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The distribution of rational numbers on Cantor's middle thirds set

Abstract : We give a heuristic argument predicting that the number N * (T) of ra-tionals 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. 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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Submitted on : Tuesday, April 21, 2020 - 10:51:48 PM
Last modification on : Monday, November 15, 2021 - 7:30:02 PM


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  • HAL Id : hal-02550072, version 1
  • ARXIV : 1909.01198



Alexander D. Rahm, Noam Solomon, Tara Trauthwein, Barak Weiss. The distribution of rational numbers on Cantor's middle thirds set. 2010. ⟨hal-02550072⟩



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