Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T21:54:25.371Z Has data issue: false hasContentIssue false
  • English
  • Français

On the ergodicity of Weyl sum cocycles

Published online by Cambridge University Press:  01 December 2007

GERNOT GRESCHONIG,
MAHESH NERURKAR and
DALIBOR VOLNÝ
GERNOT GRESCHONIG
Affiliation:
Faculty of Mathematics, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria (email: gernot.greschonig@univie.ac.at)
MAHESH NERURKAR
Affiliation:
Department of Mathematical Sciences, Rutgers University – Camden, Armitage Hall, 311 North 5th Street, Camden, NJ 08102, USA (email: nmahesh@crab.rutgers.edu)
DALIBOR VOLNÝ
Affiliation:
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Avenue de l’Université, F-76801 Saint Etienne du Rouvray, France (email: dalibor.volny@univ-rouen.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We present the quadratic Weyl sums $\sum _{k=0}^{n-1} e^{2\pi i(k^2\theta +2kx)}$ with θ,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 irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[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 $\mathbb {C}$.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007, pp. 1851 - 1863
Copyright
Copyright © Cambridge University Press 2007

References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Aaronson, J., Lemańczyk, M. and Volný, D.. A cut salad of cocycles. Fund. Math. 157(2–3) (1998), 99–119.CrossRefGoogle Scholar
[3]Fayad, B.. On the ergodicity of the Weyl sums cocycle. Acta Arith. 125(4) (2006), 305–316.CrossRefGoogle Scholar
[4]Forrest, A. H.. The limit points of Weyl sums and other continuous cocycles. J. London Math. Soc. (2) 54(3) (1996), 440–452.CrossRefGoogle Scholar
[5]Forrest, A. H.. Symmetric cocycles and classical exponential sums. Colloq. Math. 84/85(1) (2000), 125–145.CrossRefGoogle Scholar
[6]Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
[7]Hardy, G. H. and Littlewood, J. E.. The trigonometric series associated with the elliptic θ-functions. Acta Math. 37 (1914), 193–239.CrossRefGoogle Scholar
[8]Schmidt, K.. Cocycles on Ergodic Transformation Groups (Macmillan Lectures in Mathematics, 1). Macmillan Company of India, Ltd., Delhi, 1977.Google Scholar
[9]Volný, D.. Completely squashable smooth ergodic cocycles over irrational rotations. Topol. Methods Nonlinear Anal. 22(2) (2003), 331–344.CrossRefGoogle Scholar
[10]Volný, D. and Nerurkar, M.. On the ergodicity of Weyl sum cocycles. Preprint.Google Scholar