  Following K. Mahler's suggestion for further research on intrinsic
approximation on the Cantor ternary set, we obtain a Dirichlet type theorem for
the limit sets of rational iterated function systems. We further investigate
the behavior of these approximation functions under random translations. We
connect the information regarding the distribution of rationals on the limit
set encoded in the system to the distribution of rationals in reduced form by
proving a Khinchin type theorem. Finally, using a result of S. Ramanujan, we
prove a theorem motivating a conjecture regarding the distribution of rationals
in reduced form on the Cantor ternary set.
