Approximating elements of the middle third Cantor set with dyadic rationals

Let C be the middle third Cantor set and μ be the \(\frac{\log\;2}{\log\;3}\) $\frac{\log 2}{\log 3}$ -dimensional Hausdorff measure restricted to C. In this paper we study approximations of elements of C by dyadic rationals. Our main result implies that for μ almost every x ∈ C we have \(\# \left\{{1 \le n \le N:\left| {x - {p \over {{2^n}}}} \right| \le {1 \over {{n^{0.01}} \cdot {2^n}}}\,{\rm{for}}\,{\rm{some}}\,p \in \mathbb{N}} \right\}\sim2\sum\limits_{n = 1}^N {{n^{- 0.01}}}.\) $$\mathrm{\#}\left\{1\le n\le N:\mid x-\frac{p}{2^n}\mid \le \frac{1}{n^{0.01}\cdot 2^n}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}p\in \mathbb{N}\right\}\sim 2\sum _{n=1}^N{n}^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.