The Distribution of Rational Numbers on Cantor’s Middle Thirds Set-Reference-Cited by-同舟云学术

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

Published:2020-12-01 Issue:2 Volume:15 Page:73-92
ISSN:2309-5377
Container-title:Uniform distribution theory
language:en
Short-container-title:

Author:

Rahm Alexander D.¹,Solomon Noam²,Trauthwein Tara³,Weiss Barak⁴

Affiliation:

1. University of French Polynesia , GAATI Laboratory , Faa’a , French Polynesia

2. Massachusetts Institute of Technology , Cambridge , MA, U.S.A

3. University of Luxembourg , Department of Mathematics , Esch-sur-Alzette , Luxembourg

4. Tel Aviv University , Department of Mathematics , Tel-Aviv , Israel

Abstract

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.

Publisher

Walter de Gruyter GmbH

Link

https://www.sciendo.com/pdf/10.2478/udt-2020-0011

Reference14 articles.

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