Subdiffusive behavior generated by irrational rotations

The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums $y_n(x)=\sum_{k=0}^{n-1}ψ(x+kα)$, where $ψ$ is a periodic function of bounded variation and $α$ an irrational number. It is known that no diffusion process will be observed. Nevertheless, we find a picewise constant function $ψ$ and an increasing sequence of integer $(n_j)_j$ such that the limit distribution of the sequence $(y_{n_j}/\sqrt j)_j$ is Gaussian (with stricly positive variance). If $α$ is of constant type, we show that the sequence $(n_j)_j$ may be taken to grow exponentially (this is close to optimal in some sense, and one has $||y_{n_j}||_{\mathrm L^2}\sim \max_{0\le k\le n_j}||y_k||_{\mathrm L^2}$ as $j\to\infty$). We give an heuristic link with the theory of expanding maps of the interval.

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