The structure of arbitrary Conze-Lesigne systems
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| Format: | Preprint |
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2021
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| _version_ | 1866910334269259776 |
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| author | Jamneshan, Asgar Shalom, Or Tao, Terence |
| author_facet | Jamneshan, Asgar Shalom, Or Tao, Terence |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_02056 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The structure of arbitrary Conze-Lesigne systems Jamneshan, Asgar Shalom, Or Tao, Terence Dynamical Systems 37A35 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 $G$. |
| title | The structure of arbitrary Conze-Lesigne systems |
| topic | Dynamical Systems 37A35 |
| url | https://arxiv.org/abs/2112.02056 |