Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log\sum_{k=1}^n \Re R_\alpha^k z}{\log n}=\frac12$$ for all $z\in S^1$?

Birkhoff's ergodic theorem works for all $z\in S^1 $ since $R_\alpha$ is uniquely ergodic. Does the CLT?