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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 constanttype, 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.

Diffuse Behaviour of Ergodic Sums Over Rotations