<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

Online veröffentlicht: 25. Dez. 2020

Seitenbereich: 73 - 92

Eingereicht: 23. Mai 2020

Akzeptiert: 27. Sept. 2020

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

Schlüsselwörter
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

Online veröffentlicht: 25. Dez. 2020

Seitenbereich: 73 - 92

Eingereicht: 23. Mai 2020

Akzeptiert: 27. Sept. 2020

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

SchlüsselwörterRational 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.

Schlüsselwörter
Rational numbers in the Cantor set