Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T18:04:13.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Rates of divergence of non-conventional ergodic averages

Published online by Cambridge University Press:  23 June 2009

ANTHONY QUAS  and
MÁTÉ WIERDL
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria BC, Canada, V8W 3R4 (email: aquas@uvic.ca)
MÁTÉ WIERDL
Affiliation:
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA (email: mw@csi.hu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We first study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x)=∑ n≤Nf(Tanx). We characterize the maximal rate of growth and identify a number of sequences such as an=2n, along which the maximal rate of growth is achieved. To point out though the general character of our techniques, we then turn to Khintchine’s strong uniform distribution conjecture that the averages (1/N)∑ n≤Nf(nx mod 1) converge pointwise to ∫ f for integrable functions f. We give a simple, intuitive counterexample and prove that, in fact, divergence occurs at the maximal rate.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 233 - 262
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]Akcoglu, M., Bellow, A., Jones, R. L., Losert, V., Reinhold-Larsson, K. and Wierdl, M.. The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergod. Th. & Dynam. Sys. 16 (1996), 207–253.CrossRefGoogle Scholar
[2]Akcoglu, M., Jones, R. and Rosenblatt, J.. The worst sums in ergodic theory. Michigan Math. J. 47 (2000), 265–285.CrossRefGoogle Scholar
[3]Bellow, A.. On ‘bad universal’ sequences in ergodic theory. Measure Theory and its Applications (Sherbrooke, Quebec, 1982) (Lecture Notes in Mathematics, 1033). Springer, Berlin, 1983, pp. 74–78.CrossRefGoogle Scholar
[4]Bennett, C. and Sharpley, R.. Interpolation of Operators. Academic Press, New York, 1988.Google Scholar
[5]Besicovitch, A. S.. On the linear independence of fractional powers of integers. J. London Math. Soc. 15 (1940), 3–6.CrossRefGoogle Scholar
[6]Bourgain, J.. Almost sure convergence and bounded entropy. Israel J. Math. 63 (1988), 79–97.CrossRefGoogle Scholar
[7]Demeter, C.. The best constants associated with some weak maximal inequalities in ergodic theory. Canad. J. Math. 56 (2004), 449–471.CrossRefGoogle Scholar
[8]Garsia, A.. Topics in Almost Everywhere Convergence. Markham Publishing Co., Chicago, IL, 1970.Google Scholar
[9]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers. Oxford University Press, Oxford, 1939.Google Scholar
[10]Jones, R. and Wierdl, M.. Convergence and divergence of ergodic averages. Ergod. Th. & Dynam. Sys. 14 (1994), 515–535.CrossRefGoogle Scholar
[11]Khintchine, A.. Ein Satz über Kettenbrüche mit arithmetischen Anwendungen. Math. Z. 18 (1923), 289–306.CrossRefGoogle Scholar
[12]Krengel, U.. Ergodic Theorems. de Gruyter, Berlin, 1985.CrossRefGoogle Scholar
[13]Lacroix, Y.. On strong uniform distribution II. Acta Arith. 84 (1998), 279–290.CrossRefGoogle Scholar
[14]Lacroix, Y.. On strong uniform distribution III. Monatsh. Math. 143 (2004), 13–19.CrossRefGoogle Scholar
[15]Marstrand, J. M.. On Khintchine’s conjecture about strong uniform distribution. J. London Math. Soc. 21 (1970), 540–556.CrossRefGoogle Scholar
[16]Nair, R.. On strong uniform distribution. Acta Arith. 56 (1990), 183–193.CrossRefGoogle Scholar
[17]Ornstein, D. and Weiss, B.. The Shannon–McMillan–Breiman theorem for a class of amenable groups. Israel J. Math 44 (1983), 53–60.CrossRefGoogle Scholar
[18]Rosenblatt, J. and Wierdl, M.. Pointwise ergodic theorems via harmonic analysis. Ergodic Theory and its Connections with Harmonic Analysis. Eds. K. Petersen and I. Salama. Cambridge University Press, Cambridge, 1995.Google Scholar
[19]Sawyer, S.. Maximal inequalities of weak type. Ann. Math. 84 (1966), 157–174.CrossRefGoogle Scholar
[20]Stein, E. M. and Weiss, N. J.. On the convegence of Poisson integrals. Trans. Amer. Math. Soc. 140 (1969), 35–53.CrossRefGoogle Scholar