{
  "abstract": "We study sets in R^n that contain a unit k-dimensional disk in each direction from a subset of the Grassmannian G(n,k) of Hausdorff dimension at least t; we call them (n,k,t)-sets. In the range 2k>n, we prove that every (n,k,t)-set has positive Lebesgue measure whenever t>k(n-k-1)+1. This follows from a sharp Sobolev-to-L^2 estimate for the k-plane maximal operator over a Frostman measure on G(n,k), which also yields an L^p-L^2 estimate for every p>n/k with respect to Haar measure. In the range 2k<=n, where translations can no longer be ignored, we obtain the same threshold via a lifting argument, conditional on the quantitative Kakeya estimate in dimension n-k+1. Using the quantitative Kakeya estimate of Wang and Zahl in dimension 3, we deduce that (4,2,t)-sets have positive measure for t>3. The threshold t>k(n-k-1)+1 is sharp, in the sense that (n,k,t)-sets of measure zero exist at t=k(n-k-1)+1. These results improve the previous threshold t>n/2+k(n-k-1) of Fraser, Harris and Kroon, and give the first restricted-direction results in the range 2k<=n.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33734",
  "author_total": 1,
  "authors": [
    {
      "name": "Longhui Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8asj1reahkgdn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33734",
  "fulltext_url": null,
  "id": "8asj1reahkgdn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33734",
  "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",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223811676",
  "pdf_url": "https://arxiv.org/pdf/2610.33734",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8asj1reahkgdn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33734",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33734"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the positive Lebesgue measure of (n,k,t)-sets",
  "title_text": "On the positive Lebesgue measure of (n,k,t)-sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8asj1reahkgdn",
  "venue": "arXiv",
  "versions": []
}