We show that there is a measure-preserving system (X,B,μ,T) together with functions F0,F1,F2∈L∞(μ) such that the correlation sequence CF0,F1,F2(n)=∫XF0⋅TnF1⋅T2nF2dμ is not an approximate integral combination of 2-step nilsequences.

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Ergodic Theory and Dynamical Systems
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Briët, J, & Green, B. (2021). Multiple correlation sequences not approximable by nilsequences. Ergodic Theory and Dynamical Systems, 42(9), 2711–2722. doi:10.1017/etds.2021.66