{
  "abstract": "Let $Γ\\subset \\operatorname{SL}(2, \\mathbb{R})$ be discrete, co-finite with cusps, and let ${\\frak b}\\in \\mathbb{R}\\cup\\{\\infty\\}$ be a cusp of $Γ$. Let $f: Γ{\\frak b} \\to \\mathbb{R}$ be a quantum modular form in the sense that the maps $$ ϕ_γ(x) = f(x) - f(γx) \\qquad (x\\in Γ{\\frak b}, γ\\in Γ) $$ extend to Hölder-regular functions. Under bounded moments assumptions of the shape $\\sum_n n^{-2} \\int |ϕ_{γ^n}(x)|^αd x < \\infty$ for some $α>0$ and every fixed parabolic element $γ\\in Γ$, we prove asymptotic formulas for the characteristic functions of the values $f(x)$ for $x\\in Γ{\\frak b}$, ordered by growing denominators. These yield Gaussian limits when $α>2$ and other stable limit laws under suitable further assumptions. This generalizes to Fuchsian groups the work of Baladi-Vallée on rational orbits of the Gauss map. In companion papers, we use this result, and in particular its flexible assumptions on $ϕ_γ$, to deduce limit theorems for automorphic $L$-series at the central point. One ingredient in our proof is a particular cuspidal acceleration of the Bowen-Series map, whose inverse branches admit a convenient parametrization, and for which we describe explicitly a natural extension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41973",
  "author_total": 3,
  "authors": [
    {
      "name": "Sandro Bettin"
    },
    {
      "name": "Sary Drappeau"
    },
    {
      "name": "Jungwon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b202j6kantc10.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41973",
  "fulltext_url": null,
  "id": "b202j6kantc10",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41973",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b202j6kantc10.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Limit theorems for cuspidal orbits on Fuchsian groups",
  "title_text": "Limit theorems for cuspidal orbits on Fuchsian groups",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/b202j6kantc10",
  "venue": "arXiv",
  "versions": []
}