Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^{n} f(T^k x)$.

I've done some googling and found statements like "generic smooth functions $f$ do not obey a CLT for irrational rotation", but was unable to find a definitive cohesive reference. The specific example of $f$ I'm interested in are of the form $\approx \log \mathrm{Re}z$.

Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^{n} f(T^k x)$.

I've done some googling and found statements like "generic smooth functions $f$ do not obey a CLT for irrational rotation", but was unable to find a definitive cohesive reference. The specific example of sums I'm interested in are of the form $\approx \sum_{k}\log \left\vert \frac{f(T^{2k}x)}{f(T^{2k+1}x)}\right\vert$ for some smooth $f$.