On a conjecture of Velani

Abstract

We investigate the approximation of elements of the middle-third Cantor set by dyadic rationals, and natural extensions. We prove a conjecture of Velani, completing the metric theory for the Cantor measure. Our key method enables a quantitative study of the incommensurability of binary and ternary expansions of integers, initiated by Furstenberg. The method has other applications, e.g. ${\sum }_k|\hat{\mu }(2^k){|}^{\sigma }<\infty$ for any $\sigma >0$, where $\mu$ is the Cantor measure.