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

Abstract

For $0<\lambda <1/2$, let $K_{\lambda }$ be the attractor of the iterated function system $\{\lambda x,\lambda x+1-\lambda \}$, and put $C_{\lambda }=K_{\lambda }\times K_{\lambda }$. 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 $E_{\lambda }$ has the cardinality of the continuum for $(\sqrt{3}-1)/2<\lambda <1/2$, and ${\dim }_{\mathrm{H}}{E}_{\lambda }>0$ for $\sqrt{2}-1<\lambda <1/2$. We also establish quantitative lower bounds for ${\dim }_{\mathrm{H}}{E}_{\lambda }$ for $\lambda$ near $1/2$; in particular, ${\dim }_{\mathrm{H}}{E}_{\lambda }$ approaches $1$ as $\lambda \uparrow 1/2$. For the Minkowski sum problem, we prove that $A_{\lambda }$ has nonempty interior throughout the previously open range $1/4<\lambda <1/3$, answering a question of Simon and Taylor. Together with earlier results of Simon and Taylor, our theorem yields the complete classification: $A_{\lambda }$ has nonempty interior in ${\mathbb{R}}^2$ if and only if $1/4<\lambda <1/2$. More generally, we prove that $C_{\lambda }+\Gamma$ has nonempty interior for every regular $C^1$ closed curve $\Gamma$ whenever $1/4<\lambda <1/2$.

Revised to acknowledge the priority of Kan Jiang and Zhikun Xie for Theorem 1.1(i); abstract,reference,and introduction updated accordingly. Mathematical statements and proofs unchanged