{
  "abstract": "Let $K \\subset\\mathbb{R}^d$, $d\\ge2$, be a connected compact set and $ϕ(t)=tL(t)$ be a gauge function with $L$ non-increasing. We study intersections of $K$ with rectifiable curves and chord-arc curves. We show that if $\\int_0^1\\frac{dt}{ϕ(t)}<\\infty$, then there exists a rectifiable curve $γ\\subset\\mathbb{R}^d$ such that $\\mathcal H^ϕ(K \\capγ)>0$. Moreover, for every $0<s<1$, there exists a chord-arc curve $γ$ such that $\\textrm{H.dim }(K \\capγ)\\ge s$. Conversely, we show that if $K$ is uniformly porous on lines and $\\int_0^1\\frac{dt}{ϕ(t)}=\\infty$, then $\\mathcal H^ϕ(K \\capγ)=0$ for every rectifiable curve $γ$ and $\\textrm{H.dim }(K \\capγ)<1$ for every chord-arc curve $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41510",
  "author_total": 2,
  "authors": [
    {
      "name": "Oleg Ivrii"
    },
    {
      "name": "Truong Vu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2f6arer6fbam1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41510",
  "fulltext_url": null,
  "id": "2f6arer6fbam1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41510",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41510",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2f6arer6fbam1.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41510",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41510"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Intersections of fractal sets with rectifiable curves",
  "title_text": "Intersections of fractal sets with rectifiable curves",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/2f6arer6fbam1",
  "venue": "arXiv",
  "versions": []
}