[hal-00701757, v1] Subdiffusive behavior generated by irrational rotations

We study asymptotic distributions of the sums yn(x) = ∑n−1 k=0 ψ(x + kα) with respect to the Lebesgue measure, where α ∈ R−Q and where ψ is the 1-periodic function of bounded variation such that ψ(x) = 1 if x ∈ [0, 1/2[ and ψ(x) = −1 if x ∈ [1/2, 1[. For every α ∈ R − Q, we find a sequence (nj)j ⊂ N such that ynj/ √ j is asymptotically normally distributed. For n ≥ 1, let zn ∈ (ym)m≤n be such that ||zn||L2 = maxm≤n ||ym||L2 . If α is of constant type, we show that zn/||zn||L2 is also asymptotically normally distributed. We give an heuristic link with the theory of expanding maps of the interval.