{
  "abstract": "Two transverse segments have local Hausdorff and upper box dimension one at their crossing, although they carry two independent tangent directions. We define the Point-Cross Dimension by assigning each tangent line the largest local upper box dimension captured by intersection with a regular \\(C^1\\) curve in that direction, then taking the supremum of their sums over finite independent families of tangent lines. Every Bouligand direction admits such a recurrent curve. The resulting profile is invariant under closure and covariant under local \\(C^1\\)-diffeomorphisms. Each directional weight is attained, and the profile is upper semicontinuous. An integral representation by ranks of weight superlevel sets gives submodularity and, together with projection bounds, exact Cartesian-product additivity. Germs with the same projective tangent support, local upper box dimension, and rank of segment directions can have different Point-Cross Dimensions. Further computations give exact directional profiles for chirps and the Point-Cross Dimension of a self-affine carpet. Finally, approximation of a ray with error \\(O(t^d)\\), for \\(d>1\\), forces directional weight at least \\(1-1/d\\), and this bound is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08612",
  "author_total": 1,
  "authors": [
    {
      "name": "Nadir Maaroufi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4vxtsjyfn27qb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08612",
  "fulltext_url": null,
  "id": "4vxtsjyfn27qb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08612",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7167901554",
  "pdf_url": "https://arxiv.org/pdf/2607.08612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4vxtsjyfn27qb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.08612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08612"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The point-cross dimension",
  "title_text": "The point-cross dimension",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/4vxtsjyfn27qb",
  "venue": "arXiv",
  "versions": []
}