We exploit dynamical properties of diagonal actions to derive results in
Diophantine approximations. In particular, we prove that the continued fraction
expansion of almost any point on the middle third Cantor set (with respect to
the natural measure) contains all finite patterns (hence is well approximable).
Similarly, we show that for a variety of fractals in [0,1]^2, possessing some
symmetry, almost any point is not Dirichlet improvable (hence is well
approximable) and has property C (after Cassels). We then settle by similar
methods a conjecture of M. Boshernitzan saying that there are no irrational
numbers x in the unit interval such that the continued fraction expansions of
{nx mod1 : n is a natural number} are uniformly eventually bounded.