{
  "abstract": "In this paper, we consider Kakeya sets with the additional restriction that centers of the unit line segments belong to a given set. In particular, for every uncountable Borel set $A\\subset\\mathbb{R}^d$, $d\\geq 2$, we construct a compact subset of $\\mathbb{R}^d$ of Lebesgue measure zero that contains, in every direction, a unit line segment whose center lies in $A$. Notice that every such Kakeya set (not even necessarily compact) must have positive Lebesgue measure if $A$ is countable. So our result shows that the countability is in fact the only obstruction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06556",
  "author_total": 7,
  "authors": [
    {
      "name": "Cheng Chen"
    },
    {
      "name": "Jianbin Li"
    },
    {
      "name": "Longhui Li"
    },
    {
      "name": "Bochen Liu"
    },
    {
      "name": "Ruichen Sun"
    },
    {
      "name": "Haoran Wang"
    },
    {
      "name": "Saijie Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cnyp10e10tmx3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06556",
  "fulltext_url": null,
  "id": "cnyp10e10tmx3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06556",
  "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": "W7211991905",
  "pdf_url": "https://arxiv.org/pdf/2609.06556",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cnyp10e10tmx3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06556",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06556"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Some constructions of restricted Kakeya sets",
  "title_text": "Some constructions of restricted Kakeya sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/cnyp10e10tmx3",
  "venue": "arXiv",
  "versions": []
}