On the ergodicity of the Weyl sums cocycle

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