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Averaging along cubes. (English) Zbl 1147.37301

Brin, Michael (ed.) et al., Modern dynamical systems and applications. Dedicated to Anatole Katok on his 60th birthday. Cambridge: Cambridge University Press (ISBN 0-521-84073-2/hbk). 123-144 (2004).
Summary: We study the convergence of an average of eight functions, where the average is taken along cubes whose sizes tend to \(+\infty\) and show that this reduces to proving an ergodic theorem for translations on a 2-step nilsystem. We derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density.
For the entire collection see [Zbl 1051.00012].

MSC:

37A05 Dynamical aspects of measure-preserving transformations
28D05 Measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems
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