AI 中文总结
本文否定解决了弗兰齐基纳基斯和库卡关于幂零系统中多重相关序列消失的猜想，通过构造反例反驳了另外两个相关猜想，为此开发了\(G/\Gamma\)的傅里叶分析框架并可扩展到一般幂零系统。
AI 中文摘要
我们否定地解决了弗兰齐基纳基斯和库卡关于幂零系统中多重相关序列消失的一个猜想。具体而言，我们证明了存在一个遍历性的三步幂零系统\((G/\Gamma, \mu_{G/\Gamma}, R_\alpha)\)以及在\(L^\infty(\mu_{G/\Gamma})\)中与康泽 - 莱西格因子\(L^2(G/G_3\Gamma)\)正交的有界函数\(f_0, f_1, f_2\)，其相关的多重相关序列\(a(n) = \int_{G/\Gamma} f_0(x) f_1(\alpha^n x) f_2(\alpha^{2n} x) \, d\mu_{G/\Gamma}(x)\)不衰减到零。同一个反例也反驳了弗兰齐基纳基斯和库卡的另一个猜想以及莱布曼的一个猜想。为构造此反例，我们为\(G/\Gamma\)（其中\(G\)是由4个生成元生成的自由三步幂零李群）开发了一个傅里叶分析框架，该方法自然地扩展到一般幂零系统。
英文摘要
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.
Comments48 pages
