{
  "abstract": "The set of $τ$-approximable numbers, $\\mathcal W(τ)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\\Bad^γ$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $Ψ$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $ψ$, we prove that $\\mathcal W(ψ)\\cap\\Bad^γ$ is strong $Ψ$-rapid winning for a natural gauge $Ψ$ determined by $ψ$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $Ψ$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $Ψ$. In particular, for $ψ(q)=q^{-τ}, τ\\ge1,$ we recover the exact Jarník--Besicovitch dimension, that is, $$ \\HD\\bigl(\\mathcal W(τ)\\cap\\Bad^γ\\bigr)=\\frac{2}{τ+1}.$$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25349",
  "author_total": 2,
  "authors": [
    {
      "name": "Mumtaz Hussain"
    },
    {
      "name": "David Simmons"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dke2z13jddj69.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25349",
  "fulltext_url": null,
  "id": "dke2z13jddj69",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25349",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7204475771",
  "pdf_url": "https://arxiv.org/pdf/2608.25349",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dke2z13jddj69.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2608.25349",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25349"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Well and badly approximable sets, and rapid winning",
  "title_text": "Well and badly approximable sets, and rapid winning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/dke2z13jddj69",
  "venue": "arXiv",
  "versions": []
}