "Dzmitry Badziahin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We provide an upper bound on the efficient irrationality exponents of cubic algebraics $x$ with the minimal polynomial $x^3 - tx^2 - a$. In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville in the case $|t| > 19.71 a^{4/3}$. Moreover, under the condition $|t| > 86.58 a^{4/3}$, we provide an explicit lower bound on the expression $||qx||$ for all large $q\\in\\mathbb{Z}$. These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-06T05:55:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J68",
      "11J70",
      "11J82"
    ],
    "keywords": [
      "effective irrationality exponents",
      "cubic irrationals",
      "efficient irrationality exponents",
      "explicit lower bound",
      "cubic algebraics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}