Approximating elements of the middle third Cantor set with dyadic rationals

Abstract

AbstractLet C be the middle third Cantor set and μ be the $$\frac{\log\;2}{\log\;3}$$ 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 $$\# \left\{{1 \le n \le N:\left| {x - {p \over {{2^n}}}} \right| \le {1 \over {{n^{0.01}} \cdot {2^n}}}\,{\rm{for}}\,{\rm{some}}\,p \in \mathbb{N}} \right\}\sim2\sum\limits_{n = 1}^N {{n^{- 0.01}}}.$$ # { 1 ≤ n ≤ N : ∣ x − p 2 n ∣ ≤ 1 n 0.01 ⋅ 2 n f o r s o m e p ∈ N } ∼ 2 ∑ 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.