Non-vanishing of multiple correlation sequences

We resolve in the negative a conjecture of Frantzikinakis and Kuca concerning the vanishing of multiple correlation sequences in nilsystems. Specifically, we prove the existence of an ergodic $3$-step nilsystem $(G\mathrm{/}\mathrm{\Gamma },{\mu }_{G\mathrm{/}\mathrm{\Gamma }},R_{\alpha })$ and bounded functions $f_0,f_1,f_2\in L^{\mathrm{\infty }}({\mu }_{G\mathrm{/}\mathrm{\Gamma }})$ orthogonal to the Conze–Lesigne factor $L^2(G\mathrm{/}{G}_3\mathrm{\Gamma })$, whose associated multiple correlation sequence $$a(n)={\int }_{G\mathrm{/}\mathrm{\Gamma }}{f}_0(x)f_1({\alpha }^{n}x)f_2({\alpha }^{2n}x)d{\mu }_{G\mathrm{/}\mathrm{\Gamma }}(x)$$ does not decay to zero. The same counterexample also refutes another conjecture of Frantzikinakis and Kuca and a conjecture of Leibman. To construct this counterexample, we develop a framework for Fourier analysis on $G\mathrm{/}\mathrm{\Gamma }$ where $G$ is the free $3$-step nilpotent Lie group on $4$ generators, a methodology that extends naturally to general nilsystems.