1 Introduction

The theory of higher-order Fourier analysis, initiated by Gowers in his celebrated work on Szemerédi’s theorem [Reference Gowers15], has generated various fascinating developments in analysis and combinatorics in the last two decades. Many of these developments aim to understand the relation between the central objects in this theory, namely the uniformity norms (or Gowers norms), and structures involving nilpotent groups. This relation emerged early on, especially in the work [Reference Host and Kra27] of Host and Kra which introduced seminorms in ergodic theory analogous to the uniformity norms, and proved the ergodic structure theorem, establishing a deep connection between these seminorms and nilmanifolds [Reference Host and Kra27, Theorem 10.1] (see also [Reference Host and Kra29, Ch. 16]). This inspired further progress, notably in the work of Green and Tao [Reference Green and Tao17] and Green, Tao and Ziegler [Reference Green, Tao and Ziegler19] in arithmetic combinatorics, developing this connection between Gowers norms and nilmanifolds, leading to the proof by Green, Tao and Ziegler of the inverse theorem for Gowers norms on finite cyclic groups [Reference Green, Tao and Ziegler20].

The search for further conceptual clarification of the above-mentioned connection also led to the discovery of interesting structures closely related to the uniformity norms, starting with the parallelepiped structures introduced by Host and Kra [Reference Host and Kra28], leading to the concept of nilspaces defined by Antolín Camarena and the third named author in [Reference Camarena and Szegedy6].

Nilspace-related topics have now grown into an active research area, including detailed treatments by the first named author [Reference Candela7, Reference Candela8] and by Gutman, Manners and Varjú [Reference Gutman, Manners and Varjú23–Reference Gutman, Manners and Varjú25], as well as further applications in arithmetic combinatorics, ergodic theory, probability theory and topological dynamics [Reference Candela, González-Sánchez and Szegedy9, Reference Candela and Szegedy12, Reference Glasner, Gutman and Ye14, Reference Gutman and Lian22–Reference Gutman, Manners and Varjú25].

Initial applications of nilspaces in higher-order Fourier analysis were obtained in [Reference Szegedy36], where they were combined with analysis on ultraproducts to prove regularity and inverse theorems for the Gowers norms on various families of compact abelian groups. In [Reference Host and Kra29, end of Ch. 17], Host and Kra suggested that the nilspace approach from [Reference Szegedy36] might be unified with the analysis of characteristic factors for uniformity seminorms from [Reference Host and Kra27]. A measure-theoretic framework enabling such a unification was introduced in [Reference Candela and Szegedy12], based on the notion of cubic couplings, inspired by the Host–Kra measures from [Reference Host and Kra27, §3]. Applications of this framework included an extension of the ergodic structure theorem to nilpotent group actions (see [Reference Candela and Szegedy12, Theorem 5.12] and [Reference Candela, González-Sánchez and Szegedy9, Theorem 5.1]), and an extension of the inverse theorem to all compact abelian groups and also to nilmanifolds [Reference Candela and Szegedy13].

In this paper we aim to demonstrate the utility of the above-mentioned framework in another principal setting for higher-order Fourier analysis, in which this approach had not been applied previously, namely the characteristic-p setting. Here, the uniformity norms are studied on vector spaces $\mathbb {F}_p^n$ over a field $\mathbb {F}_p$ of fixed prime order p, with dimension n allowed to tend to infinity. This direction was fostered notably by Green [Reference Green and Webb21], who promoted these vector spaces as useful models for various problems in arithmetic combinatorics that were originally posed in the integer setting, the latter setting being usually modelled by cyclic groups $\mathbb {Z}_N$ of large prime order N allowed to tend to infinity. The usefulness of the vector space models relies mainly on the fact that they provide much richer algebraic structure than is available in the integer setting. The characteristic-p setting has been very fruitful for higher-order Fourier analysis, with many interesting results in arithmetic combinatorics and in ergodic theory (for more background on this setting, see, for instance, the survey [Reference Wolf39]). Among these results, the present paper is related mainly to the work of Bergelson, Tao and Ziegler on ergodic actions of the (additive group of the) vector space $\mathbb {F}_p^\omega =\bigoplus _{i\in \mathbb {N}}\mathbb {F}_p$ (see [Reference Bergelson, Tao and Ziegler2, Reference Bergelson, Tao and Ziegler3]), and the related inverse theorems for Gowers norms proved by Tao and Ziegler in [Reference Tao and Ziegler37, Reference Tao and Ziegler38]. Let us now describe the approach to these topics in this paper.

In the integer setting, a decisive conceptual step was to identify nilmanifolds as adequate spaces with which to define basic harmonics (nilsequences) that could yield a useful inverse theorem for the Gowers norms. In characteristic p, the greater algebraic richness of this setting made it possible to obtain an inverse theorem with the corresponding harmonics being global phase polynomials, easily definable directly on the initial spaces $\mathbb {F}_p^n$ . Thus, inverse theorems in characteristic p have hitherto been obtained without a conceptual step similar to the above-mentioned process involving nilmanifolds. On the other hand, this has left open several questions that are relevant in order to clarify and strengthen the connections between the integer setting and the characteristic-p setting. These questions also lead to new results in the latter setting in itself, and can be subsumed under the following initial and more general question.

In this paper, we show that the nilspace approach offers a useful answer to this question. More precisely, we identify and study the class of compact nilspaces that emerges when the above-mentioned framework from [Reference Candela and Szegedy12, Reference Candela and Szegedy13] is applied in the characteristic-p setting. We call these structures p-homogeneous nilspaces. We show that these nilspaces yield a description of the characteristic factors for uniformity seminorms which is strong enough to give new proofs of central results in higher-order Fourier analysis in this setting, such as the inverse theorem for the Gowers norms from [Reference Tao and Ziegler37, Reference Tao and Ziegler38], and also new results concerning the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems.

To explain our results, first let us briefly recall the strategy used in [Reference Candela and Szegedy12, Reference Candela and Szegedy13] to prove the inverse theorem for Gowers norms in the integer setting, as it is overall the same process that will lead to p-homogeneous nilspaces in characteristic p. To this end, let us recall that a nilspace is a set $\operatorname {\mathrm {X}}$ equipped with a sequence of cube sets $\operatorname {\mathrm {C}}^n(\operatorname {\mathrm {X}})\subset \operatorname {\mathrm {X}}^{\{0,1\}^n}$ , $n\geq 0$ , whose elements are called the n-cubes on $\operatorname {\mathrm {X}}$ , satisfying three axioms, the most subtle of which is the completion axiom, which states that any n-corner on $\operatorname {\mathrm {X}}$ (roughly speaking, an n-cube missing one vertex) can always be completed to an n-cube. We say that $\operatorname {\mathrm {X}}$ is a k-step nilspace if every $(k+1)$ -corner on $\operatorname {\mathrm {X}}$ has a unique completion. Instead of recalling these definitions in detail, it is more helpful intuitively at this point just to recall the standard example of how any abelian group $\operatorname {\mathrm {Z}}$ can be viewed as a 1-step nilspace, denoted by $\mathcal {D}_1(\operatorname {\mathrm {Z}})$ : for each n, the cube set $\operatorname {\mathrm {C}}^n(\mathcal {D}_1(\operatorname {\mathrm {Z}}))$ consists of the standard n-cubes, of the form $(x+v(1)h_1+\cdots +v(n)h_n)_{v\in \{0,1\}^n}\in \operatorname {\mathrm {Z}}^{\{0,1\}^n},$ for any elements $x,h_i\in \operatorname {\mathrm {Z}}$ . Given nilspaces $\operatorname {\mathrm {X}}$ , $\operatorname {\mathrm {Y}}$ , a nilspace morphism from $\operatorname {\mathrm {X}}$ to $\operatorname {\mathrm {Y}}$ is a cube-preserving map $\operatorname {\mathrm {X}}\to \operatorname {\mathrm {Y}}$ , and the set of all such morphisms is denoted by $\hom (\operatorname {\mathrm {X}},\operatorname {\mathrm {Y}})$ . A nilspace $\operatorname {\mathrm {X}}$ is compact if the set $\operatorname {\mathrm {X}}$ is equipped with a compact second-countable Hausdorff topology which is compatible with the cubic structure in the sense that each cube set $\operatorname {\mathrm {C}}^n(\operatorname {\mathrm {X}})$ is compact in the product topology on $\operatorname {\mathrm {X}}^{\{0,1\}^n}\!.$ We refer to [Reference Candela7, Reference Candela8] for more background on nilspaces.

The main result in [Reference Candela and Szegedy12] is a structure theorem describing the characteristic factors, for a general type of uniformity seminorms, in terms of compact nilspaces [Reference Candela and Szegedy12, Theorem 4.2]. The strategy in [Reference Candela and Szegedy13] is based on the fact that when this structure theorem is applied to the uniformity seminorms on ultraproducts of cyclic groups $\mathbb {Z}_N$ (for increasing primes N), the resulting characteristic factors are completely described by the class of compact nilspaces $\operatorname {\mathrm {X}}$ admitting morphisms $\mathcal {D}_1(\mathbb {Z}_N)\to \operatorname {\mathrm {X}}$ that are increasingly balanced as N grows (the notion of balanced morphism, recalled in detail in §2, involves a quantitative form of equidistribution which also requires the n-cubic power of the morphism to be equidistributed in $\operatorname {\mathrm {C}}^n(\operatorname {\mathrm {X}})$ for large n). A key step in this proof of the inverse theorem on $\mathbb {Z}_N$ is then to show that the relevant nilspaces arising in this way are all connected nilmanifolds, more precisely, toral nilspaces [Reference Candela and Szegedy13, Theorem 6.1].

As signalled by Question 1.1, previously there was no class of compact nilspaces clearly identified as playing a role in characteristic p similar to the role of nilmanifolds in the integer setting. However, the above strategy indicates such a class in a natural way, namely, the class of compact nilspaces $\operatorname {\mathrm {X}}$ with the property of admitting increasingly balanced morphisms from (the additive group of) $\mathbb {F}_p^n$ into $\operatorname {\mathrm {X}}$ as n grows. As we shall see, this property yields one of various equivalent ways of defining p-homogeneous compact nilspaces. Moreover, one of the main results in this paper shows that these nilspaces can also be identified by a much simpler and purely algebraic property. Owing to its simplicity, we use this property to define p-homogeneous nilspaces, as follows.

To state the announced result linking Definition 1.2 to balanced morphisms from $\mathbb {F}_p^n$ , we need the notion of a finite-rank nilspace. This involves the fact that a k-step compact nilspace $\operatorname {\mathrm {X}}$ can always be decomposed as a k-fold compact abelian bundle [Reference Candela8, Definition 2.1.8], where, for each $i\in [k]$ , the nilspace factor $\operatorname {\mathrm {X}}_i$ is an extension (in the sense of [Reference Candela7, Definition 3.3.13]) of $\operatorname {\mathrm {X}}_{i-1}$ by a compact abelian group, called the ith structure group of $\operatorname {\mathrm {X}}$ and denoted by $\operatorname {\mathrm {Z}}_i$ or $\operatorname {\mathrm {Z}}_i(\operatorname {\mathrm {X}})$ ; see [Reference Candela8, Proposition 2.1.9]. If every structure group has finite rank, then $\operatorname {\mathrm {X}}$ is called a compact finite-rank nilspace, which we abbreviate to ‘ nilspace’. The inverse limit theorem for nilspaces [Reference Camarena and Szegedy6] states that every compact nilspace can be decomposed as an inverse limit of cfr nilspaces (see also [Reference Candela8, Theorem 2.7.3]). This often enables the study of a class of compact nilspaces to be reduced in a very useful way to the study of the cfr members of the class. We can now state the announced result.

The proof of Theorem 1.3 occupies §2 and involves several steps, using, in particular, a recent refinement of the generalized Von Neumann theorem [Reference Candela, González-Sánchez, Szegedy, Nešetřil, Perarnau, Rué and Serra10, Reference Candela, González-Sánchez and Szegedy11] (see also the recent work of Manners [Reference Manners32]).

With Theorem 1.3, the theory of p-homogeneous nilspaces can be developed using the property in Definition 1.2, a property that has rather strong consequences, which we begin to develop in §3. The simplest examples of p-homogeneous nilspaces are the 1-step nilspaces based on finite elementary abelian p-groups, that is, the nilspaces $\mathcal {D}_1(\mathbb {Z}_p^n)$ . In §3, the property in Definition 1.2 is used, in particular, to identify a larger family of examples of p-homogeneous nilspaces, within the standard class of so-called group nilspaces. Recall that a k-step group nilspace is constructed by taking any group G equipped with a filtration $G_\bullet $ of finite degree k, and equipping G with the associated Host–Kra cube sets  $\operatorname {\mathrm {C}}^n(G_\bullet ), n\geq 0$ ; we call this nilspace the group nilspace associated with the filtered group $(G,G_\bullet )$ (see, for instance, [Reference Host and Kra29, Ch. 6] for more background on Host–Kra cubes, and [Reference Candela7, §2.2.1] for the group nilspace construction).

We say that a filtration $G_\bullet =(G_i)_{i\geq 0}$ is p-homogeneous if for all i, for all $g\in G_i$ , we have $g^p\in G_{i+p-1}$ . We obtain the following description of p-homogeneous group nilspaces.

In particular, because morphisms between group nilspaces are the same thing as polynomial maps between the corresponding filtered groups (see, e.g., [Reference Candela7, §2.2.2]), Theorem 1.4 implies that given any filtered group $(G,G_{\bullet })$ , the filtration $G_\bullet $ is p-homogeneous if and only if for every polynomial $f\in \operatorname {\mathrm {poly}}(\mathbb {Z}^n,G_{\bullet })$ the restriction $f|_{[0,p-1]^n}$ yields a polynomial map in $\operatorname {\mathrm {poly}}(\mathbb {Z}_p^n,G_{\bullet })$ . Theorem 1.4 also implies in a simple way that for every p-homogeneous k-step nilspace defined on a finite cyclic group, the group must, in fact, be isomorphic to a subgroup of $\mathbb {Z}_{p^r}$ for $r=\lfloor ({k-1})/({p-1})\rfloor +1$ ; see Proposition 3.9. These groups $\mathbb {Z}_{p^r}$ underlie the non-classical polynomials of degree k on $\mathbb {F}_p^n$ , identified by Tao and Ziegler in [Reference Tao and Ziegler38] as adequate harmonics for an inverse theorem in characteristic p that is valid even in the so-called low-characteristic setting, that is, for $p\leq k$ . In the present approach, these cyclic groups also play a key role, but rather as basic objects used to describe more general p-homogeneous nilspaces. Using these general descriptions (detailed below), the inverse theorem can be deduced relatively easily, as explained at the end of this introduction.

Concerning more general p-homogeneous nilspaces (not necessarily group nilspaces), our main results in §3 include the following proposition. In particular, this indicates that p-homogeneous nilspaces are natural generalizations of elementary abelian p-groups.

When the property of structure groups in this proposition is combined with a certain lifting property for morphisms from elementary abelian p-groups into $\operatorname {\mathrm {X}}$ , we obtain a useful sufficient condition for $\operatorname {\mathrm {X}}$ to be p-homogeneous; see Proposition 3.12.

In §4, we prove a structure theorem for p-homogeneous nilspaces, which is also a key ingredient in our applications. The theorem describes general finite p-homogeneous nilspaces as images, under nilspace fibrations, of members of a much simpler class of p-homogeneous nilspaces, defined as follows. (To recall the notion of nilspace fibrations, also known by the original term fibre surjective morphisms, see [Reference Gutman, Manners and Varjú24, Definition 7.1] and [Reference Candela7, Definition 3.3.7]; essentially, the role of fibrations for compact nilspaces generalizes the role of continuous surjective homomorphisms for compact abelian groups.)

Definition 1.6. Let p be a prime, let $k,\ell \in \mathbb {N}$ with $k\geq \ell $ and let $r=r(k,\ell ,p):=\lfloor ({k-\ell })/({p-1})\rfloor +1$ . We define $\operatorname {\mathrm {\cal {U}}}_{k,\ell }$ to be the k-step p-homogeneous group nilspace (when the prime p needs to be specified we write $\operatorname {\mathrm {\cal {U}}}^{(p)}_{k,\ell }$ , but usually we omit this superscript) consisting of the cyclic group $G=\mathbb {Z}_{p^r}$ equipped with the p-homogeneous degree-k filtration $(G_i)_{i\geq 0}$ where $G_i = \mathbb {Z}_{p^r}$ for $i\in [0,\ell ]$ and $G_i=p^{\lfloor ({i-\ell -1})/({p-1})\rfloor +1}\mathbb {Z}_{p^r}$ for $i\ge \ell +1$ , that is, the filtration

$$ \begin{align*}\begin{array}{cccccccccccccccc} G_1 & & G_\ell & & G_{\ell+1} & & G_{\ell+p-1} & & G_{\ell+p}\\ \parallel & & \parallel & & \parallel & & \parallel & & \parallel\\ \mathbb{Z}_{p^r}& = \cdots = & \mathbb{Z}_{p^r} & \ge & p\mathbb{Z}_{p^r} & = \cdots = & p\mathbb{Z}_{p^r} &\geq & p^2\mathbb{Z}_{p^r}\;\cdots. \end{array} \end{align*} $$

We define $\mathcal {Q}_{p,k}$ to be the set of all p-homogeneous nilspaces $\operatorname {\mathrm {Y}}$ such that for some integers $a_\ell \ge 0$ ( $\ell \in [k]$ ), we have that $\operatorname {\mathrm {Y}}$ is isomorphic to the product nilspace $\prod _{\ell = 1}^k \operatorname {\mathrm {\cal {U}}}_{k,\ell }^{\,a_\ell }$ (the definition of product nilspaces, or of powers $\operatorname {\mathrm {X}}^a$ of a nilspace $\operatorname {\mathrm {X}}$ , is the natural one; see [Reference Candela7, §3.1.1]).

The cyclic groups underlying the nilspaces $\operatorname {\mathrm {\cal {U}}}_{k,\ell }$ agree with those underlying the non-classical polynomials in [Reference Tao and Ziegler38, Lemma 1.6(vi)], as mentioned previously.

We can now state the structure theorem, which establishes that the abelian group nilspaces in $\mathcal {Q}_{p,k}$ suffice to describe all p-homogeneous cfr nilspaces via fibrations.

This theorem is a refinement of (and was inspired by) results on finite nilspaces that were obtained by the third named author in [Reference Szegedy35], in particular [Reference Szegedy35, Theorem 6]. It is a central ingredient in our proofs of the regularity and inverse theorems, discussed below.

The second main result in §4 refines Theorem 1.7 for $k\leq p$ , as follows. This uses the so-called degree- $\ell $ nilspace structure on any abelian group $\operatorname {\mathrm {Z}}$ , denoted by $\mathcal {D}_\ell (\operatorname {\mathrm {Z}})$ , which is a standard way to turn $\operatorname {\mathrm {Z}}$ into an $\ell $ -step nilspace; see [Reference Candela7, Definition 2.2.30].

Theorem 1.8 is a key tool in our applications in ergodic theory. Note that the theorem covers the high-characteristic case $k<p$ , but also the additional case $k=p$ . For $k>p$ , we do not yet know what would be a corresponding qualitatively optimal refinement of Theorem 1.7. This leads to questions that we leave open in this paper; see Remark 4.12.

We close this introduction with a summary of the main applications.

In §5, we treat the applications in ergodic theory. Our results here concern the ergodic measure-preserving actions of $\mathbb {F}_p^\omega $ studied by Bergelson, Tao and Ziegler in [Reference Bergelson, Tao and Ziegler2, Reference Bergelson, Tao and Ziegler3], and specifically the Host–Kra factors of such $\mathbb {F}_p^\omega $ -systems, that is, the characteristic factors for uniformity seminorms on these systems. (Recall from [Reference Bergelson, Tao and Ziegler2, Definition 1.1] the notion of a G-system for a locally compact abelian group G; see also [Reference Candela and Szegedy12, Definition 5.9] for a definition of Host–Kra factors valid for G-systems more generally.) In the setting of $\mathbb {Z}$ -systems, the ergodic structure theorem from [Reference Host and Kra27] describes the Host–Kra factors as inverse limits of nilsystems. In the characteristic-p setting, the Host–Kra factors have hitherto been described in terms of Weyl systems, which were defined in [Reference Bergelson, Tao and Ziegler3, Definition 1.5, Theorem 1.6] specifically for this setting (see also [Reference Bergelson, Tao and Ziegler2, Theorem 4.8]). The different approach in the present paper unifies the descriptions of Host–Kra factors in these two settings via the common notion of a nilspace system, introduced in [Reference Candela and Szegedy12]. Let us recall that a (k-step) nilspace system $(\operatorname {\mathrm {X}},G)$ is a specific type of G-system, consisting of a compact (k-step) nilspace $\operatorname {\mathrm {X}}$ , and a topological group G acting continuously on $\operatorname {\mathrm {X}}$ via a group homomorphism $G \to \operatorname {\mathrm {\Theta }}(\operatorname {\mathrm {X}})$ , where $\operatorname {\mathrm {\Theta }}(\operatorname {\mathrm {X}})$ is the translation group of $\operatorname {\mathrm {X}}$ . (The translation group $\operatorname {\mathrm {\Theta }}(\operatorname {\mathrm {X}})$ is a group of automorphisms naturally defined on any nilspace $\operatorname {\mathrm {X}}$ , which can be viewed as a generalization of the regular action of abelian groups on themselves; see [Reference Candela7, §3.2.4]. The nilspace system can also be specified as a triple $(\operatorname {\mathrm {X}},G,\phi )$ , if the homomorphism $\phi :G\to \operatorname {\mathrm {\Theta }}(\operatorname {\mathrm {X}})$ needs to be emphasized; we can also add to the data an explicit filtration on G, preserved by $\phi $ , writing $(\operatorname {\mathrm {X}},(G,G_\bullet ),\phi )$ and calling this a filtered nilspace system; see [Reference Candela and Szegedy12, Definition 5.10].) Such a system can be viewed as a topological dynamical system, and if we equip $\operatorname {\mathrm {X}}$ with its Haar probability measure, then the nilspace system becomes a measure-preserving G-system. Nilspace systems were shown in [Reference Candela, González-Sánchez and Szegedy9, Reference Candela and Szegedy12] to yield extensions of the ergodic structure theorem (in particular, ergodic nilspace $\mathbb {Z}$ -systems are inverse limits of nilsystems [Reference Candela, González-Sánchez and Szegedy9, Theorem 5.1]). Let us mention that there are other descriptions of the characteristic factors of ergodic G-systems. There is, for instance, the concept of nilpotent system introduced in [Reference Shalom34, Definition 1.29] for the 2-step case with $G=\bigoplus _{p\in P}\mathbb {F}_p$ (where P is a multiset of primes) and, more generally, for any countable abelian group G in the 2-step case, there is a description of the 2-factor as a double coset space [Reference Shalom33, Theorem 1.21].

Let us say that a nilspace system is p-homogeneous if the underlying compact nilspace is p-homogeneous. We then have the following result.

In fact, a property markedly stronger than ergodicity holds here: the action of $\mathbb {F}_p^\omega $ on $\operatorname {\mathrm {X}}$ is uniquely ergodic (the notion of unique ergodicity may be recalled from [Reference Hasselblatt and Katok26, p. 87, §4.3.a]) and, for every n, the standard cube set $\operatorname {\mathrm {C}}^n(\mathbb {F}_p^\omega )$ also has a uniquely ergodic action on $\operatorname {\mathrm {C}}^n(\operatorname {\mathrm {X}})$ ; see Theorem 5.3. This stronger form of ergodicity follows from the main results in [Reference Candela and Szegedy12], which we combine with Theorem 1.3 to prove Theorem 1.9.

With Theorem 1.9, the study of these Host–Kra factors can be reduced to the study of p-homogeneous nilspace systems. Focusing on the latter systems, we then obtain more precise descriptions of the factors. In particular, for $k\leq p$ we have the following result, showing that the kth factor consists of an elementary abelian p-group equipped with a specific degree-k filtration.

This relates to previous work as follows. In the high-characteristic case $k<p$ , the results of [Reference Bergelson, Tao and Ziegler2] describe the kth Host–Kra factor as a k-fold iterated abelian extension by elementary abelian p-groups, with the cocycle for the jth extension being polynomial of degree at most j; see [Reference Bergelson, Tao and Ziegler2, Corollary 8.7]. Theorem 1.10 instead describes the factor as a nilspace system on the explicit product nilspace $\prod _{\ell =1}^k \mathcal {D}_\ell (\mathbb {Z}_p^{a_\ell })$ , which is a k-fold abelian bundle where the ith factor $\prod _{\ell =1}^i \mathcal {D}_\ell (\mathbb {Z}_p^{a_\ell })$ is a splitting extension of degree i (in the nilspace sense) of the previous factor $\prod _{\ell =1}^{i-1} \mathcal {D}_\ell (\mathbb {Z}_p^{a_\ell })$ by the group $\mathbb {Z}_p^{a_i}$ , and this description holds even for $k=p$ . We note that the translation group of this product nilspace can be described more explicitly, thus obtaining a complete description of the transformations in this factor in terms of polynomial maps between filtered elementary abelian p-groups; see Theorem 5.9.

Theorem 1.9 also enables progress in a closely related direction for these applications in ergodic theory, namely the direction concerning Abramov systems. Recall that an ergodic $\mathbb {F}_p^\omega $ -system is an Abramov system of order at most k if its $L^2$ -space is the closure of the linear span of phase polynomials of degree at most k; see Definition 5.11. The following interesting question arose in the work of Bergelson, Tao and Ziegler [Reference Bergelson, Tao and Ziegler2].

In [Reference Bergelson, Tao and Ziegler2], an affirmative answer is given in the case $k<p$ ([Reference Bergelson, Tao and Ziegler2, Theorem 1.19]), and this is believed to hold also for $k\geq p$ ; see [Reference Bergelson, Tao and Ziegler2, Remark 1.21]. For $k\geq p$ , a partial affirmative answer is given in [Reference Bergelson, Tao and Ziegler2, Theorem 1.20], showing that the factor is Abramov of order $O_k(1)$ .

We extend the affirmative answer to Question 1.11 as follows.

The proof of this theorem uses a reformulation of Question 1.11 as a problem purely about nilspaces, which we discuss in §5 (see Proposition 5.14 and Question 5.18) and which seems of interest as a possible way to make further progress on this question.

Finally, §6 contains applications concerning arithmetic combinatorics, which further illustrate the relevance of p-homogeneous nilspaces to higher-order Fourier analysis in characteristic p. In particular, we use these nilspaces to give a new proof of the inverse theorem for Gowers norms on $\mathbb {F}_p^n$ from [Reference Tao and Ziegler38], as well as regularity theorems in this setting. The idea of the proof is to start from the more general inverse theorem in the nilspace approach from [Reference Candela and Szegedy12, Reference Candela and Szegedy13], which tells us that a function f with non-trivial Gowers $U^{k+1}$ norm on $\mathbb {F}_p^n$ correlates with a complex-valued function which factors through a highly balanced morphism $\phi $ from $\mathbb {F}_p^n$ to a cfr k-step nilspace $\operatorname {\mathrm {X}}$ . Thanks to Theorem 1.4, we deduce that $\operatorname {\mathrm {X}}$ is p-homogeneous. Then Theorem 1.7 enables us to lift the morphism $\phi $ through a simpler finite abelian group nilspace, belonging to the class $\mathcal {Q}_{p,k}$ . This, combined with a standard Fourier decomposition on the abelian group underlying this nilspace, yields a non-classical phase polynomial of degree at most $ k$ correlating non-trivially with the original function f, as required. Moreover, for $k\leq p$ , using Theorem 1.8 instead of Theorem 1.7 we obtain the inverse theorem with classical polynomials; see Theorem 6.6. Note that the case $k=p$ of this result was obtained only recently, independently, in [Reference Berger, Sah, Sawhney and Tidor4].

The proof strategy on $\mathbb {F}_p^n$ outlined previously is similar to that used in the integer setting in [Reference Candela and Szegedy13]; both are rooted in the general inverse theorem from [Reference Candela and Szegedy13], and the differences arise only once we have to deal with p-homogeneous nilspaces here, instead of toral nilspaces in the integer setting. So far, this general strategy does not yield effective bounds. Currently, the proofs of inverse theorems for Gowers norms with the best effective bounds work with much more specific strategies in each setting; see the recent breakthroughs of Manners in the integer setting [Reference Manners31], and of Gowers and Milićević in the characteristic-p setting [Reference Gowers and Milićević16]. It would be very interesting to know if a more general strategy can also yield effective bounds, perhaps even an effective inverse theorem for general finite abelian groups.