{
  "abstract": "We study the least periods and multiplier orders in two constructions of Mercat for continued fractions with bounded partial quotients. Our main explicit result concerns a family in $\\Q(\\sqrt{127})$ with partial quotients at most $4$. For every integer $j\\geq1$, we prove that its primitive period has length $92j+2$ and its multiplier order has conductor $y_{4j}/4$, where $(4730624+419775\\sqrt{127})^k=x_k+y_k\\sqrt{127}$. The totally positive unit group of this order has index $4j$ in that of the maximal order. These exact formulas give divisor-sum lower bounds at fixed traces and a lower bound of order $(\\log X)^2$ for the number of bounded continued fractions with discriminant at most $X$. We also prove a primitive-period refinement of Mercat's palindromic construction. Combining this refinement with Hensley's counting theorem gives, for every $m\\geq2$ and $η>0$, at least $X^{δ_m/2-η}$ quadratic fields, up to a positive constant, that admit infinitely many purely periodic continued fractions with partial quotients at most $m$. Here $δ_m$ is the Hausdorff dimension of the corresponding continued fraction Cantor set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16530",
  "author_total": 4,
  "authors": [
    {
      "name": "Shirali Kadyrov"
    },
    {
      "name": "Aiken Kazin"
    },
    {
      "name": "Alibek Orynbassar"
    },
    {
      "name": "Nurdaulet Shynarbek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6kr6ps0g9gdkb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16530",
  "fulltext_url": null,
  "id": "6kr6ps0g9gdkb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16530",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221761434",
  "pdf_url": "https://arxiv.org/pdf/2610.16530",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6kr6ps0g9gdkb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16530",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16530"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Bounded continued fractions and quadratic orders in $\\Q(\\sqrt{127})$",
  "title_text": "Bounded continued fractions and quadratic orders in Q(√(127))",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6kr6ps0g9gdkb",
  "venue": "arXiv",
  "versions": []
}