{
  "abstract": "We give an alternative proof of the $5/4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5/4$, then there exists $x\\in E$ such that the pinned distance set $$Δ_x(E):=\\{|x-y|: y\\in E\\} $$ has positive Lebesgue measure. Moreover, we are able to beat the $5/4$ barrier if the packing dimension of $E$ is $<2$. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of $E$. These conditions improve the $5/4$ threshold when the packing dimension of $E$ is $<2$, and recover the second author's result for $\\dim_H E=\\dim_P E>1$. The proof is based on $L^2$ ball inflation and does not use decoupling inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05015",
  "author_total": 3,
  "authors": [
    {
      "name": "Shengwen Gan"
    },
    {
      "name": "Bochen Liu"
    },
    {
      "name": "Shukun Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3grr5y2pc6433.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05015",
  "fulltext_url": null,
  "id": "3grr5y2pc6433",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05015",
  "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": "W7220661350",
  "pdf_url": "https://arxiv.org/pdf/2610.05015",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3grr5y2pc6433.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05015",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05015"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Lebesgue measure of distance sets and $L^2$ ball inflation",
  "title_text": "Lebesgue measure of distance sets and L^2 ball inflation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3grr5y2pc6433",
  "venue": "arXiv",
  "versions": []
}