On the ergodicity of Weyl sum cocycles

Abstract

AbstractWe present the quadratic Weyl sums$\sum _{k=0}^{n-1} e^{2\pi i(k^2\theta +2kx)}$withθ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrationalθ∈[0,1), and that for a denseGδset ofθ∈[0,1) the corresponding skew product is ergodic. For each of thoseθ, there exists a denseGδset of full measure ofx∈[0,1) for which the sequence$\sum _{k=0}^{n-1} e^{2\pi i(k^2\theta +2kx)}$,n=1,2,… , is dense in$\mathbb {C}$.