Random walks, spectral gaps, and Khintchine's theorem on fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle 1/3 set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of R-d (for any d >= 1) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle 1/5 Cantor set"; i.e. the set of numbers whose base 5 expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space Ld+1 of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of S-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on Ld+1.