{
  "id": "2310.11698",
  "version": "v1",
  "published": "2023-10-18T04:02:34.000Z",
  "updated": "2023-10-18T04:02:34.000Z",
  "title": "Complex numbers with a prescribed order of approximation and Zaremba's conjecture",
  "authors": [
    "Gerardo González Robert",
    "Mumtaz Hussain",
    "Nikita Shulga"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Given $b=-A\\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\\mathcal A=\\{0,1,\\ldots, A^2\\}$. We prove that, for any real $\\tau\\geq 2$ and any non-empty proper subset $J(b)$ of $\\mathcal A$, there are uncountably many complex numbers (including transcendental numbers) that can be expressed as a power series in $b$ with coefficients in $J(b)$ and with the irrationality exponent (in terms of Gaussian integers) equal to $\\tau$. One of the key ingredients in our construction is the `Folding Lemma' applied to Hurwitz continued fractions. This motivates a Hurwitz continued fraction analogue of the well-known Zaremba's conjecture. We prove several results in support of this conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-18T04:02:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex number",
      "prescribed order",
      "approximation",
      "power series",
      "hurwitz continued fraction analogue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}