{
  "id": "1801.09416",
  "version": "v1",
  "published": "2018-01-29T09:28:41.000Z",
  "updated": "2018-01-29T09:28:41.000Z",
  "title": "Asymptotic behaviour of the Sudler product of sines for quadratic irrationals",
  "authors": [
    "Sigrid Grepstad",
    "Mario Neumüller"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We study the asymptotic behaviour of the sequence of sine products $P_n(\\alpha) = \\prod_{r=1}^n |2\\sin \\pi r \\alpha|$ for real quadratic irrationals $\\alpha$. In particular, we study the subsequence $Q_n(\\alpha)=\\prod_{r=1}^{q_n} |2\\sin \\pi r \\alpha|$, where $q_n$ is the $n$th best approximation denominator of $\\alpha$, and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of $\\alpha$. This verifies a conjecture recently posed by Mestel and Verschueren.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-29T09:28:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "41A60"
    ],
    "keywords": [
      "asymptotic behaviour",
      "sudler product",
      "th best approximation denominator",
      "real quadratic irrationals",
      "sine products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}