Approximating elements of the middle third Cantor set with dyadic rationals

Let C be the middle third Cantor set and μ be the log2log3-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

#{1≤n≤N:|x−p2n|≤1n0.01⋅2n for some p∈N}∼2∑n=1Nn−0.01.

This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.