Subdiffusive behavior generated by irrational rotations

Abstract

AbstractWe study asymptotic distributions of the sums yn(x)=∑ k=0n−1ψ(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 $y_{n_j}/\sqrt j$ is asymptotically normally distributed. For n≥1, let zn∈(ym)m≤n be such that ‖zn‖L2=max m≤n‖ym‖L2. If α is of constant type, we show that zn/‖zn‖L2 is also asymptotically normally distributed. We give a heuristic link with the theory of expanding maps of the interval.