On planar self-similar sets with a dense set of rotations

Abstract

We prove that if $E$ is a planar self-similar set with similarity dimension $d$ whose defining maps generate a dense set of rotations, then the $d$-dimensional Hausdorff measure of the orthogonal projection of $E$ onto any line is zero. We also prove that the radial projection of $E$ centered at any point in the plane also has zero $d$-dimensional Hausdorff measure. Then we consider a special subclass of these sets and give an upper bound for the Favard length of $E(\rho)$ where $E(\rho)$ denotes the $\rho$-neighborhood of the set $E$.