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 \[ E_\lambda=C_\lambda\cap S^1 \] and the Minkowski sum \[ A_{\lambda}=C_\lambda+S^1, \] where $S^1$ is the unit circle. For the intersection problem, let $\lambda_{\infty}\approx 0.305854$ be the unique root in $(1/4,1/2)$ of $2x^3-3x^2+4x-1=0$. We prove that $E_\lambda$ is infinite for every $\lambda_{\infty}<\lambda<1/2$. In particular, $C_{1/3}\cap S^1$ is infinite, answering the first part of a question of Jiang, Kong, Li and Wang. Moreover, $E_\lambda$ has the cardinality of the continuum for $(\sqrt{3}-1)/2<\lambda<1/2$, and $\dim_{\rm H} E_\lambda>0$ for $\sqrt{2}-1<\lambda<1/2$. We also establish quantitative lower bounds for $\dim_{\rm H} E_\lambda$ for $\lambda$ near $1/2$; in particular, $\dim_{\rm H} E_\lambda$ approaches $1$ as $\lambda\uparrow1/2$. 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 $1/4<\lambda<1/2$.