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? (I only need $\sum_{k=1}^n\Re( R_\alpha^k z)\asymp c_n(\alpha)\sqrt n$ with $c_n(\alpha)=o(\sqrt n)$ , not the full power of the CLT.)