Diffuse Behaviour of Ergodic Sums Over Rotations

Abstract

For a rotation by an irrational $\alpha$ on the circle and a BV function $\phi$, we study the variance of the ergodic sums $S_L\phi (x):={\sum }_{j=0}^{L-1}\phi (x+j\alpha )$. When $\alpha$ is not of constant type, we construct sequences $(L_N)$ such that, at some scale, the ergodic sums $S_{L_N}\phi$ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane.