Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T22:32:44.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak mixing implies weak mixing of higher orders along tempered functions

Published online by Cambridge University Press:  26 February 2009

VITALY BERGELSON  and
INGER J. HÅLAND KNUTSON
VITALY BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA (email: vitaly@math.ohio-state.edu)
INGER J. HÅLAND KNUTSON
Affiliation:
Department of Mathematical Sciences, University of Agder, N-4604 Kristiansand, Norway (email: inger.j.knutson@uia.no)
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the weakly mixing PET (polynomial ergodic theorem) obtained in Bergelson [Weakly mixing PET. Ergod. Th. & Dynam. Sys.7 (1987), 337–349] to much wider families of functions. Besides throwing new light on the question of ‘how much higher-degree mixing is hidden in weak mixing’, the obtained results also show the way to possible new extensions of the polynomial Szemerédi theorem obtained in Bergelson and Leibman [Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc.9 (1996), 725–753].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1375 - 1416
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Berend, D. and Bergelson, V.. Mixing sequences in Hilbert spaces. Proc. Amer. Math. Soc. 98 (1986), 239–246.CrossRefGoogle Scholar
[2]Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7 (1987), 337–349.CrossRefGoogle Scholar
[3]Bergelson, V.. The multifarious Poincaré recurrence theorem. Descriptive Set Theory and Dynamical Systems (London Mathematical Society Lecture Note Series, 277). Eds. M. Foreman, A. Kechris, A. Louveau and B. Weiss. Cambridge University Press, Cambridge, 2000, pp. 31–57.Google Scholar
[4]Bergelson, V.. Combinatorial and Diophantine applications of ergodic theory. Appendix A by A. Leibman and Appendix B by Anthony Quas and Máté Wierdl (Handbook of Dynamical Systems, Vol. 1B). Eds. B. Hasselblatt and A. Katok. Elsevier B. V, Amsterdam, 2006, pp. 745–869.Google Scholar
[5]Bergelson, V., Boshernitzan, M. and Bourgain, J.. Some results on nonlinear recurrence. J. Anal. Math. 62 (1994), 29–46.CrossRefGoogle Scholar
[6]Bergelson, V. and Leibman, A.. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc. 9 (1996), 725–753.CrossRefGoogle Scholar
[7]Bergelson, V. and McCutcheon, R.. An ergodic IP polynomial Szemerédi theorem. Mem. Amer. Math. Soc. 146(695) (2000), viii+106 pp.Google Scholar
[8]Boos, J.. Classical and Modern Methods in Summability. Oxford University Press, Oxford, 2000.CrossRefGoogle Scholar
[9]Boshernitzan, M., Kolesnik, G., Quas, A. and Wierdl, M.. Ergodic averaging sequences. J. D’Analyse Math. 95 (2005), 63–103.CrossRefGoogle Scholar
[10]Bourbaki, N.. Fonctions d’une variable réele. Étude Locale des Fonctions, 2nd edn. Herman, Paris, 1961, Ch. 5.Google Scholar
[11]Cigler, J.. Some remarks on the distribution mod 1 of tempered sequences. Nieuw Arch. Wisk. (3) 16 (1968), 194–196.Google Scholar
[12]Frantzikinakis, N. and Wierdl, M.. A Hardy field extension of Szemerédi’s theorem. Preprint. Available at http://arxiv.org/abs/0802.2734.Google Scholar
[13]Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. d’Analyse Math. 31 (1977), 204–256.CrossRefGoogle Scholar
[14]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.Google Scholar
[15]Hardy, G. H.. Orders of Infinity: The ‘Infinitärcalcül’ of Paul du Bois-Reymond (Cambridge Tracts in Mathematics and Mathematical Physics, 12). Hafner, New York, 1971.Google Scholar
[16]Hardy, G. H.. Divergent Series. Chelsea House, New York, 1991.Google Scholar
[17]Hlawka, E.. The Theory of Uniform Distribution. AB Academic Publishers, Berkhamsted, UK, 1984.Google Scholar
[18]Jones, L. K. and Lin, M.. Ergodic theorems for weak mixing type. Proc. Amer. Math. Soc. 57 (1976), 50–52.Google Scholar
[19]Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences. Dover Publications, Inc., New York, 2006.Google Scholar
[20]Lorentz, G. G.. A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167–190.CrossRefGoogle Scholar
[21]van der Corput, J. G.. Diophantische Ungleichungen I. Zur Gleichverteilung modulo Eins. Acta Math. 56 (1931), 373–456.CrossRefGoogle Scholar