Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

Let $C$ be the middle-third Cantor set and let $\mu$ be the natural Cantor probability measure.

Let \[ \gamma=\frac{\log2}{\log3}. \] The two main results of this paper are \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=0 \qquad \text{ for } \tau>2-\gamma. \] and \[ \mu\{x\in C:\|2^n x\|<n^{-\tau}\text{ for infinitely many }n\}=1 \qquad \text{ for } \tau<\frac{1-\gamma}{2}. \] These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.