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

Let $K$ denote the middle third Cantor set and ${\cal A}:= {3^n : n = 0,1,2,

… } $\mathrm{.}Givenareal,positivefunction$\psi$let$ W_{\cal A}(\psi)$denotethesetofrealnumbers$x$ in the unit interval for which there exist infinitely many $(p,q)\in \mathbb{Z}\times \mathcal{A}$ such that $\mathrm{\mid }x-p\mathrm{/}q\mathrm{\mid }<\psi (q)$. The analogue of the Hausdorff measure version of the Duffin-Schaeffer conjecture is established for $W_{\mathcal{A}}(\psi )\cap K$. 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.