{
  "abstract": "We generalize Ren and Wang's incidence bound between points and lines in $\\R^2$ \\cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in $\\R^3$ and in $\\R^4$. We show that if $d=3$ or $d=4$, and $E\\subset \\R^d$ is a Borel set of dimension $\\dim_H(E) > d/2$, then \\begin{equation*} \\sup_{x\\in E} \\dim_H(Δ_x(E)) \\geq 2/3, \\end{equation*} where $Δ_x(E)$ is the pinned distance set of $E$ with respect to $x$. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in $\\R^d$ that contain a $t$-dimensional set of hyperplanes, each of which contains an $s$-dimensional set of points, for any $d\\ge 2$, $s \\in (d-2, d-1]$ and $t \\in (0, d]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38742",
  "author_total": 2,
  "authors": [
    {
      "name": "Marianna Csornyei"
    },
    {
      "name": "D. M. Stull"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dsm4m5hkyhs4d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38742",
  "fulltext_url": null,
  "id": "dsm4m5hkyhs4d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38742",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7215103669",
  "pdf_url": "https://arxiv.org/pdf/2609.38742",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dsm4m5hkyhs4d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38742",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38742"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hyperplane Incidences and Distance Sets in Higher Dimensions",
  "title_text": "Hyperplane Incidences and Distance Sets in Higher Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dsm4m5hkyhs4d",
  "venue": "arXiv",
  "versions": []
}