A smooth family of Cantor sets

Abstract

We show that the Cantor set $C_p$ associated to the sequence $\{1/n^p\}_n$, $p>1$, is a smooth attractor. Moreover, it is smoothly conjugate to the $2^{-p}$-middle Cantor set. We also study the convolution of Hausdorff measures supported on these sets and the structure and size of the sumset $C_p+C_q$.