{
  "abstract": "For $0<λ<1/2$, let $K_λ$ be the attractor of the iterated function system $\\{λx, λx+1-λ\\}$, and put $C_λ=K_λ\\times K_λ$. We study the intersection \\[ E_λ=C_λ\\cap S^1 \\] and the Minkowski sum \\[ A_λ=C_λ+S^1, \\] where $S^1$ is the unit circle. For the intersection problem, we prove that $E_λ$ has the cardinality of the continuum for $(\\sqrt{3}-1)/2<λ<1/2$, and $\\dim_{\\rm H} E_λ>0$ for $\\sqrt{2}-1<λ<1/2$. We also establish quantitative lower bounds for $\\dim_{\\rm H} E_λ$ for $λ$ near $1/2$; in particular, $\\dim_{\\rm H} E_λ$ approaches $1$ as $λ\\uparrow1/2$. For the Minkowski sum problem, we prove that $A_λ$ has nonempty interior throughout the previously open range $1/4<λ<1/3$, answering a question of Simon and Taylor. Together with earlier results of Simon and Taylor, our theorem yields the complete classification: $A_λ$ has nonempty interior in $\\mathbb{R}^2$ if and only if $1/4<λ<1/2$. More generally, we prove that $C_λ+Γ$ has nonempty interior for every regular $C^1$ closed curve $Γ$ whenever $1/4<λ<1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08568",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiangtao Li"
    },
    {
      "name": "Zhao Shen"
    },
    {
      "name": "Yufeng Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9c4dj5yrc75ec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08568",
  "fulltext_url": null,
  "id": "9c4dj5yrc75ec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08568",
  "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": "W7212026917",
  "pdf_url": "https://arxiv.org/pdf/2609.08568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9c4dj5yrc75ec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.08568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle",
  "title_text": "Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9c4dj5yrc75ec",
  "venue": "arXiv",
  "versions": []
}