The Distribution of Rational Numbers on Cantor’s Middle Thirds Set

We give a heuristic argument predicting that the number ) of rationals on Cantor’s middle thirds set such that gcd( )=1 and , has asymptotic growth ), for = dim . Our heuristic is related to similar heuristics and conjectures proposed by Fishman and Simmons. We also describe extensive numerical computations supporting this heuristic. Our heuristic predicts a similar asymptotic if is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of ( )is motivated by a problem of Mahler on intrinsic Diophantine approximation on