  We establish various new results on a problem proposed by K. Mahler in 1984
concerning rational approximation to fractal sets by rational numbers inside
and outside the set in question, respectively. Some of them provide a natural
continuation and improvement of recent results of Broderick, Fishman and Reich
and Fishman and Simmons. A key feature is that many of our new results apply to
more general, multi-dimensional fractal sets and require only mild assumptions
on the iterated function system. Moreover we provide a non-trivial lower bound
for the distance of a rational number $p/q$ outside the Cantor middle third set
$C$ to the set $C$, in terms of the denominator $q$. We further discuss
patterns of rational numbers in fractal sets. We want to highlight two of them:
Firstly, an upper bound for the number of rational (algebraic) numbers in a
fractal set up to a given height (and degree) for a wide class of fractal sets.
Secondly we find properties of the denominator structure of rational points in
''missing digit'' Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

Number Theory