On higher-order Fourier analysis in characteristic p

Abstract

In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$ , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$ . We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .

A Host--Kra ${\mathbf F}_2^ω$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm

“…In particular, from the previously mentioned results of [4], Conjecture 1.3 holds in the high characteristic case k ≤ p + 1; also, from [1,Theorem 1.20] one can establish a weaker version of Conjecture 1.3 in which the polynomial P is of degree at most C(p, k) rather than k for some quantity C(p, k) depending only on p, k.…”

“…. , h k ∈ G (our choice of terminology here is inspired by [4]) . This motivates the following definition:…”

“…Remark 4.6. While the above proposition shows that X 5,1 cannot be embedded into a finite filtered abelian group, [4, Theorem 1.7] does show that there is an fibration π : Y → X 5,1 (as defined in [14, Definition 7.1], [3, Definition 3.3.7]) that has the structure of a finite filtered abelian group and has good lifting properties; this result was in particular used in [4] to give an alternate proof of Conjecture 1.1 in both high and low characteristic. In fact, we can explicitly give such an extension.…”

“…In [23], a variant of the Furstenberg correspondence principle was used to show that Conjecture 1.2 implied Conjecture 1.1 for any given choice of p, k. In [1], Conjecture 1.2 was established in the high characteristic case k + 1 ≤ p; combining the two results, this also gave Conjecture 1.1 in this regime. The full case of Conjecture 1.1 was subsequently established in [23] by a different method; alternate proofs of some or all of the cases of this conjecture have since been given in [7], [8], [4], [24], [18]. In particular Conjecture 1.2 was established in [4,Theorem 1.12] in the slightly larger range k ≤ p + 1 (and an alternate proof of Conjecture 1.1 was given for all k, p).…”

“…The full case of Conjecture 1.1 was subsequently established in [23] by a different method; alternate proofs of some or all of the cases of this conjecture have since been given in [7], [8], [4], [24], [18]. In particular Conjecture 1.2 was established in [4,Theorem 1.12] in the slightly larger range k ≤ p + 1 (and an alternate proof of Conjecture 1.1 was given for all k, p). We also remark that in [1,Theorem 1.20], a weaker version of Conjecture 1.2 was established in which Poly k (X) was replaced by some unspecified subalgebra of Poly C(p,k) (X) for some constant C(p, k) depending only on p, k. We also note that several other structural results on ergodic F ω psystems are known; see in particular [4], [16].…”