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

Alexander D. Rahm (1) , Noam Solomon , Tara Trauthwein , Barak Weiss

1 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

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 .

Domains: Mathematics [math]; Algebraic Geometry [math.AG]

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