{
  "abstract": "A variant of the Falconer distance problem asks for fixed $k\\geq 1$ and $d\\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\\subset\\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\\ldots,x_{k}\\in E$ such that $\\text{Vol}_{k+1}^{(x_0,\\ldots,x_{k})}(E) = \\lbrace \\text{Vol}_{k+1}(x_0,\\ldots,x_{k},x_{k+1}) : x_{k+1}\\in E \\rbrace$ has positive Lebesgue measure. Here $\\text{Vol}_{k+1}(x_0,\\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \\leq d \\leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.22450",
  "author_total": 2,
  "authors": [
    {
      "name": "José Gaitan Montejo"
    },
    {
      "name": "Eyvindur Ari Palsson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b95y72yqneds8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.22450",
  "fulltext_url": null,
  "id": "b95y72yqneds8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.22450",
  "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/2605.22450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b95y72yqneds8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2605.22450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.22450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On volumes of simplices in intermediate dimensions",
  "title_text": "On volumes of simplices in intermediate dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/b95y72yqneds8",
  "venue": "arXiv",
  "versions": []
}