Host, Bernard; Kra, Bryna 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]. Cited in 5 Documents MSC: 37A05 Dynamical aspects of measure-preserving transformations 28D05 Measure-preserving transformations 37A15 General groups of measure-preserving transformations and dynamical systems Keywords:averaging along combinatorial cubes; measure-preserving probability system; ergodic theorem; translations on a 2-step nilsystem PDFBibTeX XMLCite \textit{B. Host} and \textit{B. Kra}, in: Modern dynamical systems and applications. Dedicated to Anatole Katok on his 60th birthday. Cambridge: Cambridge University Press. 123--144 (2004; Zbl 1147.37301)