Polynomial towers and inverse Gowers theory for bounded-exponent groups
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| Format: | Preprint |
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2026
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| author | Jamneshan, Asgar Shalom, Or Tao, Terence |
| author_facet | Jamneshan, Asgar Shalom, Or Tao, Terence |
| contents | In this paper we develop Host--Kra and inverse Gowers theory for abelian groups of bounded exponent. We show that the Host--Kra factors $Z^{\leq k}(\mathrm{X})$ associated with actions of such groups admit extensions with the structure of \emph{polynomial towers}. This new notion is a system obtained as a finite iteration of abelian extensions of the trivial system by polynomial cocycles; crucially, the intermediate extensions in this system are not required to agree with the Host--Kra factors. We prove that all such extensions are Abramov (generalizing a recent result of Candela, González-Sánchez, and Szegedy), but not necessarily Weyl, and have the structure of k-step translational systems.
Combining this structure theorem with a correspondence principle due to the first and third authors, we derive an inverse theorem for the Gowers norms on finite abelian groups of bounded exponent: large $U^{k+1}$-norm implies large correlation with a polynomial of degree $\le k$ (on the same group), even when the exponent is not square-free or is divisible by small primes. This resolves a conjecture of the first and third authors for such groups, and also answers a question of Candela, González-Sánchez, and Szegedy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial towers and inverse Gowers theory for bounded-exponent groups Jamneshan, Asgar Shalom, Or Tao, Terence Dynamical Systems Combinatorics Group Theory In this paper we develop Host--Kra and inverse Gowers theory for abelian groups of bounded exponent. We show that the Host--Kra factors $Z^{\leq k}(\mathrm{X})$ associated with actions of such groups admit extensions with the structure of \emph{polynomial towers}. This new notion is a system obtained as a finite iteration of abelian extensions of the trivial system by polynomial cocycles; crucially, the intermediate extensions in this system are not required to agree with the Host--Kra factors. We prove that all such extensions are Abramov (generalizing a recent result of Candela, González-Sánchez, and Szegedy), but not necessarily Weyl, and have the structure of k-step translational systems. Combining this structure theorem with a correspondence principle due to the first and third authors, we derive an inverse theorem for the Gowers norms on finite abelian groups of bounded exponent: large $U^{k+1}$-norm implies large correlation with a polynomial of degree $\le k$ (on the same group), even when the exponent is not square-free or is divisible by small primes. This resolves a conjecture of the first and third authors for such groups, and also answers a question of Candela, González-Sánchez, and Szegedy. |
| title | Polynomial towers and inverse Gowers theory for bounded-exponent groups |
| topic | Dynamical Systems Combinatorics Group Theory |
| url | https://arxiv.org/abs/2601.00961 |