Averaged Fourier Estimates and Dyadic Approximation on the Cantor set

Abstract

Let $C$ be the middle-third Cantor set and let $\hat{I}¼$ be the natural Cantor probability measure. Let \[ γ=\frac{\log2}{\log3}. \] The two main results of this paper are \[ μ\{x\in C:\|2^n x\|<n^{-τ}\text{ for infinitely many }n\}=0 \qquad \text{ for } τ>2-γ. \] and \[ μ\{x\in C:\|2^n x\|<n^{-τ}\text{ for infinitely many }n\}=1 \qquad \text{ for } τ<\frac{1-γ}{2}. \] These results give new progress toward Velani's conjecture on zero-one law for dyadic approximation in the middle-third Cantor set.