<abstract xmlns="http://www.w3.org/1999/xhtml">We give a heuristic argument predicting that the number <italic>N∗</italic>(<italic>T</italic>) of rationals <italic>p/q</italic> on Cantor’s middle thirds set <italic>C</italic> such that gcd(<italic>p, q</italic>)=1 and <italic>q ≤ T</italic>, has asymptotic growth <italic>O</italic>(<italic>Td</italic>+<italic>ε</italic>), for <italic>d</italic> = dim <italic>C</italic>. 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 <italic>C</italic> is replaced with any similar fractal with a description in terms of missing digits in a base expansion. Interest in the growth of <italic>N∗</italic> (<italic>T</italic>)is motivated by a problem of Mahler on intrinsic Diophantine approximation on <italic>C</italic>.</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.

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

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"The Distribution of Rational Numbers on Cantor\u2019s Middle Thirds Set"}

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...

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

Published Online: Dec 25, 2020

Page range: 73 - 92

Received: May 23, 2020

Accepted: Sep 27, 2020

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

Keywords
Rational numbers in the Cantor set

© 2020 Alexander D. Rahm et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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

Published Online: Dec 25, 2020

Page range: 73 - 92

Received: May 23, 2020

Accepted: Sep 27, 2020

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

KeywordsRational numbers in the Cantor set

© 2020 Alexander D. Rahm et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Rational numbers in the Cantor set