Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

In this article, for a large class of rational self-similar IFS’s wich contains the middle-third Cantor set, we compute the Hausdorff dimension of elements a self-similar set that are $\psi$-approximable by rational belonging to this set and satisfying that its numerator has a bounded number of distinct prime divisors. This paper is based on a previous version in which the proof of a lemma (Lemma 3.8) was incorrect.