On a conjecture of Velani

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. ${\sum }_k\mathrm{\mid }\widehat{\mu }(2^k){\mathrm{\mid }}^{\sigma }<\mathrm{\infty }$ for any $\sigma >0$, where $\mu$ is the Cantor measure.