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Weighted Birkhoff ergodic theorem with oscillating weights

  • AI-HUA FAN (a1) (a2)


We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any $q$ -multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.



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[1] El Abdalaoui, H., Kulaga-Przymus, J., Lemanczyk, M. and de la Rue, T.. The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete Contin. Dyn. Syst. 37(6) (2017), 28992944.
[2] Abramov, L. M.. Metric automorphisms with quasi-discrete spectrum. Izv. Akad. Nauk 26 (1962), 513530.
[3] Baladi, V.. Positive Transfer Operators and Decay Of Correlations (Advanced Series in Nonlinear Dynamics, 16) . World Scientific, Singapore, 2000.
[4] Bourgain, J.. Pointwise ergodic theorem for arithmetic sets. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 541.
[5] Cohen, G. and Lin, M.. Extensions of the Menchoff–Rademacher theorem with applications to ergodic theory. Israel J. Math. 148(1) (2005), 4186.
[6] Cuny, C.. On the a.s. convergence of the one-sided ergodic Hilbert transform. Ergod. Th. & Dynam. Sys. 29 (2009), 17811788.
[7] Cuny, C. and Fan, A.-H.. Study of almost everywhere convergence of series by means of martingale methods. Stochastic Process. Appl. 127 (2017), 27252750.
[8] Davenport, H.. On some infinite series involving arithmetical functions (II). Q. J. Math. Oxford 8 (1937), 313320.
[9] Davenport, H., Erdös, P. and LeVeque, W. J.. On Weyl’s criterion for uniform distribution. Michigan Math. J. 10 (1963), 311314.
[10] Durand, F. and Schneider, D.. Ergodic averages with deterministic weights. Ann. Inst. Fourier 52(2) (2002), 561583.
[11] Eisner, T.. A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353 (2015), 569572.
[12] Fan, A.-H.. Oscillating sequences of higher orders and topological systems of quasi-discrete spectrum. Preprint, 2016.
[13] Fan, A.-H.. Topological Wiener–Wintner ergodic theorem with polynomial weights. Preprint, 2016.
[14] Fan, A.-H.. Almost everywhere convergence of ergodic series. Ergod. Th. & Dynam. Sys. 37(2) (2017), 490511.
[15] Fan, A.-H.. Fully oscillating sequences and weighted multiple ergodic limit. C. R. Acad. Sci., to appear.
[16] Fan, A.-H. and Jiang, Y. P.. On Ruelle–Perron–Frobenius operators II. Comm. Math. Phys. 223(1) (2001), 143159.
[17] Fan, A.-H. and Jiang, Y. P.. Oscillating sequences, minimal mean attractability and minimal mean-Lyapunov-stability. Ergod. Th. & Dynam. Sys., doi:10.1017/etds.2016.121, Published online: 14 March 2017, pp. 1–36.
[18] Gelfond, A. O.. Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13 (1968), 259265.
[19] Hahn, F. and Parry, W.. Minimal dynamical systems with quasi-discrete spectrum. J. Lond. Math. Soc. (2) 40 (1965), 309323.
[20] Kahane, J. P.. Some Random Series of Functions (Cambridge Studies in Advanced Mathematics, 5) , 2nd edn. Cambridge University Press, 1985.
[21] Konieczny, J.. Gowers norms for the Thue–Morse and Rudin–Shapiro sequences. Preprint, 2016, arXiv:1611.09985.
[22] Krengel, U.. Ergodic Theorems. Walter de Gruyter, Berlin, 1982.
[23] Lesigne, E.. Spectre quasi-discret et théorème ergodique de Wiener–Wintner pour les polymômes. Ergod. Th. & Dynam. Sys. 13 (1993), 676684.
[24] Lesigne, E. and Mauduit, Ch.. Propriétés ergodiques des suites q-multiplicatives. Compos. Math. 100 (1996), 131169.
[25] Lesigne, E., Mauduit, Ch. and Mossé, B.. Le théorème ergodique le long d’une suite q-multiplicative. Compos. Math. 93 (1994), 4979.
[26] Mauduit, Ch., Rivat, J. and Sárközy, A.. On digits of sumsets. Canad. J. Math., online (2016).
[27] Móricz, F.. Moment inequalities and the strong law of large numbers. Z. Wahrscheinlichkeitsth. Verw. Geb. 35 (1976), 299314.
[28] Sarnak, P.. Three Lectures on the Möbius Function, Randomness and Dynamics, IAS Lecture Notes, 2009;
[29] Sarnak, P.. Möbius randomness and dynamics. Not. S. Afr. Math. Soc. 43 (2012), 8997.
[30] Tao, T.. Higher Order Fourier Analysis (Graduate Studies in Mathematics, 142) . American Mathematical Society, Providence, RI, 2012.


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