{
  "abstract": "For each $α\\in(0,1)$, we construct an explicit deterministic finite homogeneous planar iterated function system of similarities satisfying the strong separation condition, whose attractor $E_α$ has positive finite one-dimensional Hausdorff measure and, for all sufficiently small $δ>0$, satisfies $\\sup_{0\\leθ<π}H^1(π_θ(N_δ(E_α)))\\lesssim_α(\\logδ^{-1})^{-α}$. The defining system depends only on $α$ and is fixed at every scale. For these same sets, we prove the sharp analytic capacity law $γ(N_δ(E_α))\\asymp_α(\\logδ^{-1})^{-1/2}$ as $δ\\rightarrow0$. Consequently, each $E_α$ satisfies the quantitative Vitushkin comparison $\\operatorname{Fav}(N_δ(E_α))\\lesssim_αγ(N_δ(E_α))^{2α}$ for every $δ\\in(0,1)$. These comparisons are superlinear for $α>1/2$, with exponents arbitrarily close to the optimal upper limit two for this class of self-similar sets. A completely specified sixteen-map system gives $α=53/100$ and comparison exponent $53/50$. Inspired by Falconer's Venetian blind constructions, the projection argument combines exact collinear cylinder subfamilies, quantitative distribution of word orientations, and a cover indexed by first successful symbolic blocks. The rotation parameters are explicit powers of a real algebraic number, linking the construction to simultaneous Diophantine approximation. The capacity estimate follows from the curvature characterization of analytic capacity and a telescoping inequality for cylinder masses.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22908",
  "author_total": 1,
  "authors": [
    {
      "name": "Caleb Marshall"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftgdmqdh1jg93.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22908",
  "fulltext_url": null,
  "id": "ftgdmqdh1jg93",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22908",
  "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.CV"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7222956259",
  "pdf_url": "https://arxiv.org/pdf/2610.22908",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftgdmqdh1jg93.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22908",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22908"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets",
  "title_text": "Nearly Sharp Uniform Projection Bounds and Superlinear Vitushkin Inequalities for Deterministic Self-Similar Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/ftgdmqdh1jg93",
  "venue": "arXiv",
  "versions": []
}