# [math/0505074] On a problem of K. Mahler: Diophantine approximation and Cantor sets

Let K denote the middle third Cantor set and A:={3n:n=0,1,2,>…}. Given a real, positive function ψ let WA(ψ) denote the set of real numbers x in the unit interval for which there exist infinitely many $(p,q) \in \Z \times {\cal A}$ such that |x−p/q|<ψ(q). The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for WA(ψ)∩K. One of the consequences of this is that there exist very well approximable numbers, other than Liouville numbers, in K — an assertion attributed to K. Mahler.