Orbit of the diagonal in the power of a nilmanifold

Let $X$ be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group $G$, and let $a\in G$. We study the closure of the orbit of the diagonal of $X^{r}$ under the action $(a^{p_{1}(n)} ,\ldots ,a^{p_{r}(n)})$, where $p_{i}$ are integer-valued polynomials in $m$ integer variables. (Knowing this closure is crucial for finding limits of the form $\hbox {lim}_{N\rightarrow \infty }\frac{1}{N^{m}}\sum _{n\in \{1,\ldots ,N\}^{m}} \mu (T^{p_{1}(n)}A_{1}\cap \ldots \cap T^{p_{r}(n)}A_{r})$, 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 $\hbox {lim}_{N\rightarrow \infty }\frac{1}{N^{m}}\sum _{n\in \{1,\ldots ,N\}^{m}} d((A_{1}+p_{1}(n))\cap \ldots \cap (A_{r}+p_{r}(n)))$, where $A_{i}$ are subsets of Z and $d(A)$ is the density of $A$ in Z.) We give a simple 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 our description of the orbit is not explicit.