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 $f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})$ 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