Some counterexamples to the central limit theorem for random rotations

Abstract

Fix an irrational number $\alpha$, and consider a random walk on the circle in which at each step one moves to $x+\alpha$ or $x-\alpha$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of $\alpha$ implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle such that the central limit theorem fails with some analytic observable. For Diophantine angles some counterexample is given as well. An interesting question remained open.