A note on dyadic approximation in Cantor's set

Abstract

We consider the convergence theory for dyadic approximation in the middlethird Cantor set, K, for approximation functions of the form ψτ(n) = n−τ (τ ⩾ 0). In particular, we show that for values of τ beyond a certain threshold we have that almost no point in K is dyadically ψτ-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.