Orbit of the diagonal in the power of a nilmanifold

First published in Transactions of the American Mathematical Society in volume 362, issue 3, published by the American Mathematical Society.Let X be a nilmanifold, that is, a compact is, a compact homogeneous space of a nilpotent Lie group G, and let a is an element of G. We study the closure of the orbit of the diagonal of X-r under the action (a(p1(n)),..., a(pr(n))), where p(i) are integer-valued polynomials in m integer variables. (Knowing this closure is crucial for finding limits of the form lim(N ->infinity) 1/N-m Sigma(n is an element of{1,..., N}m) mu(T(p1(n))A1 boolean AND...boolean AND T(pr(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 ->infinity) 1/N-m Sigma(n is an element of{1,..., N}m) d((A(1) + p(1)(n)) boolean AND...boolean AND (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 C 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