On the ergodicity of the Weyl sums cocycle

Abstract

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 $f_{\theta }(x,y,z)=(T_{\theta }(x,y),z+e^{2\pi iy})$ generates the so called Weyl sums cocycle $a_\a(x,n) = \sum_{k=0}^{n-1} e^{2πi(k^2θ+kx)}$ since the $n^{\mathrm{th}}$ iterate of $f_\a$ writes as $f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2πiy} a_\a(2x,n))$. 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.