The article deals with orbits of the diagonal element \((a,\dots,a)\) of \(X^r\) under the action \((a^{p_1(n)},\dots,a^{p_r(n)})\), where \(p_i\) are integer-valued polynomials in \(m\) integer variables, \(X\) is a nilmanifold (a compact homogeneous space of a nilpotent Lie group \(G\)), \(a \in G\). The author gives an explicit description of the closure of the orbit of the diagonal in the case that all \(p_i\) are linear, in the case that \(G\) is connected, and in the case that the identity component of \(G\) is commutative; in the general case the description of the orbit is also presented, however, it is not explicit.
The article contains the following sections: 0. Introduction; 1. Orbits in a manifold; 2. Characteristic factors, natural factors of nilmanifolds, and complexity; 3. Spans; 4. A group of polynomial mappings to a nilpotent group; 5. A subgroup of \(G^r\) associated with a subgroup of \({\mathbb Z}^r\); 6. The orbit of the diagonal under a system of linear actions; 7. Polynomial orbits on tori; 8. A system of polynomials actions — the case of a connected group; 9. A system of polynomials actions — the case of a Weil group; 10. Contribution of a polynomial orbit of a point; 11. The general case — an algorithm; 12. A filtration of \({\mathcal G}\); 13. The general case — estimation of complexity.
As the author states, the knowledge of the orbit of the diagonal allows to calculate limits of the form \[ \lim_{N \to \infty} \;\frac1{N^m} \sum_{n \in \{1,\dots,N\}^m} \mu (T^{p_1(n)}A_1 \cap \dots \cap T^{p_r(n)}A_r), \] where \(T\) is a measure-preserving transformation of a finite measure space \((Y,\mu)\) and \(A_i\) are subsets of \(Y\) and limits of the form \[ \lim_{N \to \infty} \;\frac1{N^m} \sum_{n \in \{1,\dots,N\}^m} d((A_1 + p_1(n)) \cap \dots \cap A_r + p_r(n)), \] where \(A_i\) are subsets of \({\mathbb Z}\) and \(d(A)\) is the density of \(A\) in \({\mathbb Z}\).