A note on dyadic approximation in Cantor's set

Abstract

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, $K$, for approximation functions of the form $\ddot{I}{ˆ}_{\ddot{I}}„(n)=n^{-\ddot{I}„}$ ($\ddot{I}„\ge 0$). In particular, we show that for values of $\ddot{I}„$ beyond a certain threshold we have that almost no point in $K$ is dyadically $\ddot{I}{ˆ}_{\ddot{I}}„$-well approximable with respect to the natural probability measure on $K$. This refines a previous result in this direction obtained by the first, third, and fourth named authors (arXiv, 2020).