On the growth of averaged Weyl sums for rigid rotations

Let ω ∈ ℝ╲ℚ and $f ∈ L^2(^1)$ of zero average. We study the asymptotic behaviour of the Weyl sums $S(m,ω)f(x) = ∑^{m-1}_{k=0} f(x + kω)$ and their averages $Ŝ(m,ω)f(x) = {1/m}∑^m_{j=1} S(j,ω)f(x)$, in the $L^2$-norm. In particular, for a suitable class of Liouville rotation numbers ω ∈ ℝ╲ℚ, we are able to construct examples of functions $f ∈ H^{s}^{1}$, s > 0, such that, for all ε > 0, ||Ŝ(m,ω)f||_2 ≥ C_{ε}m^{1/(1+s)-ε}$ as m → ∞. We show in addition that, for all $f ∈ H^{s}^{1}$, $lim inf m^{-1/(1+s)} (log m)^{-1/2} ||Ŝ(m,ω)f||_2 < ∞$ for all ω ∈ ℝ╲ℚ.