References of "Trauthwein, Tara 50020387"
     in
Bookmark and Share    
Full Text
Peer Reviewed
See detailThe distribution of rational numbers on Cantor's middle thirds set
Trauthwein, Tara UL; Rahm, Alexander; Solomon, Noam et al

in Uniform distribution theory (2020), 15(2), 73-92

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 ... [more ▼]

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. [less ▲]

Detailed reference viewed: 110 (13 UL)