{
  "abstract": "Let $E\\subset\\mathbb R^2$ be analytic with $t=\\dim_{\\mathrm H}E>1$. We prove $\\dim_{\\mathrm H}ΔE\\ge 4/5+η_0$ for an absolute, non-explicit constant $η_0>0$, and \\[ \\dim_{\\mathrm H}ΔE\\ge \\frac{t^2-2t+5}{2t^2-7t+10} \\quad\\text{for }1<t\\le\\frac{11-\\sqrt{41}}{4}, \\] which reaches $7/8$ at the right endpoint. For pinned distances, $\\dim_{\\mathrm H}Δ_xE\\ge q_*=0.7813\\ldots$ for every $x\\in\\mathbb R^2$ outside an exceptional set of Hausdorff dimension at most one, with stronger bounds depending on $t$ and further exceptional-set estimates. If $E$ is compact and supports an $s$-Frostman measure $μ$, $s>1$, then \\[ \\underline{\\dim}_{\\mathrm B}Δ_xE\\ge \\max\\left\\{\\frac45+η_0,\\min\\left\\{\\frac{3s+1}{5},\\frac78\\right\\}\\right\\} \\] for $μ$-almost every $x$. The same bound holds for pinned Hausdorff dimension under a common dyadic mass law, allowing oscillation with scale. We retain information about both endpoints in the Kolmogorov complexity estimates of Stull and Fiedler--Stull. Applying the partition estimates with the endpoints reversed yields $4/5$ and the rational bound. Equality forces both information profiles to have excess information on a common interval of scales. Encoding one source measure for all pins converts this constraint into a configuration of points and circles with many rich perpendicular bisectors. Lower bounds of Katz--Tardos and Ren--Wang contradict the upper bound of Fu--Gan--Ren; compactness then gives $η_0>0$. This rigidity argument has a precedent in the work of Héra, Shmerkin and Yavicoli on Furstenberg sets, using Bourgain's projection theorem. A separate finite-scale selection proves the pinned lower box bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05500",
  "author_total": 2,
  "authors": [
    {
      "name": "Alex Iosevich"
    },
    {
      "name": "Alexia Yavicoli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftnq288bbg6k6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05500",
  "fulltext_url": null,
  "id": "ftnq288bbg6k6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05500",
  "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",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7220665524",
  "pdf_url": "https://arxiv.org/pdf/2610.05500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftnq288bbg6k6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05500"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Improved dimension bounds for Falconer's distance problem in the plane",
  "title_text": "Improved dimension bounds for Falconer's distance problem in the plane",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/ftnq288bbg6k6",
  "venue": "arXiv",
  "versions": []
}