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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)
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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.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 233 - 262
Copyright
Copyright © Cambridge University Press 2009

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