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

Abstract

Let $K$ denote the middle third Cantor set and $\mathcal{A}:=\{3^n:n=0,1,2,>...\}$. Given a real, positive function $\psi$ let $W_{\mathcal{A}}(\psi )$ denote the set of real numbers $x$ in the unit interval for which there exist infinitely many $(p,q)\in \mathbb{Z}\times \mathcal{A}$ such that $|x-p/q|<\psi (q)$. 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.

20 pages