Central limit theorem for irrational rotations

Question

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.

Answer 1

I am surprised by the question. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$, $$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big| = \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \Big| \Re \Big(\frac{\alpha-\alpha^{n+1}}{1-\alpha}z\Big) \Big| \le \Big|\frac{\alpha-\alpha^{n+1}}{1-\alpha}z\Big| \le \frac{2}{|1-\alpha|}.$$ The log of this quantity is bounded above, so it can not increase like $(1/2)\log(n)$. But it is not bounded below, it can takes large negative values for the integers $n$ such that $\alpha^n$ is close to $1$.

ADDENDUM : one can check that $$\limsup_{n \to +\infty} \log\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big| \times \frac{1}{\log(n)} = 0,$$ whereas $$\liminf_{n \to +\infty} \log\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big| \times \frac{1}{\log(n)} \le -1.$$ Indeed, setting $\alpha = e^{i\theta}$ with $\theta \in \mathbb{R}$, Dirichlet principle ensures that $|\alpha^n-1| = |e^{in\theta}-1| \le \mathrm{dist}(n\theta,\mathbb{Z}) < 1/n$ for infinitely many $n$. Actually, the liminf above is $-1$ for almost every $\alpha$.