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Pré-Publication, Document De Travail Année : 2010

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

Résumé

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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Dates et versions

hal-02550072 , version 1 (21-04-2020)

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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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