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 $(G/Γ, μ_{G/Γ}, R_α)$ and bounded functions $f_0, f_1, f_2 \in L^\infty(μ_{G/Γ})$ orthogonal to the Conze--Lesigne factor $L^2(G/G_3Γ)$, whose associated multiple correlation sequence $$a(n) = \int_{G/Γ} f_0(x) f_1(α^n x) f_2(α^{2n} x) \, dμ_{G/Γ}(x)$$ 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 $G/Γ$ where $G$ is the free $3$-step nilpotent Lie group on $4$ generators, a methodology that extends naturally to general nilsystems.

Disclosure

“Kuca for useful suggestions on an earlier version of this manuscript, leading in particular to Appendix A. The role of AI AI was used in this paper for the following purposes: (1) Claude (Opus 4.8) was used to program codes for various computations, some of which ended up being useful for this paper (see Appendix B). (2) Claude and Gemini (Pro 3.1) were used for copy-editing.”

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2026-08-23T18:54:25.710950Z

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