A rigorous proof of Khintchine’s theorem in fractal geometry.

By combining homogeneous dynamical systems, fractal measure theory, andergodic methods, this paper rigorously proves a version of Khintchine’s theorem forone-dimensional self-similar fractals satisfying the open set condition (OSC). Usinga fractal version of the Borel-Cantelli lemma and the mass distribution principle,the relationship between Diophantine approximation on fractal sets and changesin Hausdorff dimension is explicitly demonstrated. Numerical experiments usingthe Cantor set validate the theoretical results, while potential extensions to higher?dimensional and random fractals are explored.