Free nilspaces, double-coset nilspaces, and Gowers norms
Pablo Candela, Diego González-Sánchez, Balázs Szegedy
TL;DR
This work unifies compact finite-rank nilspaces with classical topological-group actions by proving that every CFR nilspace can be realized as a quotient F/Γ of a free nilspace F by a higher-order lattice action, and as a double-coset K\backslash G / Γ with G a degree-k nilpotent Lie group. It then extends these representations to all k-step nilspaces via pro-free nilspaces and pro-Lie groups, yielding a pro-object framework that connects nilspace theory with topological dynamics and Lie theory. The paper develops the topological theory of locally-compact nilspaces, including open-mapping results, Cartan bundles, and the Lie-structure of translation groups, and uses these to prove split-extension results for Lie-fibered extensions of free nilspaces. These structural insights enable new inverse-theorem formulations for Gowers norms on finite abelian groups, expressed in terms of double-coset spaces and free-nilspace data, and provide a pathway toward resolving the Jamneshan–Tao conjecture. Overall, the results broaden the classical toolkit for higher-order Fourier analysis by embedding nilspace structure in the language of free nilspaces, groupable congruences, and double cosets, with concrete consequences for inverse theorems and regularity lemmas.
Abstract
Compact finite-rank nilspaces have become central in the nilspace approach to higher-order Fourier analysis, notably through their role in a general form of the inverse theorem for the Gowers norms. This paper studies these nilspaces per se, and in connection with further refinements of this inverse theorem that have been conjectured recently. Our first main result states that every compact finite-rank nilspace is obtained by taking a free nilspace (a nilspace based on an abelian group of the form $\mathbb{Z}^{r}\times \mathbb{R}^s$) and quotienting this by a discrete group action of a specific type, describable in terms of polynomials. We call these group actions "higher-order lattice actions", as they generalize actions of lattices in $\mathbb{Z}^r\times \mathbb{R}^s$. The second main result (which relies on the first one) represents every compact finite-rank nilspace as a double-coset space $K\backslash G / Γ$ where $G$ is a nilpotent Lie group of a specific kind. Our third main result extends the aforementioned results to $k$-step compact nilspaces (not necessarily of finite rank), by representing any such nilspace as a quotient of infinite products of free nilspaces and also as double coset spaces $K\backslash G/Γ$ where $G$ is a degree-$k$ nilpotent pro-Lie group. These results open the study of compact nilspaces to areas more classical than nilspace theory, such as the theory of topological group actions. The results also require developing the theory of topological non-compact nilspaces, for which we provide groundwork in this paper. Applications include new inverse theorems for Gowers norms on any finite abelian group. These theorems are purely group theoretic in that the correlating harmonics are based on double-coset spaces. This yields progress towards the Jamneshan-Tao conjecture.