Year: 2026
Type: preprint
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 $1/4<λ<1/2$.
Source: arXiv (Cornell University)
Authors: Li, Jiangtao, Shen, Zhao, Wu, Yufeng
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G6276659556
OpenAlex ID: W7212026917