完全不连通的Host–Kra–Ziegler因子的结构，以及有界挠有限阿贝尔群上$U^k$ Gowers均匀性范数的逆定理
The structure of totally disconnected Host--Kra--Ziegler factors, and the inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion
AI总结：
本文研究完全不连通的Host–Kra–Ziegler因子的结构，证明其为Weyl型$\boldsymbol{\text{Z^ω}}$系统的广义因子，进而利用对应原理得到有界挠有限阿贝尔群上$U^k$ Gowers均匀性范数的弱逆定理。
摘要：
设$\boldsymbol{\text{Γ}}$为可数阿贝尔群，$k\boldsymbol{\text{≥1}}$，$\boldsymbol{\text{X=(X,𝒳,μ,T)}}$是按Host–Kra–Ziegler意义下的$k$阶遍历$\boldsymbol{\text{Γ}}$系统。若$\boldsymbol{\text{Γ}}$系统$\boldsymbol{\text{X}}$的所有结构群均完全不连通，则称其为完全不连通的。本文证明：任何$k$阶完全不连通$\boldsymbol{\text{Γ}}$系统都是具有Weyl系统结构的$\boldsymbol{\text{Z^ω}}$系统的广义因子。作为该结构定理的推论，$k$阶完全不连通$\boldsymbol{\text{Γ}}$系统可表示为幂零Polish群的双陪集上的平移。借助本文两位作者提出的对应原理，该表示可用于建立有界挠有限阿贝尔群上$U^k$ Gowers均匀性范数的（弱）逆定理。
英文摘要：
Let $Γ$ be a countable abelian group, let $k\geq 1$, and let $\mathrm{X}=(X,\mathcal{X},μ,T)$ be an ergodic $Γ$-system of order $k$ in the sense of Host--Kra--Ziegler. The $Γ$-system $\mathrm{X}$ is said to be totally disconnected if all its structure groups are totally disconnected. We show that any totally disconnected $Γ$-system of order $k$ is a generalized factor of a $\mathbb{Z}^ω$-system with the structure of a Weyl system. As a consequence of this structure theorem, we show that totally disconnected $Γ$-systems of order $k$ are represented by translations on double cosets of nilpotent Polish groups. By a correspondence principle of two of us, we can use this representation to establish a (weak) inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion.
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