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

Abstract : Abstract We give a heuristic argument predicting that the number N ∗ ( T ) of rationals p/q on Cantor’s middle thirds set C such that gcd( p, q )=1 and q ≤ T , has asymptotic growth O ( T d + ε ), for d = dim C . 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 C is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of N ∗ ( T )is motivated by a problem of Mahler on intrinsic Diophantine approximation on C .