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Weighted means, strict ergodicity, and uniform distributions. (English. Russian original) Zbl 1121.37009
Math. Notes 78, No. 3, 329-337 (2005); translation from Mat. Zametki 78, No. 3, 358-367 (2005).
Author’s summary: We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesàro 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 polynomial with irrational coefficient.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 40G99 Special methods of summability 11K06 General theory of distribution modulo $$1$$
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##### References:
 [1] G. Hardy, Divergent Series, Oxford, 1949. · Zbl 0032.05801 [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. · Zbl 0183.47203 [3] V. V. Kozlov, ”Summation of divergent series and ergodic theorems,” Trudy Sem. Petrovsk., 22 (2002), 142–168. [4] G. Baxter, ”An ergodic theorem with weighted averages,” J. Math. Mech., 13 (1964), no. 3, 481–488. · Zbl 0125.06701 [5] R. V. Chacon, ”Ordinary means imply recurrent means,” Bull. Amer. Math. Soc., 70 (1964), 796–797. · Zbl 0196.43902 [6] J. C. Oxtoby, ”Ergodic sets,” Bull. Amer. Math. Soc., 58 (1952), no. 2, 116–136. · Zbl 0046.11504 [7] H. Furstenberg, ”Strict ergodicity and transformation of the torus,” Amer. J. Math., 83 (1961), no. 4, 573–601. · Zbl 0178.38404 [8] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974. · Zbl 0281.10001 [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. · Zbl 1073.37049 [10] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow, 1980. [11] H. Weyl, ”Uber die Gleichverteilung von Zahlen mod Eins,” Math. Ann., 77 (1915/16), 313–352. · JFM 46.0278.06 [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. · Zbl 1076.37500 [13] J. Cigler, ”Methods of summability and uniform distribution mod 1,” Compos. Math., 16 (1964), 44–51. · Zbl 0135.10901 [14] A. F. Doroidar, ”A note on the generalized uniform distribution (mod 1),” J. Natur. Sci. Math., 11 (1971), 185–189. [15] P. Bohl, ”Uber eine Differentialgleichung der Storungstheorie,” J. Reine Angew. Math., 131 (1906), no. 4, 268–321. · JFM 37.0343.02 [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. [17] G. Halasz, ”Remarks on the remainder in Birkhoff’s ergodic theorem,” Acta Math. Acad. Sci. Hungar., 28 (1976), nos. 3–4, 289–395. · Zbl 0336.28005 [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. · Zbl 1103.60009
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