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Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group ${\Gamma}$ acting in a measure-preserving fashion on a probability space ${(X,\mu)}$, thus each group element ${\gamma \in \Gamma}$ gives rise to a measure-preserving map ${T^\gamma: X \rightarrow X}$. Define the third Gowers-Host-Kra seminorm ${\|f\|_{U^3(X)}}$ of a function ${f \in L^\infty(X)}$ via the formula

$\displaystyle \|f\|_{U^3(X)}^8 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2,h_3 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2,\omega_3 \in \{0,1\}}$

$\displaystyle {\mathcal C}^{\omega_1+\omega_2+\omega_3} f(T^{\omega_1 h_1 + \omega_2 h_2 + \omega_3 h_3} x)\ d\mu(x)$

where ${\Phi_n}$ is a Folner sequence for ${\Gamma}$ and ${{\mathcal C}: z \mapsto \overline{z}}$ is the complex conjugation map. One can show that this limit exists and is independent of the choice of Folner sequence, and that the ${\| \|_{U^3(X)}}$ seminorm is indeed a seminorm. A Conze-Lesigne system is an ergodic measure-preserving system in which the ${U^3(X)}$ seminorm is in fact a norm, thus ${\|f\|_{U^3(X)}>0}$ whenever ${f \in L^\infty(X)}$ is non-zero. Informally, this means that when one considers a generic parallelepiped in a Conze–Lesigne system ${X}$, the location of any vertex of that parallelepiped is more or less determined by the location of the other seven vertices. These are the important systems to understand in order to study “complexity two” patterns, such as arithmetic progressions of length four. While not all systems ${X}$ are Conze-Lesigne systems, it turns out that they always have a maximal factor ${Z^2(X)}$ that is a Conze-Lesigne system, known as the Conze-Lesigne factor or the second Host-Kra-Ziegler factor of the system, and this factor controls all the complexity two recurrence properties of the system.

The analogous theory in complexity one is well understood. Here, one replaces the ${U^3(X)}$ norm by the ${U^2(X)}$ norm

$\displaystyle \|f\|_{U^2(X)}^4 := \lim_{n \rightarrow \infty} {\bf E}_{h_1,h_2 \in \Phi_n} \int_X \prod_{\omega_1,\omega_2 \in \{0,1\}} {\mathcal C}^{\omega_1+\omega_2} f(T^{\omega_1 h_1 + \omega_2 h_2} x)\ d\mu(x)$

and the ergodic systems for which ${U^2}$ is a norm are called Kronecker systems. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group ${Z}$ equipped with Haar probability measure and a translation action ${T^\gamma \colon z \mapsto z + \phi(\gamma)}$ for some homomorphism ${\phi: \Gamma \rightarrow Z}$ with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).

We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) nilsystems, in which the space ${X}$ is a quotient ${G/\Lambda}$ of a two-step nilpotent Lie group ${G}$ by a lattice ${\Lambda}$ (equipped with Haar probability measure), and the action is given by a translation ${T^\gamma x = \phi(\gamma) x}$ for some group homomorphism ${\phi: \Gamma \rightarrow G}$. For instance, the Heisenberg ${{\bf Z}}$-nilsystem

$\displaystyle \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix}$

with a shift of the form

$\displaystyle Tx = \begin{pmatrix} 1 & \alpha & 0 \\ 0 & 1 & \beta \\ 0 & 0 & 1 \end{pmatrix} x$

for ${\alpha,\beta}$ two real numbers with ${1,\alpha,\beta}$ linearly independent over ${{\bf Q}}$, is a Conze-Lesigne system. As the base case of a well known result of Host and Kra, it is shown in fact that all Conze-Lesigne ${{\bf Z}}$-systems are inverse limits of nilsystems (previous results in this direction were obtained by Conze-Lesigne, Furstenberg-Weiss, and others). Similar results are known for ${\Gamma}$-systems when ${\Gamma}$ is finitely generated, thanks to the thesis work of Griesmer (with further proofs by Gutman-Lian and Candela-Szegedy). However, this is not the case once ${\Gamma}$ is not finitely generated; as a recent example of Shalom shows, Conze-Lesigne systems need not be the inverse limit of nilsystems in this case.

Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which ${G}$ is a locally compact Polish group rather than a Lie group:

Theorem 1 (Classification of Conze-Lesigne systems) Let ${\Gamma}$ be a countable abelian group, and ${X}$ an ergodic measure-preserving ${\Gamma}$-system. Then ${X}$ is a Conze-Lesigne system if and only if it is the inverse limit of translational systems ${G/\Lambda}$, where ${G}$ is a nilpotent locally compact Polish group of nilpotency class two, and ${\Lambda}$ is a lattice in ${G}$ (and also a lattice in the commutator group ${[G,G]}$), with ${G/\Lambda}$ equipped with the Haar probability measure and a translation action ${T^\gamma x = \phi(\gamma) x}$ for some homomorphism ${\phi: \Gamma \rightarrow G}$.

In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm ${U^3(G)}$ for an arbitrary finite abelian group ${G}$ (with no restrictions on the order of ${G}$, in particular our result handles the case of even and odd ${|G|}$ in a unified fashion). In principle, having a higher order version of this theorem will similarly allow us to derive inverse theorems for ${U^{s+1}(G)}$ norms for arbitrary ${s}$ and finite abelian ${G}$; we hope to investigate this further in future work.

We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension ${Z \rtimes_\rho K}$, where ${Z}$ is a Kronecker system (a rotational system on a compact abelian group ${Z = (Z,+)}$ and translation action ${\phi: \Gamma \rightarrow Z}$), ${K = (K,+)}$ is another compact abelian group, and the cocycle ${\rho = (\rho_\gamma)_{\gamma \in \Gamma}}$ is a collection of measurable maps ${\rho_\gamma: Z \rightarrow K}$ obeying the cocycle equation

$\displaystyle \rho_{\gamma_1+\gamma_2}(x) = \rho_{\gamma_1}(T^{\gamma_2} x) + \rho_{\gamma_2}(x) \ \ \ \ \ (1)$

for almost all ${x \in Z}$. Furthermore, ${\rho}$ is of “type two”, which means in this concrete setting that it obeys an additional equation

$\displaystyle \rho_\gamma(x + z_1 + z_2) - \rho_\gamma(x+z_1) - \rho_\gamma(x+z_2) + \rho_\gamma(x) \ \ \ \ \ (2)$

$\displaystyle = F(x + \phi(\gamma), z_1, z_2) - F(x,z_1,z_2)$

for all ${\gamma \in \Gamma}$ and almost all ${x,z_1,z_2 \in Z}$, and some measurable function ${F: Z^3 \rightarrow K}$; roughly speaking this asserts that ${\phi_\gamma}$ is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where ${Z}$ is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group ${K}$ is a circle ${{\bf T} = {\bf R}/{\bf Z}}$, one can use some Fourier analysis to convert (2) into the more tractable Conze-Lesigne equation

$\displaystyle \rho_\gamma(x+z) - \rho_\gamma(x) = F_z(x+\phi(\gamma)) - F_z(x) + c_z(\gamma) \ \ \ \ \ (3)$

for all ${\gamma \in \Gamma}$, all ${z \in Z}$, and almost all ${x \in Z}$, where for each ${z}$, ${F_z: Z \rightarrow K}$ is a measurable function, and ${c_z: \Gamma \rightarrow K}$ is a homomorphism. (For technical reasons it is often also convenient to enforce that ${F_z, c_z}$ depend in a measurable fashion on ${z}$; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group ${K}$ (as long as one has the measurability in ${z}$ mentioned previously), but the converse turns out to fail for some groups ${K}$, such as solenoid groups (e.g., inverse limits of ${{\bf R}/2^n{\bf Z}}$ as ${n \rightarrow \infty}$), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$. Putting together the ${K={\bf T}}$ and ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$ cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian Lie groups ${K}$ (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system ${X}$ as a translational system ${G/\Lambda}$, where the Host-Kra group ${G}$ is the set of all pairs ${(z, F_z)}$ that solve an equation of the form (3) (with these pairs acting on ${X \equiv Z \rtimes_\rho K}$ by the law ${(z,F_z) \cdot (x,k) := (x+z, k+F_z(x))}$), and ${\Lambda}$ is the stabiliser of a point in this system. This then establishes the theorem in the case when ${K}$ is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of ${X}$ form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)

There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of ${G}$ on ${X}$ is technically only a near-action rather than a genuine action, and as such one cannot directly define ${\Lambda}$ to be the stabiliser of a point without running into multiple problems. To fix this, one has to pass to a topological model of ${X}$ in which the action becomes continuous, and the stabilizer becomes well defined, although one then has to work a little more to check that the action is still transitive. This can be done via Gelfand duality; we proceed using a mixture of a construction from this book of Host and Kra, and the machinery in this recent paper of Asgar and myself.

Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case ${K = \frac{1}{N}{\bf Z}/{\bf Z}}$. As this group embeds into the torus ${{\bf T}}$, it is easy to use existing methods obtain (3) but with the homomorphism ${c_z}$ and the function ${F_z}$ taking values in ${{\bf R}/{\bf Z}}$ rather than in ${\frac{1}{N}{\bf Z}/{\bf Z}}$. The main task is then to fix up the homomorphism ${c_z}$ so that it takes values in ${\frac{1}{N}{\bf Z}/{\bf Z}}$, that is to say that ${Nc_z}$ vanishes. This only needs to be done locally near the origin, because the claim is easy when ${z}$ lies in the dense subgroup ${\phi(\Gamma)}$ of ${Z}$, and also because the claim can be shown to be additive in ${z}$. Near the origin one can leverage the Steinhaus lemma to make ${c_z}$ depend linearly (or more precisely, homomorphically) on ${z}$, and because the cocycle ${\rho}$ already takes values in ${\frac{1}{N}{\bf Z}/{\bf Z}}$, ${N\rho}$ vanishes and ${Nc_z}$ must be an eigenvalue of the system ${Z}$. But as ${Z}$ was assumed to be separable, there are only countably many eigenvalues, and by another application of Steinhaus and linearity one can then make ${Nc_z}$ vanish on an open neighborhood of the identity, giving the claim.

Tamar Ziegler and I have just uploaded to the arXiv two related papers: “Concatenation theorems for anti-Gowers-uniform functions and Host-Kra characteoristic factors” and “polynomial patterns in primes“, with the former developing a “quantitative Bessel inequality” for local Gowers norms that is crucial in the latter.

We use the term “concatenation theorem” to denote results in which structural control of a function in two or more “directions” can be “concatenated” into structural control in a joint direction. A trivial example of such a concatenation theorem is the following: if a function ${f: {\bf Z} \times {\bf Z} \rightarrow {\bf R}}$ is constant in the first variable (thus ${x \mapsto f(x,y)}$ is constant for each ${y}$), and also constant in the second variable (thus ${y \mapsto f(x,y)}$ is constant for each ${x}$), then it is constant in the joint variable ${(x,y)}$. A slightly less trivial example: if a function ${f: {\bf Z} \times {\bf Z} \rightarrow {\bf R}}$ is affine-linear in the first variable (thus, for each ${y}$, there exist ${\alpha(y), \beta(y)}$ such that ${f(x,y) = \alpha(y) x + \beta(y)}$ for all ${x}$) and affine-linear in the second variable (thus, for each ${x}$, there exist ${\gamma(x), \delta(x)}$ such that ${f(x,y) = \gamma(x)y + \delta(x)}$ for all ${y}$) then ${f}$ is a quadratic polynomial in ${x,y}$; in fact it must take the form

$\displaystyle f(x,y) = \epsilon xy + \zeta x + \eta y + \theta \ \ \ \ \ (1)$

for some real numbers ${\epsilon, \zeta, \eta, \theta}$. (This can be seen for instance by using the affine linearity in ${y}$ to show that the coefficients ${\alpha(y), \beta(y)}$ are also affine linear.)

The same phenomenon extends to higher degree polynomials. Given a function ${f: G \rightarrow K}$ from one additive group ${G}$ to another, we say that ${f}$ is of degree less than ${d}$ along a subgroup ${H}$ of ${G}$ if all the ${d}$-fold iterated differences of ${f}$ along directions in ${H}$ vanish, that is to say

$\displaystyle \partial_{h_1} \dots \partial_{h_d} f(x) = 0$

for all ${x \in G}$ and ${h_1,\dots,h_d \in H}$, where ${\partial_h}$ is the difference operator

$\displaystyle \partial_h f(x) := f(x+h) - f(x).$

(We adopt the convention that the only ${f}$ of degree less than ${0}$ is the zero function.)

We then have the following simple proposition:

Proposition 1 (Concatenation of polynomiality) Let ${f: G \rightarrow K}$ be of degree less than ${d_1}$ along one subgroup ${H_1}$ of ${G}$, and of degree less than ${d_2}$ along another subgroup ${H_2}$ of ${G}$, for some ${d_1,d_2 \geq 1}$. Then ${f}$ is of degree less than ${d_1+d_2-1}$ along the subgroup ${H_1+H_2}$ of ${G}$.

Note the previous example was basically the case when ${G = {\bf Z} \times {\bf Z}}$, ${H_1 = {\bf Z} \times \{0\}}$, ${H_2 = \{0\} \times {\bf Z}}$, ${K = {\bf R}}$, and ${d_1=d_2=2}$.

Proof: The claim is trivial for ${d_1=1}$ or ${d_2=1}$ (in which ${f}$ is constant along ${H_1}$ or ${H_2}$ respectively), so suppose inductively ${d_1,d_2 \geq 2}$ and the claim has already been proven for smaller values of ${d_1-1}$.

We take a derivative in a direction ${h_1 \in H_1}$ along ${h_1}$ to obtain

$\displaystyle T^{-h_1} f = f + \partial_{h_1} f$

where ${T^{-h_1} f(x) = f(x+h_1)}$ is the shift of ${f}$ by ${-h_1}$. Then we take a further shift by a direction ${h_2 \in H_2}$ to obtain

$\displaystyle T^{-h_1-h_2} f = T^{-h_2} f + T^{-h_2} \partial_{h_1} f = f + \partial_{h_2} f + T^{-h_2} \partial_{h_1} f$

$\displaystyle \partial_{h_1+h_2} f = \partial_{h_2} f + T^{-h_2} \partial_{h_1} f.$

Since ${f}$ has degree less than ${d_1}$ along ${H_1}$ and degree less than ${d_2}$ along ${H_2}$, ${\partial_{h_1} f}$ has degree less than ${d_1-1}$ along ${H_1}$ and less than ${d_2}$ along ${H_2}$, so is degree less than ${d_1+d_2-2}$ along ${H_1+H_2}$ by induction hypothesis. Similarly ${\partial_{h_2} f}$ is also of degree less than ${d_1+d_2-2}$ along ${H_1+H_2}$. Combining this with the cocycle equation we see that ${\partial_{h_1+h_2}f}$ is of degree less than ${d_1+d_2-2}$ along ${H_1+H_2}$ for any ${h_1+h_2 \in H_1+H_2}$, and hence ${f}$ is of degree less than ${d_1+d_2-1}$ along ${H_1+H_2}$, as required. $\Box$

While this proposition is simple, it already illustrates some basic principles regarding how one would go about proving a concatenation theorem:

• (i) One should perform induction on the degrees ${d_1,d_2}$ involved, and take advantage of the recursive nature of degree (in this case, the fact that a function is of less than degree ${d}$ along some subgroup ${H}$ of directions iff all of its first derivatives along ${H}$ are of degree less than ${d-1}$).
• (ii) Structure is preserved by operations such as addition, shifting, and taking derivatives. In particular, if a function ${f}$ is of degree less than ${d}$ along some subgroup ${H}$, then any derivative ${\partial_k f}$ of ${f}$ is also of degree less than ${d}$ along ${H}$, even if ${k}$ does not belong to ${H}$.

Here is another simple example of a concatenation theorem. Suppose an at most countable additive group ${G}$ acts by measure-preserving shifts ${T: g \mapsto T^g}$ on some probability space ${(X, {\mathcal X}, \mu)}$; we call the pair ${(X,T)}$ (or more precisely ${(X, {\mathcal X}, \mu, T)}$) a ${G}$-system. We say that a function ${f \in L^\infty(X)}$ is a generalised eigenfunction of degree less than ${d}$ along some subgroup ${H}$ of ${G}$ and some ${d \geq 1}$ if one has

$\displaystyle T^h f = \lambda_h f$

almost everywhere for all ${h \in H}$, and some functions ${\lambda_h \in L^\infty(X)}$ of degree less than ${d-1}$ along ${H}$, with the convention that a function has degree less than ${0}$ if and only if it is equal to ${1}$. Thus for instance, a function ${f}$ is an generalised eigenfunction of degree less than ${1}$ along ${H}$ if it is constant on almost every ${H}$-ergodic component of ${G}$, and is a generalised function of degree less than ${2}$ along ${H}$ if it is an eigenfunction of the shift action on almost every ${H}$-ergodic component of ${G}$. A basic example of a higher order eigenfunction is the function ${f(x,y) := e^{2\pi i y}}$ on the skew shift ${({\bf R}/{\bf Z})^2}$ with ${{\bf Z}}$ action given by the generator ${T(x,y) := (x+\alpha,y+x)}$ for some irrational ${\alpha}$. One can check that ${T^h f = \lambda_h f}$ for every integer ${h}$, where ${\lambda_h: x \mapsto e^{2\pi i \binom{h}{2} \alpha} e^{2\pi i h x}}$ is a generalised eigenfunction of degree less than ${2}$ along ${{\bf Z}}$, so ${f}$ is of degree less than ${3}$ along ${{\bf Z}}$.

We then have

Proposition 2 (Concatenation of higher order eigenfunctions) Let ${(X,T)}$ be a ${G}$-system, and let ${f \in L^\infty(X)}$ be a generalised eigenfunction of degree less than ${d_1}$ along one subgroup ${H_1}$ of ${G}$, and a generalised eigenfunction of degree less than ${d_2}$ along another subgroup ${H_2}$ of ${G}$, for some ${d_1,d_2 \geq 1}$. Then ${f}$ is a generalised eigenfunction of degree less than ${d_1+d_2-1}$ along the subgroup ${H_1+H_2}$ of ${G}$.

The argument is almost identical to that of the previous proposition and is left as an exercise to the reader. The key point is the point (ii) identified earlier: the space of generalised eigenfunctions of degree less than ${d}$ along ${H}$ is preserved by multiplication and shifts, as well as the operation of “taking derivatives” ${f \mapsto \lambda_k}$ even along directions ${k}$ that do not lie in ${H}$. (To prove this latter claim, one should restrict to the region where ${f}$ is non-zero, and then divide ${T^k f}$ by ${f}$ to locate ${\lambda_k}$.)

A typical example of this proposition in action is as follows: consider the ${{\bf Z}^2}$-system given by the ${3}$-torus ${({\bf R}/{\bf Z})^3}$ with generating shifts

$\displaystyle T^{(1,0)}(x,y,z) := (x+\alpha,y,z+y)$

$\displaystyle T^{(0,1)}(x,y,z) := (x,y+\alpha,z+x)$

for some irrational ${\alpha}$, which can be checked to give a ${{\bf Z}^2}$ action

$\displaystyle T^{(n,m)}(x,y,z) := (x+n\alpha, y+m\alpha, z+ny+mx+nm\alpha).$

The function ${f(x,y,z) := e^{2\pi i z}}$ can then be checked to be a generalised eigenfunction of degree less than ${2}$ along ${{\bf Z} \times \{0\}}$, and also less than ${2}$ along ${\{0\} \times {\bf Z}}$, and less than ${3}$ along ${{\bf Z}^2}$. One can view this example as the dynamical systems translation of the example (1) (see this previous post for some more discussion of this sort of correspondence).

The main results of our concatenation paper are analogues of these propositions concerning a more complicated notion of “polynomial-like” structure that are of importance in additive combinatorics and in ergodic theory. On the ergodic theory side, the notion of structure is captured by the Host-Kra characteristic factors ${Z^{ of a ${G}$-system ${X}$ along a subgroup ${H}$. These factors can be defined in a number of ways. One is by duality, using the Gowers-Host-Kra uniformity seminorms (defined for instance here) ${\| \|_{U^d_H(X)}}$. Namely, ${Z^{ is the factor of ${X}$ defined up to equivalence by the requirement that

$\displaystyle \|f\|_{U^d_H(X)} = 0 \iff {\bf E}(f | Z^{

An equivalent definition is in terms of the dual functions ${{\mathcal D}^d_H(f)}$ of ${f}$ along ${H}$, which can be defined recursively by setting ${{\mathcal D}^0_H(f) = 1}$ and

$\displaystyle {\mathcal D}^d_H(f) = {\bf E}_h T^h f {\mathcal D}^{d-1}( f \overline{T^h f} )$

where ${{\bf E}_h}$ denotes the ergodic average along a Følner sequence in ${G}$ (in fact one can also define these concepts in non-amenable abelian settings as per this previous post). The factor ${Z^{ can then be alternately defined as the factor generated by the dual functions ${{\mathcal D}^d_H(f)}$ for ${f \in L^\infty(X)}$.

In the case when ${G=H={\bf Z}}$ and ${X}$ is ${G}$-ergodic, a deep theorem of Host and Kra shows that the factor ${Z^{ is equivalent to the inverse limit of nilsystems of step less than ${d}$. A similar statement holds with ${{\bf Z}}$ replaced by any finitely generated group by Griesmer, while the case of an infinite vector space over a finite field was treated in this paper of Bergelson, Ziegler, and myself. The situation is more subtle when ${X}$ is not ${G}$-ergodic, or when ${X}$ is ${G}$-ergodic but ${H}$ is a proper subgroup of ${G}$ acting non-ergodically, when one has to start considering measurable families of directional nilsystems; see for instance this paper of Austin for some of the subtleties involved (for instance, higher order group cohomology begins to become relevant!).

One of our main theorems is then

Proposition 3 (Concatenation of characteristic factors) Let ${(X,T)}$ be a ${G}$-system, and let ${f}$ be measurable with respect to the factor ${Z^{ and with respect to the factor ${Z^{ for some ${d_1,d_2 \geq 1}$ and some subgroups ${H_1,H_2}$ of ${G}$. Then ${f}$ is also measurable with respect to the factor ${Z^{.

We give two proofs of this proposition in the paper; an ergodic-theoretic proof using the Host-Kra theory of “cocycles of type ${ (along a subgroup ${H}$)”, which can be used to inductively describe the factors ${Z^{, and a combinatorial proof based on a combinatorial analogue of this proposition which is harder to state (but which roughly speaking asserts that a function which is nearly orthogonal to all bounded functions of small ${U^{d_1}_{H_1}}$ norm, and also to all bounded functions of small ${U^{d_2}_{H_2}}$ norm, is also nearly orthogonal to alll bounded functions of small ${U^{d_1+d_2-1}_{H_1+H_2}}$ norm). The combinatorial proof parallels the proof of Proposition 2. A key point is that dual functions ${F := {\mathcal D}^d_H(f)}$ obey a property analogous to being a generalised eigenfunction, namely that

$\displaystyle T^h F = {\bf E}_k \lambda_{h,k} F_k$

where ${F_k := T^k F}$ and ${\lambda_{h,k} := {\mathcal D}^{d-1}( T^h f \overline{T^k f} )}$ is a “structured function of order ${d-1}$” along ${H}$. (In the language of this previous paper of mine, this is an assertion that dual functions are uniformly almost periodic of order ${d}$.) Again, the point (ii) above is crucial, and in particular it is key that any structure that ${F}$ has is inherited by the associated functions ${\lambda_{h,k}}$ and ${F_k}$. This sort of inheritance is quite easy to accomplish in the ergodic setting, as there is a ready-made language of factors to encapsulate the concept of structure, and the shift-invariance and ${\sigma}$-algebra properties of factors make it easy to show that just about any “natural” operation one performs on a function measurable with respect to a given factor, returns a function that is still measurable in that factor. In the finitary combinatorial setting, though, encoding the fact (ii) becomes a remarkably complicated notational nightmare, requiring a huge amount of “epsilon management” and “second-order epsilon management” (in which one manages not only scalar epsilons, but also function-valued epsilons that depend on other parameters). In order to avoid all this we were forced to utilise a nonstandard analysis framework for the combinatorial theorems, which made the arguments greatly resemble the ergodic arguments in many respects (though the two settings are still not equivalent, see this previous blog post for some comparisons between the two settings). Unfortunately the arguments are still rather complicated.

For combinatorial applications, dual formulations of the concatenation theorem are more useful. A direct dualisation of the theorem yields the following decomposition theorem: a bounded function which is small in ${U^{d_1+d_2-1}_{H_1+H_2}}$ norm can be split into a component that is small in ${U^{d_1}_{H_1}}$ norm, and a component that is small in ${U^{d_2}_{H_2}}$ norm. (One may wish to understand this type of result by first proving the following baby version: any function that has mean zero on every coset of ${H_1+H_2}$, can be decomposed as the sum of a function that has mean zero on every ${H_1}$ coset, and a function that has mean zero on every ${H_2}$ coset. This is dual to the assertion that a function that is constant on every ${H_1}$ coset and constant on every ${H_2}$ coset, is constant on every ${H_1+H_2}$ coset.) Combining this with some standard “almost orthogonality” arguments (i.e. Cauchy-Schwarz) give the following Bessel-type inequality: if one has a lot of subgroups ${H_1,\dots,H_k}$ and a bounded function is small in ${U^{2d-1}_{H_i+H_j}}$ norm for most ${i,j}$, then it is also small in ${U^d_{H_i}}$ norm for most ${i}$. (Here is a baby version one may wish to warm up on: if a function ${f}$ has small mean on ${({\bf Z}/p{\bf Z})^2}$ for some large prime ${p}$, then it has small mean on most of the cosets of most of the one-dimensional subgroups of ${({\bf Z}/p{\bf Z})^2}$.)

There is also a generalisation of the above Bessel inequality (as well as several of the other results mentioned above) in which the subgroups ${H_i}$ are replaced by more general coset progressions ${H_i+P_i}$ (of bounded rank), so that one has a Bessel inequailty controlling “local” Gowers uniformity norms such as ${U^d_{P_i}}$ by “global” Gowers uniformity norms such as ${U^{2d-1}_{P_i+P_j}}$. This turns out to be particularly useful when attempting to compute polynomial averages such as

$\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} f(n) g(n+r^2) h(n+2r^2) \ \ \ \ \ (2)$

for various functions ${f,g,h}$. After repeated use of the van der Corput lemma, one can control such averages by expressions such as

$\displaystyle \sum_{n \leq N} \sum_{h,m,k \leq \sqrt{N}} f(n) f(n+mh) f(n+mk) f(n+m(h+k))$

(actually one ends up with more complicated expressions than this, but let’s use this example for sake of discussion). This can be viewed as an average of various ${U^2}$ Gowers uniformity norms of ${f}$ along arithmetic progressions of the form ${\{ mh: h \leq \sqrt{N}\}}$ for various ${m \leq \sqrt{N}}$. Using the above Bessel inequality, this can be controlled in turn by an average of various ${U^3}$ Gowers uniformity norms along rank two generalised arithmetic progressions of the form ${\{ m_1 h_1 + m_2 h_2: h_1,h_2 \le \sqrt{N}\}}$ for various ${m_1,m_2 \leq \sqrt{N}}$. But for generic ${m_1,m_2}$, this rank two progression is close in a certain technical sense to the “global” interval ${\{ n: n \leq N \}}$ (this is ultimately due to the basic fact that two randomly chosen large integers are likely to be coprime, or at least have a small gcd). As a consequence, one can use the concatenation theorems from our first paper to control expressions such as (2) in terms of global Gowers uniformity norms. This is important in number theoretic applications, when one is interested in computing sums such as

$\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} \mu(n) \mu(n+r^2) \mu(n+2r^2)$

or

$\displaystyle \sum_{n \leq N} \sum_{r \leq \sqrt{N}} \Lambda(n) \Lambda(n+r^2) \Lambda(n+2r^2)$

where ${\mu}$ and ${\Lambda}$ are the Möbius and von Mangoldt functions respectively. This is because we are able to control global Gowers uniformity norms of such functions (thanks to results such as the proof of the inverse conjecture for the Gowers norms, the orthogonality of the Möbius function with nilsequences, and asymptotics for linear equations in primes), but much less control is currently available for local Gowers uniformity norms, even with the assistance of the generalised Riemann hypothesis (see this previous blog post for some further discussion).

By combining these tools and strategies with the “transference principle” approach from our previous paper (as improved using the recent “densification” technique of Conlon, Fox, and Zhao, discussed in this previous post), we are able in particular to establish the following result:

Theorem 4 (Polynomial patterns in the primes) Let ${P_1,\dots,P_k: {\bf Z} \rightarrow {\bf Z}}$ be polynomials of degree at most ${d}$, whose degree ${d}$ coefficients are all distinct, for some ${d \geq 1}$. Suppose that ${P_1,\dots,P_k}$ is admissible in the sense that for every prime ${p}$, there are ${n,r}$ such that ${n+P_1(r),\dots,n+P_k(r)}$ are all coprime to ${p}$. Then there exist infinitely many pairs ${n,r}$ of natural numbers such that ${n+P_1(r),\dots,n+P_k(r)}$ are prime.

Furthermore, we obtain an asymptotic for the number of such pairs ${n,r}$ in the range ${n \leq N}$, ${r \leq N^{1/d}}$ (actually for minor technical reasons we reduce the range of ${r}$ to be very slightly less than ${N^{1/d}}$). In fact one could in principle obtain asymptotics for smaller values of ${r}$, and relax the requirement that the degree ${d}$ coefficients be distinct with the requirement that no two of the ${P_i}$ differ by a constant, provided one had good enough local uniformity results for the Möbius or von Mangoldt functions. For instance, we can obtain an asymptotic for triplets of the form ${n, n+r,n+r^d}$ unconditionally for ${d \leq 5}$, and conditionally on GRH for all ${d}$, using known results on primes in short intervals on average.

The ${d=1}$ case of this theorem was obtained in a previous paper of myself and Ben Green (using the aforementioned conjectures on the Gowers uniformity norm and the orthogonality of the Möbius function with nilsequences, both of which are now proven). For higher ${d}$, an older result of Tamar and myself was able to tackle the case when ${P_1(0)=\dots=P_k(0)=0}$ (though our results there only give lower bounds on the number of pairs ${(n,r)}$, and no asymptotics). Both of these results generalise my older theorem with Ben Green on the primes containing arbitrarily long arithmetic progressions. The theorem also extends to multidimensional polynomials, in which case there are some additional previous results; see the paper for more details. We also get a technical refinement of our previous result on narrow polynomial progressions in (dense subsets of) the primes by making the progressions just a little bit narrower in the case of the density of the set one is using is small.

Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.

As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:

Theorem 1 (Inverse theorem for Gowers norms) Let ${N \geq 1}$ and ${s \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z} \rightarrow [-1,1]}$ is a function supported on ${[N] := \{1,\dots,N\}}$ such that

$\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$

Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that

$\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.$

This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:

Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} f(T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}x)\ d\mu(x)$

$\displaystyle > 0$

for all non-zero ${f \in L^\infty(X)}$ (all ${L^p}$ spaces are real-valued in this post). Then ${(X,T)}$ is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree ${\leq s}$ nilsystems, that is to say systems of the form ${(G/\Gamma, x \mapsto gx)}$ for some degree ${\leq s}$ filtered nilmanifold ${G/\Gamma}$ and a group element ${g \in G}$ that acts ergodically on ${G/\Gamma}$.

It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of ${{\bf Z}}$-actions, the connection is less clear.

One can split Theorem 2 into two components:

Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has

$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} f\ d\mu$

$\displaystyle > 0$

for all non-zero ${f \in L^\infty(X)}$, where ${T^h f := f \circ T^h}$. Then ${(X,T)}$ is a factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems.

Theorem 4 (Pro-nilsystems closed under factors) Let ${s \geq 1}$ be an integer. Then any factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems, is again an inverse limit of ergodic degree ${\leq s}$ nilsystems.

Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)

The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:

Proposition 5 Theorem 1 implies Theorem 3.

As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.

Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our paper “An inverse theorem for the uniformity seminorms associated with the action of $F^\infty_p$“. This paper establishes the ergodic inverse theorems that are needed in our other recent paper to establish the inverse conjecture for the Gowers norms over finite fields in high characteristic (and to establish a partial result in low characteristic), as follows:

Theorem. Let ${\Bbb F}$ be a finite field of characteristic p.  Suppose that $X = (X,{\mathcal B},\mu)$ is a probability space with an ergodic measure-preserving action $(T_g)_{g \in {\Bbb F}^\omega}$ of ${\Bbb F}^\omega$.  Let $f \in L^\infty(X)$ be such that the Gowers-Host-Kra seminorm $\|f\|_{U^k(X)}$ (defined in a previous post) is non-zero.

1. In the high-characteristic case $p \geq k$, there exists a phase polynomial g of degree <k (as defined in the previous post) such that $|\int_X f \overline{g}\ d\mu| > 0$.
2. In general characteristic, there exists a phase polynomial of degree <C(k) for some C(k) depending only on k such that $|\int_X f \overline{g}\ d\mu| > 0$.

This theorem is closely analogous to a similar theorem of Host and Kra on ergodic actions of ${\Bbb Z}$, in which the role of phase polynomials is played by functions that arise from nilsystem factors of X.  Indeed, our arguments rely heavily on the machinery of Host and Kra.

The paper is rather technical (60+ pages!) and difficult to describe in detail here, but I will try to sketch out (in very broad brush strokes) what the key steps in the proof of part 2 of the theorem are.  (Part 1 is similar but requires a more delicate analysis at various stages, keeping more careful track of the degrees of various polynomials.)