Reference : The distribution of rational numbers on Cantor's middle thirds set
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
The distribution of rational numbers on Cantor's middle thirds set
Trauthwein, Tara mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
Rahm, Alexander mailto [Université de la Polynésie Française > Laboratoire de mathématiques GAATI]
Solomon, Noam mailto [Immunai > 180 Varick St, 6th FI, New York]
Weiss, Barak mailto [Tel Aviv University > Department of Mathematics]
Uniform distribution theory
[en] Cantor's middle third set ; Diophantine approximation
[en] 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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