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

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