Résumé en anglais

We study asymptotic distributions of the sums yn(x)=∑ k=0n−1ψ(x+kα) with respect to the Lebesgue measure, where αxs2208xs211D−xs211A and where ψ is the 1-periodic function of bounded variation such that ψ(x)=1 if xxs2208[0,1/2[ and ψ(x)=−1 if xxs2208[1/2,1[. For every αxs2208xs211D−xs211A, we find a sequence (nj)jxs2282xs2115 such that $y_{n_j}/\sqrt j$ is asymptotically normally distributed. For n≥1, let znxs2208(ym)m≤n be such that xs2016znxs2016L2=max m≤nxs2016ymxs2016L2. If α is of constant type, we show that zn/xs2016znxs2016L2 is also asymptotically normally distributed. We give a heuristic link with the theory of expanding maps of the interval.