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

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.