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.)

ANSWER. $$ \sum_{k=1}^{n} \text{Re}(\alpha^k)=\sum_{k=1}^{n}T_j(\cos\text{arg}\alpha) \asymp n, $$ where $T_j(t)=\cos\arccos jt$ is the $j$th Chebysh"ev polynomial. The lower bound comes from $\alpha$ being badly approximable by Liouville/Roth theorems.

Sqrt[2]],{k,10000}] = -0.939269... Sum[Cos[kSqrt[2]],{k,100000}] =-0.401409... Sum[Cos[k*Sqrt[2]],{k,1000000}] = 0.218388... It surely seems that these are not growing like n. (I'm aware that this is just one example, but try your favorite and see what happens!) $\endgroup$2more comments