{
  "abstract": "We prove that every planar Jordan curve of finite \\(p\\)-variation, with \\(1\\leq p<2\\), inscribes a rectangle of every prescribed similarity class. In particular, every such curve inscribes a square. The proof combines the recent criterion of Asano and Ike with a variation-controlled approximation argument. We show that for every \\(q>p\\), a Jordan curve of finite \\(p\\)-variation can be approximated in the \\(q\\)-variation topology by smooth Jordan embeddings. The construction uses simple polygonal interpolants of Boedihardjo and Geng, an elementary interpolation inequality between variation seminorms, and a variation-controlled smoothing of polygonal embeddings. Young integration then gives locally uniform convergence of the associated primitives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04918",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiangfei Li"
    },
    {
      "name": "Yichen Pan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqde63k7qjq98.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04918",
  "fulltext_url": null,
  "id": "bqde63k7qjq98",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04918",
  "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.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.04918",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqde63k7qjq98.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04918",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04918"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rectangular Pegs on Jordan Curves of Finite $p$-Variation",
  "title_text": "Rectangular Pegs on Jordan Curves of Finite p-Variation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/bqde63k7qjq98",
  "venue": "arXiv",
  "versions": []
}