{
  "abstract": "Given an integer $n\\ge 2$ and points $\\mathbf{x}, \\mathbf{y}$ lying in the $n$--dimensional torus $\\mathbb{T}^n$ equipped with its usual metric denoted by $\\left\\langle\\;\\cdot\\;\\right\\rangle$, let \\[ τ_n(\\mathbf x,\\mathbf y) = \\sup\\left\\{ τ\\ge0: \\begin{array}{l} \\text{there exist infinitely many } H\\ge1\\text{ for which}\\\\[1mm] \\|A\\|\\le H \\text{ and } \\langle A\\mathbf x-\\mathbf y\\rangle\\le H^{-τ} \\text{ for some } A\\in\\mathrm{SL}_n(\\mathbb Z) \\end{array} \\right\\} \\] be the Diophantine exponent of asymptotic approximation of $\\mathbf{y}$ by the orbit of $\\mathbf{x}$ under the action of the integral special linear group $\\mathrm{SL}_n(\\mathbb Z)$. Let also \\[ \\widehatτ_n\\!\\left(\\mathbf{x}, \\mathbf{y}\\right) = \\sup\\left\\{ τ\\ge0: \\begin{array}{l} \\text{for all but finitely many } H\\ge1, \\text{ there is } A\\in\\mathrm{SL}_n(\\mathbb Z)\\\\[1mm] \\text{such that } \\|A\\|\\le H \\text{ and } \\langle A\\mathbf x-\\mathbf y\\rangle\\le H^{-τ} \\end{array} \\right\\} \\] denote the corresponding exponent of uniform approximation. The main result of this paper proves that for almost all $\\mathbf{x}, \\mathbf{y}\\in\\mathbb{T}^n$ (with respect to the induced Lebesgue measure), $$\\widehatτ_n\\!\\left(\\mathbf{x}, \\mathbf{y}\\right)\\;=\\; τ_n\\!\\left(\\mathbf{x}, \\mathbf{y}\\right)\\;=\\; n-1.$$ This is the first result to establish the exact generic approximation exponents for an action of a lattice subgroup of $\\mathrm{SL}_n(\\mathbb R)$ induced by the standard Euclidean action.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36397",
  "author_total": 2,
  "authors": [
    {
      "name": "Faustin Adiceam"
    },
    {
      "name": "Régis Aolo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afq0et2q74jyf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36397",
  "fulltext_url": null,
  "id": "afq0et2q74jyf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36397",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7224114834",
  "pdf_url": "https://arxiv.org/pdf/2610.36397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afq0et2q74jyf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus",
  "title_text": "Exact Generic Exponents of Approximation for the Action of the Integral Special Linear Group on the Torus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/afq0et2q74jyf",
  "venue": "arXiv",
  "versions": []
}