{
  "id": "1703.07672",
  "version": "v1",
  "published": "2017-03-22T14:25:58.000Z",
  "updated": "2017-03-22T14:25:58.000Z",
  "title": "Convergents as approximants in continued fraction expansions of complex numbers with Eisenstein integers",
  "authors": [
    "S. G. Dani"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\frak E$ denote be the ring of Eisenstein integers. Let $z\\in \\mathbb C$ and $p_n,q_n \\in \\frak E$ be such that $\\{p_n/q_n\\}$ is the sequence of convergents corresponding to the continued fraction expansion of $z$ with respect to the nearest integer algorithm. Then we show that for any $q\\in \\frak E$ such that $1\\leq |q|\\leq |q_n|$ and any $p\\in \\frak E$, $|qz-p|\\geq \\frac12 |q_nz-p_n|$. This enables us to conclude that $z\\in \\mathbb C$ is badly approximable, in terms of Eisenstein integers, if and only if the corresponding sequence of partial quotients is bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-22T14:25:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continued fraction expansion",
      "eisenstein integers",
      "complex numbers",
      "convergents",
      "approximants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}