On the growth of averaged Weyl sums for rigid rotations

Let ω ∈ ℝ╲ℚ and ${\displaystyle f\in L^2(^1)}$ of zero average. We study the asymptotic behaviour of the Weyl sums ${\displaystyle S(m,\omega )f\left(x\right)={\sum }^{m-1}\_\{k=0\}f(x+k\omega )}$ and their averages ${\displaystyle Ŝ(m,\omega )f\left(x\right)=\left\{\frac{1}{m}\right\}{\sum }^m\_\{j=1\}S(j,\omega )f\left(x\right)}$, in the ${\displaystyle L^2}$-norm. In particular, for a suitable class of Liouville rotation numbers ω ∈ ℝ╲ℚ, we are able to construct examples of functions ${\displaystyle f\in H^s^\left\{1\right\}}$, s > 0, such that, for all ε > 0, ||Ŝ(m,ω)f||_2 ≥ C_{ε}m^{1/(1+s)-ε}${\displaystyle asm\to \infty .Weshow\in additiont\widehat{,}f\text{or}all}$f ∈ H^{s}^{1}${\displaystyle ,}$lim inf m^{-1/(1+s)} (log m)^{-1/2} ||Ŝ(m,ω)f||_2 < ∞!$! for all ω ∈ ℝ╲ℚ.