A quantitative shrinking target result on Sturmian sequences for rotations

Let $R_{\alpha }$ be an irrational rotation of the circle, and code the orbit of any point $x$ by whether $R_{\alpha }^i(x)$ belongs to $[0,\alpha )$ or $[\alpha ,1)$ – this produces a Sturmian sequence. A point is undetermined at step $j$ if its coding up to time $j$ does not determine its coding at time $j+1$. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of $\alpha$ and $x$.