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Journal

Uniform Distribution Theory (UDT)


Volume & Issue

UDT. Volume 15. Issue 2


Published on

December, 2020


Pages

73-92


DOI

https://doi.org/10.2478/udt-2020-0011

Article

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


Authors

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


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.


Citation

Rahm, A.D.., Solomon, N., Trauthwein, T. & Weiss, B. (2020). The Distribution of Rational Numbers on Cantor’s Middle Thirds Set. Uniform distribution theory, 15(2), 73-92. https://doi.org/10.2478/udt-2020-0011