On a problem of K. Mahler: Diophantine approximation and Cantor sets

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Let K denote the middle third Cantor set and A:= {3ⁿ: n = 0,1,2, >... }. Given a real, positive function ψ let W A(ψ) denote the set of real numbers x in the unit interval for which there exist infinitely many (p,q) ∈ × A such that |x - p/q| < ψ(q). The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for W A(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.