  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 $\mathbb{R}^d$
(for any $d\geq 1$) 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 $\mathcal{L}_{d+1}$ 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 $\mathcal{L}_{d+1}$.
