Approximating elements of the middle third Cantor set with dyadic rationals

Abstract

Let $C$ be the middle third Cantor set and $\hat{I}¼$ be the $\frac{\log 2}{\log 3}$-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 $\hat{I}¼$ almost every $x\in C$ we have $$\#\left\{1\le n\le N:\left|x-\frac{p}{2^n}\right|\le \frac{1}{n^{0.01}\cdot 2^n}\text{ for some }p\in \mathbb{N}\right\}\sim 2\sum _{n=1}^N{n}^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.