Consider is a nilpotent Lie group, a compact homogeneous space of and the nilmanifold on which acts by left translations. A sequence in of the form where the and 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 for any continuous function and for any Folner sequence in
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.