{
  "abstract": "Let $[a_1(x),a_2(x),\\ldots,a_n(x),\\ldots]$ be the continued fraction expansion of irrational $x\\in[0,1)$, and let $q_n(x)$ be the denominator of the $n$-th convergent. In this paper, we study how the growth rate of $a_{n+1}(x)$ on a prescribed logarithmic size interacts with its upper growth rate relative to $q_n(x)$. For $ψ:\\mathbb N\\to\\mathbb{R}_{\\ge0}$ satisfying $ψ(n)\\to \\infty$ and $α, β\\in [0, \\infty]$, define the joint level set \\[F_{α,β}:=\\Big\\{x\\in [0,1)\\colon \\liminf_{n\\to\\infty}\\frac{\\log (a_{n+1}(x))}{ψ(n)}=α,\\ \\limsup_{n\\to\\infty}\\frac{\\log (a_{n+1}(x))}{\\log q_n(x)}=β\\Big\\}. \\] We determine the Hausdorff dimension of $F_{α,β}$ for all values of $ α$ and $β$. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang--Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song--Tan--Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25812",
  "author_total": 2,
  "authors": [
    {
      "name": "Wanjin Cheng"
    },
    {
      "name": "Jing Feng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9eycpe6ccv868.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25812",
  "fulltext_url": null,
  "id": "9eycpe6ccv868",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25812",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214091495",
  "pdf_url": "https://arxiv.org/pdf/2609.25812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9eycpe6ccv868.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25812"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Irrationality Exponents and Partial Quotient Growth in Continued Fractions",
  "title_text": "Irrationality Exponents and Partial Quotient Growth in Continued Fractions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9eycpe6ccv868",
  "venue": "arXiv",
  "versions": []
}