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 $$\gamma =\frac{\log 2}{\log 3}\mathrm{.}$$ The two main results of this paper are $$\mu \{x\in C:\mathrm{\parallel }{2}^{n}x\mathrm{\parallel }<n^{-\tau }\text{ for infinitely many }n\}=0\text{ for }\tau >2-\gamma \mathrm{.}$$ and $$\mu \{x\in C:\mathrm{\parallel }{2}^{n}x\mathrm{\parallel }<n^{-\tau }\text{ for infinitely many }n\}=1\text{ for }\tau <\frac{1-\gamma }{2}\mathrm{.}$$ These results give new progress toward Velani’s conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.