Multiple correlation sequences not approximable by nilsequences

Abstract We show that there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such the correlation sequence $C_{F_0, F_2}(n) = \int _X F_0 \cdot T^n F_1 T^{2n} \, d\mu $ not an approximate integral combination of $2$ -step nilsequences.