{
  "abstract": "We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $ψ:\\N\\to(0,\\infty)$ be a non-increasing approximation function and let $τ_ψ$ denote its lower order. For $m,n\\ge1$ and $\\boldsymbolθ\\in\\mathbb R^m$, denote $\\Exact_{m,n}(ψ,\\boldsymbolθ)$ the set of $m\\times n$ matrices that are $ψ$-approximable with shift $\\boldsymbolθ$, but are not $cψ$-approximable for any $0<c<1$. It is shown that \\[ \\hdim \\Exact_{m,n}(ψ,\\boldsymbolθ)=m(n-1)+\\frac{m+n}{1+τ_ψ},\\ \\ \\ {\\text{for}}\\ τ_ψ>\\frac{n}{m}. \\] We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\\left\\{\\frac{\\mathbf p+\\boldsymbolθ}{q}: (\\mathbf p, q)\\in \\mathbb{Z}^{m+1}\\right\\},\\ \\ \\ \\bigg\\{\\big\\{A\\in \\Mat_{m, n}(\\R): A\\mathbf q-\\mathbf p-\\boldsymbolθ=0\\big\\}: (\\mathbf p, \\mathbf q)\\in \\mathbb{Z}^{m+n}\\bigg\\}.$$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03442",
  "author_total": 3,
  "authors": [
    {
      "name": "Yubin He"
    },
    {
      "name": "Bixuan Li"
    },
    {
      "name": "Baowei Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dptt1p5ewk3s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03442",
  "fulltext_url": null,
  "id": "2dptt1p5ewk3s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03442",
  "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": "W7220371333",
  "pdf_url": "https://arxiv.org/pdf/2610.03442",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dptt1p5ewk3s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03442",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03442"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hausdorff dimension for exactly approximable matrices",
  "title_text": "Hausdorff dimension for exactly approximable matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/2dptt1p5ewk3s",
  "venue": "arXiv",
  "versions": []
}