A Krengel-type theorem for finitely-generated nilpotent groups (Q2757008)

A unitary action \(\{U_n\}_{n\in\mathbb Z}\) on a complex Hilbert space is weakly mixing if it has no eigenvectors, and a vector \(f\) in the Hilbert space is called weakly wandering if there is an infinite set \(S\) in \(\mathbb Z\) with \(\langle U_nf,U_mf\rangle=0\) for \(n\neq m\). \textit{U. Krengel} [Trans. Am. Math. Soc. 164, 199--226 (1972; Zbl 0227.47005)] proved that any weakly mixing action has a dense set of weakly wandering vectors. This result was extended to actions of many Abelian groups by \textit{V. Bergelson, I. Kornfeld} and \textit{B. Mityagin} [J. Funct. Anal. 126, No. 2, 274--304 (1994; Zbl 0841.22004)]. Here the result is extended to actions of finitely-generated nilpotent groups, with the infinite set \(S\) replaced by an appropriate nilpotent analogue of a symmetric \(IP\)-set. The proofs require results about the structure of unitary actions of nilpotent groups.