Given an approximation functionψ, the classical Khintchine's theorem gives a dichotomy for the Lebesgue measure ofψ-approximable real vectors. In 1959, Schmidt proved a quantitative form of Khintchine's theorem, obtained the asymptotic number of solutions for the Diophantine inequality of typical real vectors.

In a recent joint work with Benard and He, we extend the classical Khintchine's theorem to self-similar measures on the real line, and obtain a quantitative version of Khintchine's theorem for those measures. The proof is based on dynamics on homogeneous spaces and certain effective double equidistribution theorem.