Polynomial towers and inverse Gowers theory for bounded-exponent groups

TL;DR

The paper develops Host–Kra and inverse Gowers theory for abelian groups of bounded exponent by introducing polynomial towers, a multi-layer extension where each cocycle is polynomial of controlled degree. It proves that Host–Kra factors admit extensions that are Abramov and can be organized as exact polynomial towers, though not necessarily Weyl towers, and shows these towers are translational systems in the finite-exponent setting. A technical core combines exactness, large spectrum, and purity to straighten cocycles and integrate polynomial ones, yielding a robust inductive construction of the main tower and its translational realization. Leveraging a correspondence principle, the authors derive an inverse theorem for Gowers norms on finite abelian groups of bounded exponent: large $U^{k+1}$ implies strong correlation with a degree-$\le k$ polynomial, even without enlarging the group. This resolves conjectures in the bounded-exponent regime and provides a framework connecting ergodic structure with finite-group additive combinatorics.

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, 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.

Figures (9)

• Figure 1.1: A tower of height $j$.
• Figure 1.2: The Host--Kra tower of height $\leq k$.
• Figure 1.3: Key implications between various properties of ergodic $\Gamma$-systems when $\Gamma$ is an arbitrary group of bounded exponent, for a given $k \geq 1$. For the dashed line, one needs to pass to an extension of the original system.
• Figure 1.4: The Host--Kra tower for $\mathrm{X}_{(k)}$. For $k \geq 3$, this is not a Weyl tower (or even a polynomial tower of order $\leq k$).
• Figure 1.5: A polynomial tower of order $\leq k$ and height $2$ for $\mathrm{X}_{(k)}$.

Theorems & Definitions (134)

• Definition 1.1: Measure preserving systems
• Definition 1.2: Cocycles of abelian group actions
• Example 1.3
• Remark 1.4
• Definition 1.5: Cocycles and skew products
• Example 1.6
• Remark 1.7
• Definition 1.8: Polynomials
• Proposition 1.9: Order and Abramov systems
• proof