Mathematical Notes

, Volume 78, Issue 3–4, pp 329–337 | Cite as

Weighted Means, Strict Ergodicity, and Uniform Distributions

  • V. V. Kozlov


We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients.

Key words

Weyl theorem Cesaro and Voronoi convergences Borel measure Oxtoby's theorem strictly ergodic transformation Riesz summation method 


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  1. 1.
    G. Hardy, Divergent Series, Oxford, 1949.Google Scholar
  2. 2.
    D. L. Hanson and G. Pledger, “On the mean ergodic theorem for weighted averages,” Z. Wahrscheinlichkeitstheorie Verw. Geb., 13 (1969), no. 13, 141–149.CrossRefGoogle Scholar
  3. 3.
    V. V. Kozlov, “Summation of divergent series and ergodic theorems,” Trudy Sem. Petrovsk., 22 (2002), 142–168.Google Scholar
  4. 4.
    G. Baxter, “An ergodic theorem with weighted averages,” J. Math. Mech., 13 (1964), no. 3, 481–488.Google Scholar
  5. 5.
    R. V. Chacon, “Ordinary means imply recurrent means,” Bull. Amer. Math. Soc., 70 (1964), 796–797.Google Scholar
  6. 6.
    J. C. Oxtoby, “Ergodic sets,” Bull. Amer. Math. Soc., 58 (1952), no. 2, 116–136.Google Scholar
  7. 7.
    H. Furstenberg, “Strict ergodicity and transformation of the torus,” Amer. J. Math., 83 (1961), no. 4, 573–601.Google Scholar
  8. 8.
    L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974.Google Scholar
  9. 9.
    V. V. Kozlov and T. Madsen, “The Poincare rotation numbers and the Riesz and Voronoi means,” Mat. Zametki [Math. Notes], 74 (2003), no. 2, 314–315.Google Scholar
  10. 10.
    I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow, 1980.Google Scholar
  11. 11.
    H. Weyl, “Uber die Gleichverteilung von Zahlen mod Eins,” Math. Ann., 77 (1915/16), 313–352.CrossRefMathSciNetGoogle Scholar
  12. 12.
    V. V. Kozlov, “On uniform distributions on the torus,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (2004), no. 2, 22–29.Google Scholar
  13. 13.
    J. Cigler, “Methods of summability and uniform distribution mod 1,” Compos. Math., 16 (1964), 44–51.Google Scholar
  14. 14.
    A. F. Doroidar, “A note on the generalized uniform distribution (mod 1),” J. Natur. Sci. Math., 11 (1971), 185–189.Google Scholar
  15. 15.
    P. Bohl, “Uber eine Differentialgleichung der Storungstheorie,” J. Reine Angew. Math., 131 (1906), no. 4, 268–321.Google Scholar
  16. 16.
    V. V. Kozlov, “On integrals of quasiperiodic functions,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1978), no. 1, 106–115.Google Scholar
  17. 17.
    G. Halasz, “Remarks on the remainder in Birkhoff's ergodic theorem,” Acta Math. Acad. Sci. Hungar., 28 (1976), nos. 3–4, 289–395.Google Scholar
  18. 18.
    A. A. Sorokin, “On oscillations of Riesz and Voronoi means,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (2005), no. 2, 13–17.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. V. Kozlov
    • 1
  1. 1.V. A. Steklov Mathematics InstituteRussian Academy of SciencesMoscowRussia

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