Approximating elements of the middle third Cantor set with dyadic rationals

TL;DR

This paper investigates how well elements of the middle third Cantor set can be approximated by dyadic rationals, showing that for almost every point, the number of good approximations grows proportionally to a specific sum, improving previous results.

Contribution

The paper provides a new almost sure approximation rate for elements of the Cantor set by dyadic rationals, surpassing prior sub-logarithmic bounds.

Findings

Almost every point in the Cantor set has approximations growing as 2 times the sum of n^{-0.01}.

Improves upon previous sub-logarithmic approximation bounds.

Establishes a precise asymptotic for the count of good dyadic rational approximations.

Abstract

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