{
  "abstract": "We introduce a family of metric outer measures whose critical exponents form a packing spectrum interpolating between Hausdorff and packing dimensions. For analytic sets, we characterize each spectral value as the supremum of the Hausdorff dimension of a product with a compact testing set minus the testing set's upper intermediate dimension at the complementary parameter. The associated product bounds refine the classical Hausdorff--packing inequalities and can give exact Hausdorff additivity when both factors have distinct endpoint dimensions. Below the packing endpoint, we prove a variational principle in terms of windowed local mass exponents and sharp parameter estimates implying local Lipschitz continuity; a jump at the endpoint is possible. We characterize all attainable profiles by monotonicity, continuity below the packing endpoint, and a sharp upper right Dini-derivative bound. Every admissible profile is realized by a compact binary digit set whose uniform measure attains the finite-window and classical packing variational suprema. Complementary digit positions yield reflected upper intermediate-dimension profiles and a single factor attaining the Hausdorff product bounds at every parameter, with an Ahlfors regular product measure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10375",
  "author_total": 1,
  "authors": [
    {
      "name": "Peizhi Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1w19kqghb8qgm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10375",
  "fulltext_url": null,
  "id": "1w19kqghb8qgm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10375",
  "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.MG"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220955602",
  "pdf_url": "https://arxiv.org/pdf/2610.10375",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1w19kqghb8qgm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.10375",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10375"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The packing spectrum: interpolation between Hausdorff and packing dimensions",
  "title_text": "The packing spectrum: interpolation between Hausdorff and packing dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/1w19kqghb8qgm",
  "venue": "arXiv",
  "versions": []
}