Diffuse Behaviour of Ergodic Sums Over Rotations

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