{
  "id": "math/0107043",
  "version": "v2",
  "published": "2001-07-06T04:14:18.000Z",
  "updated": "2001-10-15T00:09:01.000Z",
  "title": "On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle",
  "authors": [
    "Jimmy Mc Laughlin",
    "Doug Bowman"
  ],
  "comment": "27 pages. Some interesting new examples",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let the continued fraction expansion of any irrational number $t \\in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \\[ S=\\{t \\in (0,1): a_{i+1}(t) \\geq \\phi^{d_{i}(t)} \\text{infinitely often}\\}, \\] where $\\phi = (\\sqrt{5}+1)/2$. Let $Y_{S} =\\{\\exp(2 \\pi i t): t \\in S \\}$. It is shown that if $y \\in Y_{S}$ then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points, $G \\subset Y_{S}$, such that if $y \\in G$, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| > 1 and that R(y) does converge generally if y is a primitive 5m-th root of unity, some $m \\in \\mathbb{N}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-10-15T00:09:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55"
    ],
    "keywords": [
      "rogers-ramanujan continued fraction",
      "unit circle",
      "continued fraction expansion",
      "divergence",
      "uncountable set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......7043M"
    }
  }
}