Polynomial towers and inverse Gowers theory for bounded-exponent groups

Abstract

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, Gonz\'alez-S\'anchez, 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, Gonz\'alez-S\'anchez, and Szegedy.