Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle

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× K_λ. We study the intersection E_λ=C_λ∩ S¹ and the Minkowski sum A_λ=C_λ+S¹, where S¹ is the unit circle. For the intersection problem, we prove that E_λ has the cardinality of the continuum for (√3−1)/2<λ<1/2, and dim_H E_λ>0 for √2−1<λ<1/2. We also establish quantitative lower bounds for dim_H E_λ for λ near 1/2; in particular, dim_H E_λ approaches $1$ as λ↑1/2. 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 ℝ² if and only if 1/4<λ<1/2. More generally, we prove that C_λ+Γ has nonempty interior for every regular C¹ closed curve Γ whenever 1/4<λ<1/2.

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BibTeX RIS

BibTeX

@misc{li2026intersections,
  title = {{Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle}},
  author = {Jiangtao Li and Zhao Shen and Yufeng Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08568},
  eprint = {2609.08568},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08568},
}

RIS

TY  - UNPB
TI  - Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle
AU  - Jiangtao Li
AU  - Zhao Shen
AU  - Yufeng Wu
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.08568
UR  - https://arxiv.org/abs/2609.08568
AB  - 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× K_λ. We study the intersection E_λ=C_λ∩ 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 (√3−1)/2<λ<1/2, and dim_H E_λ>0 for √2−1<λ<1/2. We also establish quantitative lower bounds for dim_H E_λ for λ near 1/2; in particular, dim_H E_λ approaches $1$ as λ↑1/2. 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 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.
ER  -

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