Subdiffusive behavior generated by irrational rotations

We study asymptotic distributions of the sums y(n)(x) = Sigma(n-1)(k=0) psi(x + k alpha) with respect to the Lebesgue measure, where alpha is an element of R - Q and where psi is the 1-periodic function of bounded variation such that psi(x) = 1 if x is an element of [0, 1/2[ and psi(x) = -1 if x is an element of [1 /2, 1[. For every alpha is an element of R - Q, we find a sequence (n(j))(j) subset of N such that y(nj)/root j is asymptotically normally distributed. For n >= 1, let z(n) is an element of (y(m))(m <= n) be such that parallel to z(n)parallel to(L2) = max(m <= n) parallel to y(m)parallel to(L2). If a is of constant type, we show that z(n)/parallel to z(n)parallel to(L2) is also asymptotically normally distributed. We give a heuristic link with the theory of expanding maps of the interval.