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

Rahm, Alexander D.; Solomon, Noam; Trauthwein, Tara; Weiss, Barak

doi:10.2478/udt-2020-0011

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.

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

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