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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 √ j is asymptotically normally distributed. For n ≥ 1, let zn ∈ (ym) m≤n be such that ||zn|| L 2 = max m≤n ||ym|| L 2 . If α is of constant type, we show that zn/||zn|| L 2 is also asymptotically normally distributed. We give an heuristic link with the theory of expanding maps of the interval.doi:10.1017/s0143385708000680 fatcat:venfbrz3knhq5gugvtjab66ydu