Multiple recurrence theorem for nilpotent group actions (Q1343332)

The author considers the action of a finitely generated nilpotent group \(\Gamma\) on a compact metric space \(X\). He shows that if \(T_1, \dots, T_t\) are members of \(\Gamma\) and \(p_{i,j}\) \((i = 1, \dots, s\) and \(j = 1, \dots, t)\) are a sequence of polynomials mapping 0 to 0 and the integers into themselves, then there exists a sequence \(n_1, n_2, \dots\) and an \(x \in X\) such that for each \(i\), \(T_1^{p_{i,1} (n_k)} \circ \dots \circ T_t^{p_{i,t} (n_k)} (x) \to x\) as \(k \to \infty\). Theorems of this type are known as recurrence theorems and often have combinatorial corollaries (as this theorem does). There are known examples of metabelian, but non-nilpotent groups \(\Gamma\) for which this does not hold so this is in some sense the best possible. The main technique used is that of PET-induction, an induction scheme which covers multiple polynomials, which was introduced by \textit{V. Bergelson} [Ergodic Theory Dyn. Syst. 7, 337-349 (1987; Zbl 0645.28012)].