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On the ergodicity of the Weyl sums cocycle. (English) Zbl 1153.11044

Summary: For \(\theta \in [0,1]\), we consider the map \(T_\theta: \mathbb T^2 \to \mathbb T^2\) given by \(T_\theta(x,y)=(x+\theta,y+2x+\theta)\). The skew product \(f_\theta: \mathbb T^2 \times \mathbb C \to \mathbb T^2 \times \mathbb C\) given by \(f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})\) generates the so-called Weyl sums cocycle \(a_\theta(x,n) = \sum_{k=0}^{n-1} e^{2\pi i(k^2\theta+kx)}\) since the \(n\)th iterate of \(f_\theta\) writes as \(f_\theta^n(x,y,z)=(T_\theta^n(x,y),z+e^{2\pi iy} a_\theta(2x,n))\). In this note, we improve the study developed by A. H. Forrest in [J. Lond. Math. Soc. (2) 54, No. 3, 440–452 (1996; Zbl 0865.11057), Colloq. Math. 84–85, Pt. 1, 125–145 (2000; Zbl 0978.11041)] around the density for \(x \in \mathbb T\) of the complex sequence \({\{a_\theta(x,n)\}}_{n\in \mathbb N}\), by proving the ergodicity of \(f_\theta\) for a class of numbers \(\theta\) that contains a residual set of positive Hausdorff dimension in \([0,1]\). The ergodicity of \(f_\theta\) implies the existence of a residual set of full Haar measure of \(x \in \mathbb T\) for which the sequence \({\{ a_\theta(x,n) \}}_{n \in \mathbb N}\) is dense.

MSC:

11K60 Diophantine approximation in probabilistic number theory
11L15 Weyl sums
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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