On a conjecture of Velani

Source 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. $\sum_{k} |\hatμ(2^k)|^σ 0$, where $μ$ is the Cantor measure.