A quantitative shrinking target result on Sturmian sequences for rotations

Abstract

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