{
  "abstract": "Can the complete macroscopic spectrum of return times determine the Hausdorff dimension of the corresponding set of initial points? We classify all common lower and upper macroscopic intermediate-dimension spectra of zero-density integer sets and realize every admissible spectrum on a full-dimensional family of zero-frequency orbits under logarithmic-power bounds on inverse-branch contraction. For first-return branches with stretched-exponential contraction $e^{-n^δ}$, $0<δ<1$, complete spectrum levels exhibit a sharp zero/full dimension transition at the counting endpoint $δ$. At the critical exponent, exact laws $N_y(t)\\sim c_0t^δ(\\log t)^β$ yield zero dimension for $β\\le0$ and full dimension for $β>0$, even with the complete spectrum fixed. An explicit two-branch intermittent interval map realizes this transition through genuine first returns. Above the critical exponent, imposing an exact counting law and a complete spectrum does not reduce the dimension of an intrinsic mean-energy level in an affine return tower. For nonlinear branches, we obtain quantitative bounds and an exact conditional-pressure formula under thermodynamic regularity assumptions, verified for an explicit non-affine family.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06416",
  "author_total": 1,
  "authors": [
    {
      "name": "Jinjun Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dd6xepmv66pe4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06416",
  "fulltext_url": null,
  "id": "dd6xepmv66pe4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06416",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06416",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dd6xepmv66pe4.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06416",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06416"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Macroscopic return-time spectra and Hausdorff dimension transitions",
  "title_text": "Macroscopic return-time spectra and Hausdorff dimension transitions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/dd6xepmv66pe4",
  "venue": "arXiv",
  "versions": []
}