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.