Khintchine dichotomy for self-similar measures

TL;DR

This paper extends Khintchine's theorem to all self-similar measures on the real line, including the Hausdorff measure on the Cantor set, using advanced equidistribution techniques in homogeneous dynamics.

Contribution

It introduces a Khintchine-type theorem for self-similar measures and proves effective equidistribution of certain random walks on SL(2,R)/SL(2,Z).

Findings

Established Khintchine's theorem analogue for self-similar measures.

Provided new results for Hausdorff measure on the Cantor set.

Connected number theory with homogeneous dynamics techniques.

Abstract

We establish the analogue of Khintchine's theorem for 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 $\text{SL}_{2}(\mathbb{R})/\text{SL}_{2}(\mathbb{Z})$, a result of independent interest.