*(English)*Zbl 1080.37003

Consider $G$ is a nilpotent Lie group, $X$ a compact homogeneous space of $G$ and $X=G/{\Gamma}$ the nilmanifold on which $G$ acts by left translations. A sequence ${\left\{g\left(n\right)\right\}}_{n\in \mathbb{Z}}$ in $G$ of the form $g\left(n\right)={a}_{1}^{{p}_{1}\left(n\right)}\cdots {a}_{m}^{{p}_{m}\left(n\right)}$ where the ${a}_{i}\in G$ and ${p}_{i}$ are polynomials taking on integer values on the integers is called a polynomial sequence. The author establishes the following pointwise convergence result for continuous functions on such a nilmanifold:

For any $x\in X,$ for any continuous function $f\in C\left(X\right)$ and for any Folner sequence ${{\Phi}}_{N}$ in $\mathbb{Z}$

exists. The proof is done by studying carefully the distribution of the orbit of a point on a compact nilmanifold.

This pointwise result and its multidimensional extension obtained by the author [ibid. 25, 215–225 (2005; Zbl 1080.37004)] are useful tools as they allow one to reduce the study of the norm and pointwise convergence of several nonconventional averages to orthocomplements of characteristic factors.