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

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 ( 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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Submitted on : Friday, November 5, 2021 - 11:09:47 PM
Last modification on : Monday, November 15, 2021 - 7:30:02 PM

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Alexander Rahm, Noam Solomon, Tara Trauthwein, Barak Weiss. The Distribution of Rational Numbers on Cantor’s Middle Thirds Set. Uniform Distribution Theory, Mathematical Institute of the Slovak Academy of Sciences, 2020, 15 (2), pp.73-92. ⟨10.2478/udt-2020-0011⟩. ⟨hal-03417923⟩

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