Non-vanishing of multiple correlation sequences

Abstract

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 $(G/\hat{I}“,\hat{I}{¼}_{G/\hat{I}“},R_{\hat{I}}\pm )$ and bounded functions $f_0,f_1,f_2\in L^{\infty }(\hat{I}{¼}_{G/\hat{I}“})$ orthogonal to the Conze--Lesigne factor $L^2(G/G_3\hat{I}“)$, whose associated multiple correlation sequence $$a(n)={\int }_{G/\hat{I}“}{f}_0(x)f_1(\hat{I}{\pm }^{n}x)f_2(\hat{I}{\pm }^{2n}x)d\hat{I}{¼}_{G/\hat{I}“}(x)$$ 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 $G/\hat{I}“$ where $G$ is the free $3$-step nilpotent Lie group on $4$ generators, a methodology that extends naturally to general nilsystems.

48 pages