The structure of arbitrary Conze-Lesigne systems

Abstract

Let $Γ$ be a countable abelian group. An (abstract) $Γ$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $Γ$ - is said to be a Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor $\mathrm{Z}^2(\mathrm{X})$. The main result of this paper is a structural description of such Conze-Lesigne systems for arbitrary countable abelian $Γ$, namely that they are the inverse limit of translational systems $G_n/Λ_n$ arising from locally compact nilpotent groups $G_n$ of nilpotency class $2$, quotiented by a lattice $Λ_n$. Results of this type were previously known when $Γ$ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3(G)$ norm for arbitrary finite abelian groups $G$.

Theorems & Definitions (50)

• Remark 1.1
• Definition 1.2: Translational and rotational $\Gamma$-systems
• Lemma 1.3: Basic facts about systems of order $k$
• proof
• Theorem 1.4: Classification of Kronecker systems
• proof
• Theorem 1.5: Classification of Host--Kra--Ziegler $\mathbb{Z}$-systems
• Theorem 1.6: Classification of Host--Kra--Ziegler $\mathbb{F}_p^\omega$-systems
• proof
• Theorem 1.7: Partial classifications of Host--Kra--Ziegler systems