{
  "abstract": "For the unicritical family $f_{c,j}(z)=z^j+c$, the Multibrot sets $M_j$ converge geometrically to the closed unit disk as $j\\to\\infty$, but this limit does not quantify their finite-degree area behavior. We estimate Area($M_j$) using escape-time pixel counting while separating the effects of the iteration threshold $K$ and spatial sampling density $ρ$. The finite-iteration sets form a nested decreasing sequence, while grid refinement is treated empirically. Across $K\\in\\{25,100,500,1000\\}$ and $ρ\\in\\{128,256,512,1024\\}$, the final grid refinement changes the reported areas by at most $1.1\\times10^{-3}$. At $K=1000$ and $ρ=1024$, the estimates are well described by $C(j)\\approxπ-2.49216j^{-0.57103}$. Independently, Laurent coefficients of the exterior conformal map and Gronwall's area formula yield finite-truncation upper bounds $U_{1000}(j)\\approxπ-2.07405j^{-0.643427}$. Across all tested degrees, the grid estimates remain below the corresponding conformal bounds, and both sequences exhibit similar power-law decay of the area deficit from $π$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26837",
  "author_total": 1,
  "authors": [
    {
      "name": "Rohan Senapati"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bft3xzxxrqd1d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26837",
  "fulltext_url": null,
  "id": "bft3xzxxrqd1d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26837",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214155500",
  "pdf_url": "https://arxiv.org/pdf/2609.26837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bft3xzxxrqd1d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "A Computational and Conformal Analysis of Multibrot Sets",
  "title_text": "A Computational and Conformal Analysis of Multibrot Sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/bft3xzxxrqd1d",
  "venue": "arXiv",
  "versions": []
}