The structure of arbitrary Conze-Lesigne systems

Asgar Jamneshan; Or Shalom; Terence Tao

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arXiv · 2112.02056

The structure of arbitrary Conze-Lesigne systems

Asgar Jamneshan·Or Shalom·Terence Tao

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Abstract

Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\Gamma$ - 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 $\Gamma$, namely that they are the inverse limit of translational systems $G_n/\Lambda_n$ arising from locally compact nilpotent groups $G_n$ of nilpotency class $2$, quotiented by a lattice $\Lambda_n$. Results of this type were previously known when $\Gamma$ 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 $G$.

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Asgar Jamneshan, Or Shalom, Terence Tao. 2021-12-03. The structure of arbitrary Conze-Lesigne systems. https://arxiv.org/abs/2112.02056

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