On the ergodicity of the Weyl sums cocycle

For $\theta \in [0,1]$, we consider the map $T_\a: \T^2 \to \T^2$ given by $T_\theta(x,y)=(x+\theta,y+2x+\theta)$. The skew product $f_\a: \T^2 \times \C \to \T^2 \times \C$ given by $f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})$ generates the so called Weyl sums cocycle $a_\a(x,n) = \sum_{k=0}^{n-1} e^{2\pi i(k^2\theta+kx)}$ since the $n^{{\rm th}}$ iterate of $f_\a$ writes as $f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2\pi iy} a_\a(2x,n))$. In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for $x \in \T$ of the complex sequence ${\{a_\a(x,n)\}}_{n\in \N}$, by proving the ergodicity of $f_\theta$ for a class of numbers $\a$ that contains a residual set of positive Hausdorff dimension in $[0,1]$. The ergodicity of $f_\a$ implies the existence of a residual set of full Haar measure of $x \in \T$ for which the sequence ${\{a_\a(x,n) \}}_{n \in \N}$ is dense.