The non-ergodic Host-Kra-Ziegler structure theorem for $\mathbb{Z}^d$-actions via measurable selections

TL;DR

The paper develops a non-ergodic version of the Host–Kra–Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions by reducing to ergodic components through a measurable selection framework. It constructs an inverse system of bundles of $k$-step nilsystems over the invariant factor $\Omega$ and proves $\mathrm Z_k(\mathrm X)\cong\varprojlim_n Y_n$, where each $Y_n$ is a bundle of nilsystems assembled from fiberwise nilfactors chosen via measurable selectors. The approach relies on graph joinings, disintegration over $\Omega$ (Kallenberg), and the Jankov–von Neumann uniformization to realize a sequence of compatible factor maps $\pi_n:X\to Y_n$ with bundle structures. This yields a geometric and measure-theoretic description of $Z_k(\mathrm X)$ in the non-ergodic setting, enabling broader applications in ergodic theory and additive combinatorics. The results extend the ergodic HKZ theory to non-ergodic contexts and pave the way for further generalizations (e.g., to finitely generated nilpotent group actions).

Abstract

We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Definition 2.2: Bundle of nilsystems
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof