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Abstract

Let ω ∈ ℝ╲ℚ and $f ∈ L^2(\mathbb S^1)$ of zero average. We study the asymptotic behaviour of the Weyl sums $S(m,\omega)f(x)=\sum_{k=0}^{m-1}f(x+k\omega)$ and their averages $\widehat S(m,\omega)f(x)=\frac1m\sum_{j=1}^{m}S(j,\omega)f(x)$, in the $L^2$-norm. In particular, for a suitable class of Liouville rotation numbers $\omega\in\mathbb R\setminus\mathbb Q$, we are able to construct examples of functions $f\in H^s(\mathbb S^1)$, $s>0$, such that, for all $\varepsilon>0$, $\|\widehat S(m,\omega)f\|_2\ge C_\varepsilon m^{1/(1+s)-\varepsilon}$ as $m\to\infty$. We show in addition that, for all $f\in H^s(\mathbb S^1)$, $\liminf m^{-1/(1+s)}(\log m)^{-1/2}\|\hat S(m,\omega)f\|_2<\infty$ for all $\omega\in\mathbb R\setminus\mathbb Q$.