Khintchine dichotomy for self-similar measures

We extend Khintchine’s theorem to all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on S L 2 ( R ) / S L 2 ( Z ) SL_{2}(\mathbb {R})/SL_{2}(\mathbb {Z}) , a result of independent interest.