{
  "abstract": "The hyperplane Furstenberg problem asks how small the Hausdorff dimension of $E\\subset\\mathbb{R}^n$ can be when a $t$-dimensional family of affine $(n-1)$-dimensional hyperplanes meets $E$ in sets of dimension at least $s$. For $0<s\\le n-1$ and $0<t\\le n$, we prove the sharp bound in $\\mathbb{R}^n$ $\\dim_{\\mathrm{H}}E\\ge\\mathbf{F}(s,t;n,n-1)$, where $\\mathbf{F}$ is Gan's Furstenberg index proposed in prime fields. We also construct examples attaining this bound for all parameters. Our main reduction theorem, Theorem 3.1, states that for $n\\ge3$, $0<s\\le n-2$, and $1<t\\le n$, every weighted $(s,t)$ configuration $(E,H)$ at scale $δ$ produces a dual weighted $(s,t-1)$ configuration $(P,\\mathcal{T})$ inside a hyperplane $H_{y_0}\\cong\\mathbb{R}^{n-1}$. The reduction uses radial projection and energy estimates for Frostman measures. Iterating this reduction and applying the Csörnyei--Stull incidence theorem yields the bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03504",
  "author_total": 1,
  "authors": [
    {
      "name": "Peiqi Nie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bxc8adt0q0zgz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03504",
  "fulltext_url": null,
  "id": "bxc8adt0q0zgz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03504",
  "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/2611.03504",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bxc8adt0q0zgz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03504",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03504"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hyperplane Furstenberg Sets in Higher Dimensions",
  "title_text": "Hyperplane Furstenberg Sets in Higher Dimensions",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/bxc8adt0q0zgz",
  "venue": "arXiv",
  "versions": []
}