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(Td + 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.