On the CLT for rotations and BV functions

TL;DR

This paper proves a Central Limit Theorem for ergodic sums of step functions under circle rotations with Diophantine conditions, showing asymptotic Gaussian behavior for normalized sums on a set of density 1.

Contribution

It establishes a CLT for ergodic sums of step functions under certain Diophantine conditions on the rotation number, extending previous results to a broader class of functions and rotation parameters.

Findings

Asymptotic Gaussian distribution of normalized ergodic sums

Validity of CLT under bounded partial quotients for lpha

Method based on decorrelation inequalities at denominators q_k

Abstract

Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi\_n (x)$ the corresponding ergodic sums $\sum\_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example when $\alpha$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $\varphi$, we show that $\varphi\_n/\|\varphi\_n\|\_2$ is asymptotically Gaussian for $n$ in a set of density 1. The method is based on decorrelation inequalities for the ergodic sums taken at times $q\_k$, where the $q\_k$'s are the denominators of $\alpha$.