Let X X be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group G G , and let a ∈ G a\in G . We study the closure of the orbit of the diagonal of X r X^{r} under the action ( a p 1 ( n ) , … , a p r ( n ) ) (a^{p_{1}(n)},\ldots ,a^{p_{r}(n)}) , where p i p_{i} are integer-valued polynomials in m m integer variables. (Knowing this closure is crucial for finding limits of the form lim N → ∞ 1 N m ∑ n ∈ { 1 , … , N } m μ ( T p 1 ( n ) A 1 ∩ … ∩ T p r ( n ) A r ) \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 T is a measure-preserving transformation of a finite measure space ( Y , μ ) (Y,\mu ) and A i A_{i} are subsets of Y Y , and limits of the form lim N → ∞ 1 N m ∑ n ∈ { 1 , … , N } m d ( ( A 1 + p 1 ( n ) ) ∩ … ∩ ( A r + p r ( n ) ) ) \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 A_{i} are subsets of Z and d ( A ) d(A) is the density of A A in Z.) We give a simple description of the closure of the orbit of the diagonal in the case that all p i p_{i} are linear, in the case that G G is connected, and in the case that the identity component of G G is commutative; in the general case our description of the orbit is not explicit.