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 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 F ω p -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 ≤ p + 1 the k-th Host-Kra factor is an Abramov system of order ≤ 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 above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on F n p .

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

“…We take the opportunity, in Subsection 2.4, to give further basic examples illustrating affine-exchangeability, which also help to motivate the subsequent material. In Section 3, we begin proving Theorem 1.5, by first refining the representation theorem [14,Theorem 1.3] in light of more recent results on higher-order Fourier analysis in characteristic p from [13]. This relies especially on the results concerning p-homogeneous nilspaces.…”

“…9 Meaning that X is the inverse limit of compact nilspaces of finite step; see Subsection 2.2. exchangeability) implies that the nilspaces obtained by applying [14,Theorem 6.7] to an affine-exchangeable measure must be 2-homogeneous nilspaces. This enables us to apply the structure theorem for 2-homogeneous nilspaces obtained in [13], which tells us that any finite 2-homogeneous nilspace is the image, under some nilspace fibration, of a certain finite filtered abelian 2-group. The next part of the argument, carried out in Section 4, involves setting up adequate inverse systems of such filtered 2-groups, in order to prove that for any compact 2-homogeneous nilspace X there is a fibration H → X.…”

“…for some k ∈ N (where 0 N\[k] denotes the constant 0 sequence with coordinates indexed by N \ [k]). Then, denoting by µ C k (F n 2 ) the Haar probability measure on the group of standard k-dimensional cubes in F n 2 , and noting that any such cube can be viewed as an affine-linear map A : F k 2 → F n 2 , we can define the measure µ L,fn on B L as the pushforward of µ C k (F n 2 ) under the map A → (f n • A(L)) L∈L (see (13)). We then say that the sequence (f n ) n∈N converges if for every finite L ⊂ F ω 2 the measures µ L,fn converge vaguely as n → ∞.…”

“…Here and throughout the sequel, the group Z n p will be identified with [0, p − 1] n equipped with addition mod p, the usual way. Definition 2.9 (p-homogeneous nilspaces [13]). Let X be a nilspace and let p ∈ N be a prime.…”

“…We recall more background on these nilspaces in Section 4. For now let us mention that the most basic examples of p-homogeneous nilspaces are given by the elementary abelian p-groups Z n p , and that more examples can be constructed easily (see [13]). These include the nilspace H introduced in Definition 1.4, which is 2-homogeneous.…”