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

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.

Rational numbers in $\times b$-invariant sets

“…Later, Nagy [6] showed that, if S = {p} for some prime p > 3, then C contains only finitely many S-integers. Recently, based on a heuristic argument as well as numerical evidence, Rahm, Solomon, Trauthwein and Weiss [7] formulated an asymptotic for the number N * (T ) of reduced rational numbers in C with denominators bounded by T .…”