{
  "abstract": "We use normal-turn estimates to study the local and global geometry of the boundaries of stable-norm balls $B_X$ for complete finite-area hyperbolic once-punctured tori $X$. We show that the boundary $\\partial B_X$ is a convex-geometric detector of exponential Diophantine approximation: rational directions give genuine corners with exponentially small exterior angles, while at an irrational direction $β$ the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach $β$ and the $\\ell^\\infty$-radius of $B_X$ in the projective direction $β$. Thus irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets. The computation of the rational corners unexpectedly yields a new McShane-type identity for the area of the dual body $B_X^*$: $$ \\operatorname{Area}(B_X^*) = 4\\sum_γ \\ell_X(γ) \\log\\coth\\left(\\frac{\\ell_X(γ)}{2}\\right), $$ where the sum is over unoriented simple closed geodesics on $X$. As another global application, the normal-turn estimates also give a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly $L\\geq2$ is at most $C_X(\\log L)^2$. For the modular torus this yields $$ \\#λ_M^{-1}(m)\\leq C(\\log\\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic bounds for Markoff fibers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.14574",
  "author_total": 3,
  "authors": [
    {
      "name": "Nhat Minh Doan"
    },
    {
      "name": "Xiaobin Li"
    },
    {
      "name": "Van Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/16kw2cxsybnc4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.14574",
  "fulltext_url": null,
  "id": "16kw2cxsybnc4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.14574",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7161264112",
  "pdf_url": "https://arxiv.org/pdf/2605.14574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/16kw2cxsybnc4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2605.14574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.14574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Stable-norm boundary geometry on once-punctured tori",
  "title_text": "Stable-norm boundary geometry on once-punctured tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/16kw2cxsybnc4",
  "venue": "arXiv",
  "versions": []
}