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Let $C$ be the middle third Cantor set and $\mu$ be the $\frac{\log 2}{\log3}$-dimensional Hausdorff measure restricted to $C$. In this paper we studyapproximations of elements of $C$ by dyadic rationals. Our main result impliesthat for $\mu$ almost every $x\in C$ we have $$\#\left\{1\leq n\leqN:\left|x-\frac{p}{2^n}\right| \leq \frac{1}{n^{0.01}\cdot 2^{n}}\textrm{ forsome }p\in\mathbb{N}\right\}\sim 2\sum_{n=1}^{N}n^{-0.01}.$$ This improves upona recent result of Allen, Chow, and Yu which gives a sub-logarithmicimprovement over the trivial approximation rate.

Approximating elements of the middle third Cantor set with dyadic rationals