###### Abstract

The purpose of this paper is to study the phenomenon of large intersections in
the framework of multiple recurrence for measure-preserving actions of
countable abelian groups. Among other things, we show:

  1. 1.

If $G$ is a countable abelian group and $\varphi,\psi:G\to G$ are
homomorphisms such that $\varphi(G)$, $\psi(G)$, and $(\psi-\varphi)(G)$ have
finite index in $G$, then for every ergodic measure-preserving system
$(X,\mathcal{B},\mu,(T_{g})_{g\in G})$, every set $A\in\mathcal{B}$, and every
$\varepsilon>0$, the set $\\{g\in G:\mu(A\cap T_{\varphi(g)}^{-1}A\cap
T_{\psi(g)}^{-1}A)>\mu(A)^{3}-\varepsilon\\}$ is syndetic.

  2. 2.

If $G$ is a countable abelian group and $r,s\in\mathbb{Z}$ are integers such
that $rG$, $sG$, and $(r\pm s)G$ have finite index in $G$, then for every
ergodic measure-preserving system $(X,\mathcal{B},\mu,(T_{g})_{g\in G})$,
every set $A\in\mathcal{B}$, and every $\varepsilon>0$, the set $\\{g\in
G:\mu(A\cap T_{rg}^{-1}A\cap T_{sg}^{-1}A\cap
T_{(r+s)g}^{-1}A)>\mu(A)^{4}-\varepsilon\\}$ is syndetic.

In particular, these extend and generalize results from [BHKr05] concerning
$\mathbb{Z}$-actions and [BTZ15] on $\mathbb{F}_{p}^{\infty}$-actions. Using
an ergodic version of the Furstenberg correspondence principle, we obtain new
combinatorial applications. We also discuss numerous examples shedding light
on the necessity of the various hypotheses above. Our results lead to a number
of interesting questions and conjectures, formulated in the introduction and
at the end of the paper.

title = Multiple Recurrence and Large Intersections for Abelian Group Actions,
author = Ethan Ackelsberg, Vitaly Bergelson, and Andrew Best, plaintextauthor
= Ethan Ackelsberg, Vitaly Bergelson, Andrew Best, keywords = multiple
recurrence, characteristic factors, cocycles, group actions, year=2021,
number=18, received=15 January 2021, published=1 October 2021,
doi=10.19086/da.28877,

[classification=text]

##  1 Introduction

The purpose of this paper is to study the phenomenon of _large intersections_
in the framework of multiple recurrence for measure-preserving systems. For
context, let us juxtapose some classical and some more recent results for
$\mathbb{Z}$-actions. First, a definition:

######  Definition 1.1.

Let $(G,+)$ be a countable abelian group. A subset $S\subset G$ is called
_syndetic_ if the union of finitely many translates of $S$ covers $G$, i.e.,
if there exist $g_{1},\ldots,g_{k}\in G$ such that
$G=\bigcup_{i=1}^{k}(g_{i}+S)$, where by $g_{i}+S$ we understand the set
$\\{g_{i}+s:s\in S\\}$.

######  Theorem 1.2 (Khintchine’s recurrence theorem [Kh35]).

For any invertible probability measure-preserving system
$(X,\mathcal{B},\mu,T)$, any $\varepsilon>0$, and any $A\in\mathcal{B}$, the
set

$\\{n\in\mathbb{Z}:\mu(A\cap T^{n}A)>\mu(A)^{2}-\varepsilon\\}$ (1)

is syndetic.

This improves on the classical Poincaré recurrence theorem in two ways: it
provides a lower bound on the size of the intersection and shows that the set
of return times is large. Notice that the bound $\mu(A)^{2}$ is optimal, since
for mixing systems, $\mu(A\cap T^{n}A)\to\mu(A)^{2}$. Colloquially, we say
that Theorem 1.2 shows _syndeticity of large intersections for single
recurrence_. A natural question to ask is whether similar improvements can be
made for results about _multiple recurrence_. Recall the Furstenberg multiple
recurrence theorem (also known as the ergodic Szemerédi theorem):

######  Theorem 1.3 (Furstenberg [Fu77]).

For any invertible probability measure-preserving system
$(X,\mathcal{B},\mu,T)$, any $A\in\mathcal{B}$ with $\mu(A)>0$, and any
positive integer $k\geq 1$,

$\displaystyle\liminf_{N-M\to\infty}{\frac{1}{N-M}\sum_{n=M}^{N-1}{\mu\left(A\cap
T^{n}A\cap\cdots\cap T^{kn}A\right)}}>0.$ (2)

An immediate consequence of the positivity of this limit is that there is a
constant $c>0$ such that the set

$\displaystyle\left\\{n\in\mathbb{Z}:\mu\left(A\cap T^{n}A\cap\cdots\cap
T^{kn}A\right)>c\right\\}$ (3)

is syndetic. In trying to find the optimal such $c>0$ for $k\geq 2$, a curious
picture emerges:

######  Theorem 1.4 ([BHKr05, Theorems 1.2 and 1.3]).

  1. (i)

For any ergodic invertible probability measure-preserving system
$(X,\mathcal{B},\mu,T)$, any $\varepsilon>0$, and any $A\in\mathcal{B}$, the
set

$\\{n\in\mathbb{Z}:\mu(A\cap T^{n}A\cap T^{2n}A)>\mu(A)^{3}-\varepsilon\\}$
(4)

is syndetic.

  2. (ii)

For any ergodic invertible probability measure-preserving system
$(X,\mathcal{B},\mu,T)$, any $\varepsilon>0$, and any $A\in\mathcal{B}$, the
set

$\\{n\in\mathbb{Z}:\mu(A\cap T^{n}A\cap T^{2n}A\cap
T^{3n}A)>\mu(A)^{4}-\varepsilon\\}$ (5)

is syndetic.

  3. (iii)

There exists an ergodic system $(X,\mathcal{B},\mu,T)$ with the following
property: for any integer $\ell\geq 1$, there is a set
$A=A(\ell)\in\mathcal{B}$ of positive measure such that

$\mu(A\cap T^{n}A\cap T^{2n}A\cap T^{3n}A\cap
T^{4n}A)\leq\frac{1}{2}\mu(A)^{\ell}$ (6)

for every integer $n\neq 0$.

In other words, parts (i) and (ii) of Theorem 1.4 show syndeticity of large
returns for double (resp. triple) recurrence, and part (iii) shows that a
natural generalization of parts (i) and (ii) for longer expressions cannot
proceed.

The results of parts (i) and (ii) were subsequently generalized by
Frantzikinakis,111We do not give a full statement of [Fr08, Theorem C], as it
deals with various polynomial configurations that are beyond the scope of this
paper. The linear patterns that appear here come as a special case. and part
(iii) was generalized by Donoso, Le, Moreira, and Sun:222Part (iii) of Theorem
1.5 also follows from our more general considerations in Section 12 (see
Corollary 12.8).

######  Theorem 1.5 ([Fr08], special case of Theorem C; [DLeMSu21], Theorem
1.5).

  1. (i)

Let $a,b\in\mathbb{Z}$ be nonzero and distinct. For any ergodic invertible
probability measure-preserving system $(X,\mathcal{B},\mu,T)$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\\{n\in\mathbb{Z}:\mu(A\cap T^{an}A\cap T^{bn}A)>\mu(A)^{3}-\varepsilon\\}$
(7)

is syndetic.

  2. (ii)

Let $a,b,c\in\mathbb{Z}$ be nonzero and distinct such that $(0,a,b,c)$ forms a
parallelogram in the sense that $0+c=a+b$. For any ergodic invertible
probability measure-preserving system $(X,\mathcal{B},\mu,T)$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\\{n\in\mathbb{Z}:\mu(A\cap T^{an}A\cap T^{bn}A\cap
T^{cn}A)>\mu(A)^{4}-\varepsilon\\}$ (8)

is syndetic.

  3. (iii)

Let $a,b,c,d\in\mathbb{Z}$ be nonzero and distinct. There exists an ergodic
invertible probability measure-preserving system $(X,\mathcal{B},\mu,T)$ such
that, for any $\ell\geq 1$, there is a set $A=A(\ell)\in\mathcal{B}$ of
positive measure such that

$\mu(A\cap T^{an}A\cap T^{bn}A\cap T^{cn}A\cap
T^{dn}A)\leq\frac{1}{2}\mu(A)^{\ell}$ (9)

for every integer $n\neq 0$.

An astute reader may observe at this point that Theorem 1.4, though stated
only for ergodic systems, must hold for certain non-ergodic systems. Indeed,
by Theorem 1.5, the set $\\{n\in\mathbb{Z}:\mu(A\cap T^{2n}A\cap
T^{4n}A)>\mu(A)^{3}-\varepsilon\\}$ is syndetic for ergodic $T$. But this is
the same as the set $\\{n\in\mathbb{Z}:\mu(A\cap S^{n}A\cap
S^{2n}A)>\mu(A)^{3}-\varepsilon\\}$ for the transformation $S=T^{2}$, and $S$
need not be ergodic. A similar argument shows that any power of an ergodic
system still satisfies the conclusion of Theorem 1.4. The essential fact about
these systems is that their ergodic decompositions have only finitely many
ergodic components. For non-ergodic systems with more complicated ergodic
decomposition, the conclusion of Theorem 1.4(i) may fail:

######  Theorem 1.6 ([BHKr05], Theorem 2.1).

There exists a non-ergodic system $(X,\mathcal{B},\mu,T)$ with the following
property: for any integer $\ell\geq 1$, there is a set
$A=A(\ell)\in\mathcal{B}$ of positive measure such that

$\mu(A\cap T^{n}A\cap T^{2n}A)\leq\frac{1}{2}\mu(A)^{\ell}$ (10)

for every integer $n\neq 0$.

It is reasonable to inquire whether the above results have a version for
actions of other groups. As a reminder, if $G=(G,+)$ is a countable abelian
group which acts on a probability space $(X,\mathcal{B},\mu)$ by measure-
preserving automorphisms $(T_{g})_{g\in G}$, then we refer to the quadruple
$\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ as a measure-preserving
$G$-system, or $G$-system for short. Along these lines, some results were
obtained in [BTZ15] for measure-preserving actions of the additive group
$\mathbb{F}_{p}^{\infty}=\bigoplus_{n=1}^{\infty}{\mathbb{F}_{p}}$, the direct
sum of countably many copies of a finite field of prime order $p$, which has a
natural vector space structure over $\mathbb{F}_{p}$.

######  Theorem 1.7 ([BTZ15, Theorems 1.12 and 1.13]).

  1. (i)

Fix a prime $p>2$ and distinct elements $c_{0},c_{1},c_{2}\in\mathbb{F}_{p}$.
For any ergodic measure-preserving $\mathbb{F}_{p}^{\infty}$-system
$(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{F}_{p}^{\infty}})$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\\{g\in\mathbb{F}_{p}^{\infty}:\mu(T_{c_{0}g}A\cap T_{c_{1}g}A\cap
T_{c_{2}g}A)>\mu(A)^{3}-\varepsilon\\}$ (11)

is syndetic.

  2. (ii)

Fix a prime $p>3$ and distinct elements
$c_{0},c_{1},c_{2},c_{3}\in\mathbb{F}_{p}$ which form a parallelogram in the
sense that $c_{i}+c_{j}=c_{k}+c_{\ell}$ for some permutation
$\\{i,j,k,\ell\\}$ of $\\{0,1,2,3\\}$. For any ergodic measure-preserving
$\mathbb{F}_{p}^{\infty}$-system
$(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{F}_{p}^{\infty}})$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\\{g\in\mathbb{F}_{p}^{\infty}:\mu(T_{c_{0}g}A\cap T_{c_{1}g}A\cap
T_{c_{2}g}A\cap T_{c_{3}g}A)>\mu(A)^{4}-\varepsilon\\}$ (12)

is syndetic.

######  Remark 1.8.

By adapting from [BHKr05] the idea of the proof of Theorem 1.6, one can find
that ergodicity is again a necessary assumption for Theorem 1.7. This is done
in Proposition 11.10.

In this article, we show that Theorems 1.4, 1.6, and 1.7 are instances of a
more comprehensive phenomenon which pertains to actions of general countable
abelian groups. The most general framework for our results is the following.
We let $G$ be a countable discrete abelian group and consider homomorphisms
$\varphi_{1},\dots,\varphi_{k}:G\to G$. Our goal is to understand under what
conditions the set

$\displaystyle\left\\{g\in G:\mu\left(A\cap T_{\varphi_{1}(g)}A\cap\cdots\cap
T_{\varphi_{k}(g)}A\right)>\mu(A)^{k+1}-\varepsilon\right\\}$ (13)

is syndetic for all ergodic $G$-systems $\left(X,\mathcal{B},\mu,(T_{g})_{g\in
G}\right)$, sets $A\in\mathcal{B}$ with $\mu(A)>0$, and $\varepsilon>0$. In
this case, we say the family $\\{\varphi_{1},\dots,\varphi_{k}\\}$ has the
_large intersections property_.

A natural approach to take when addressing this question is to study _uniform
Cesàro averages_ , a special case of which are the averages (2) appearing in
Furstenberg’s Multiple Recurrence Theorem. To develop appropriate averaging
schemes in general abelian groups, we need the notion of a Følner sequence. A
_Følner sequence_ in $G$ is a sequence of finite subsets $F_{N}\subseteq G$
such that, for every $g\in G$,

$\displaystyle\frac{|(F_{N}+g)\triangle F_{N}|}{|F_{N}|}\to 0.$ (14)

A sequence $(u_{g})_{g\in G}$ has _uniform Cesàro limit_ equal to $u$, denoted
by $\text{UC-}\lim_{g\in G}{u_{g}}=u$, if for every Følner sequence
$(F_{N})_{N\in\mathbb{N}}$,

$\displaystyle\lim_{N\to\infty}{\frac{1}{|F_{N}|}\sum_{g\in F_{N}}{u_{g}}}=u.$
(15)

For $G=\mathbb{Z}$, the uniform Cesàro limit agrees with the average appearing
in Theorem 1.3 (see Lemma 2.1). The utility of this averaging approach comes
from the following fact, which we prove in Section 2.2:

######  Lemma 1.9.

A set $S\subseteq G$ is syndetic if and only if for every Følner sequence
$(F_{N})_{N\in\mathbb{N}}$ in $G$, one has
$\bigcup_{N\in\mathbb{N}}({F_{N}}\cap S)\neq\emptyset$.

A standard technique for handling (multiple) ergodic averages is to work with
_characteristic factors_. A system
$\mathbf{Y}=\left(Y,\mathcal{D},\nu,(S_{g})_{g\in G}\right)$ is a _factor_ of
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ if there are full
measure subsets $X_{0}\subseteq X$ and $Y_{0}\subseteq Y$ and a measure-
preserving map $\pi:X_{0}\to Y_{0}$ such that $S_{g}\pi(x)=\pi(T_{g}x)$ for
every $x\in X_{0}$, $g\in G$. There is a natural correspondence between
factors and invariant sub-$\sigma$-algebras: $\mathbf{Y}$ corresponds to the
$\sigma$-algebra $\pi^{-1}(\mathcal{D})\subseteq\mathcal{B}$. In a standard
abuse of notation, we will write $\mathbb{E}\left[{f}\mid{Y}\right]$ to denote
$\mathbb{E}\left[{f}\mid{\pi^{-1}(\mathcal{D})}\right]$ for $f\in L^{2}(\mu)$.
The factor $\mathbf{Y}$ is called a _characteristic factor_ for the family
$\\{\varphi_{1},\dots,\varphi_{k}\\}$ if for all $f_{1},\dots,f_{k}\in
L^{\infty}(\mu)$,

$\displaystyle\text{UC-}\lim_{g\in
G}{\left(\prod_{i=1}^{k}{T_{\varphi_{i}(g)}f_{i}}-\prod_{i=1}^{k}{T_{\varphi_{i}(g)}\mathbb{E}\left[{f_{i}}\mid{Y}\right]}\right)}=0$
(16)

in $L^{2}(\mu)$.

While the proof of Theorem 1.4 in [BHKr05] and the proof of Theorem 1.7 in
[BTZ15] both use explicit topological descriptions of characteristic factors
for $\mathbb{Z}$-systems and $\mathbb{F}_{p}^{\infty}$-systems, respectively,
the essential ingredients can be recovered with a softer approach that applies
in greater generality. Namely, we adapt tools pioneered by Conze and Lesigne
[CoLes84] and advanced in [FuWe96] and [HKr02] to our setting. This approach
provides just enough information about $G$-systems to produce large
intersection results.

We begin by looking at the case $k=2$ of (13). It is known that for any pair
of homomorphisms $\\{\varphi,\psi\\}$, there is a constant $c>0$ such that
$\\{g\in G:\mu(A\cap T_{\varphi(g)}^{-1}A\cap T_{\psi(g)}^{-1}A)>c\\}$ is
syndetic. Indeed, this follows from the IP Szemerédi theorem of Furstenberg
and Katznelson (see [FuKa85, Theorem A]). In fact, Roth’s theorem holds for
any countable group whatsoever (see [BMc07, Theorem 1.3]).

Despite the wide generality in which the ergodic Roth theorem holds, we
encounter an obstacle to its large intersections variant even in our
significantly restricted setting: results of [Chu11] can be utilized to show
that not all pairs of homomorphisms have the large intersections property (see
Example 10.2). However, imposing a simple condition on the homomorphisms
(which we conjecture to be necessary) allows us to adapt the machinery of
[CoLes84, FuWe96] to recover large intersections. Namely, we will assume that
$\varphi,\psi:G\to G$ have the property that $\varphi(G)$, $\psi(G)$, and
$(\varphi-\psi)(G)$ all have finite index in $G$. We call such pairs
$\\{\varphi,\psi\\}$ _admissible_.333Note that this encompasses the cases
handled in [BHKr05] and [BTZ15]. Multiplication by an integer is a group
homomorphism which has finite index image in $\mathbb{Z}$ (resp.,
$\mathbb{F}_{p}^{\infty}$) so long as the integer is nonzero (resp., nonzero
modulo $p$). In this case, we will show that the _Kronecker factor_ 444For
ergodic systems, the _Kronecker factor_ is the maximal factor that can be
described (up to isomorphism) as an action by rotations on a compact abelian
group; see Section 2.4 for further discussion. $\mathbf{Z}$ is characteristic
for certain ergodic averages: for any $f_{1},f_{2}\in L^{\infty}(\mu)$,

$\text{UC-}\lim_{g\in G}{\left(T_{\varphi(g)}f_{1}\cdot
T_{\psi(g)}f_{2}-T_{\varphi(g)}\mathbb{E}\left[{f_{1}}\mid{Z}\right]\cdot
T_{\psi(g)}\mathbb{E}\left[{f_{2}}\mid{Z}\right]\right)}=0$ (17)

in $L^{2}(\mu)$. Moreover, we actually derive a limit formula (still in norm)
for expressions $\text{UC-}\lim_{g\in G}T_{\varphi(g)}f_{1}T_{\psi(g)}f_{2}$;
see Theorem 3.1 for a precise formulation. With the help of this limit
formula, we will prove the following theorem.

######  Theorem 1.10.

Let $G$ be a countable discrete abelian group, and let $\\{\varphi,\psi\\}$ be
an admissible pair of homomorphisms. For any ergodic system
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\left\\{g\in G:\mu\left(A\cap T_{\varphi(g)}^{-1}A\cap
T_{\psi(g)}^{-1}A\right)>\mu(A)^{3}-\varepsilon\right\\}$ (18)

is syndetic in $G$.

This theorem is a common generalization of Theorem 1.4(i), Theorem 1.5(i), and
Theorem 1.7(i). Once again, the assumption of ergodicity is crucial for
producing large intersections, and we give a detailed discussion with many
examples in Section 11.

Now, let us turn to analyzing (13) for $k\geq 3$ with the intention of
generalizing part (ii) of Theorem 1.4 and part (ii) of Theorem 1.7. We again
impose a condition on the family of homomorphisms to avoid counterexamples
along the lines of Example 10.2. Namely, we say a family of homomorphisms
$\\{\varphi_{1},\varphi_{2},\dots,\varphi_{k}\\}$ is _admissible_ if
$\varphi_{i}(G)$ has finite index in $G$ for every $1\leq i\leq k$ and
$(\varphi_{j}-\varphi_{i})(G)$ has finite index in $G$ for every $1\leq
i<j\leq k$. This condition appears in [Gri09, Chapter 4] in the study of
multiple ergodic averages for $\mathbb{Z}^{d}$-systems, where it is also shown
that this condition is necessary for obtaining certain results about
characteristic factors. The issue of admissibility is addressed in more detail
in Section 10.

For these longer expressions, we turn to studying the $L^{2}$-limit of
averages

$\text{UC-}\lim_{g\in G}{T_{\varphi_{1}(g)}f_{1}\cdot
T_{\varphi_{2}(g)}f_{2}\cdots T_{\varphi_{k}(g)}f_{k}}$ (19)

for $f_{1},f_{2},\dots,f_{k}\in L^{\infty}(\mu)$. We will show that the
characteristic factors for these averages are built inductively as towers of
compact extensions, starting from the Kronecker factor (Theorem 4.1).

When $k=3$, for admissible families of the form $\\{\varphi,\psi,\theta\\}$
with $\theta=\varphi+\psi$, we can show that the characteristic factor is a
skew-product of the Kronecker factor with a compact abelian group over a
cocycle satisfying a _Conze–Lesigne equation_. This can be seen as a
generalized form of the _Conze–Lesigne factor_. The relevant terminology will
be made precise in Section 6.

With the help of a limit formula valid in the case that $\varphi$ and $\psi$
are multiplication by (necessarily distinct) integers (see Theorem 7.1), we
will also prove that admissible555In a slight abuse of notation, we say that a
family of integers $\\{r_{1},\dots,r_{k}\\}$ is admissible if the family of
homomorphisms $\\{\varphi_{1},\dots,\varphi_{k}\\}$ given by
$\varphi_{i}(g)=r_{i}g$ is admissible. triples $\\{r,s,r+s\\}$ have the large
intersections property:666Theorem 1.11 was proved independently by a different
method in a recent paper by Shalom (see [Sha, Theorem 1.3]).

######  Theorem 1.11.

Let $G$ be a countable discrete abelian group. Let $r,s\in\mathbb{Z}$ be
distinct nonzero integers such that $rG$, $sG$, and $(r\pm s)G$ have finite
index in $G$. Then for any ergodic system
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\left\\{g\in G:\mu\left(A\cap T_{rg}^{-1}A\cap T_{sg}^{-1}A\cap
T_{(r+s)g}^{-1}A\right)>\mu(A)^{4}-\varepsilon\right\\}$ (20)

is syndetic in $G$.

Theorem 1.11 contains Theorem 1.4(ii), Theorem 1.7(ii), and Theorem 1.5(ii) as
special cases. However, it is still not fully satisfactory: it leaves
unaddressed the question of what happens for intersections of the form $A\cap
T_{\varphi(g)}^{-1}A\cap T_{\psi(g)}^{-1}A\cap T_{(\varphi+\psi)(g)}^{-1}A$
for homomorphisms not arising as multiplication by integers (and hence does
not fully extend Theorem 1.10). Recent results about related combinatorial
patterns in finite abelian groups suggest that additional conditions on the
family $\\{\varphi,\psi,\varphi+\psi\\}$, beyond admissibility, are needed to
guarantee large intersections (see [BerSahSawTi]).

######  Question 1.12.

What are necessary and sufficient conditions for a family of homomorphisms
$\\{\varphi,\psi,\theta\\}$ to have the large intersections property, i.e., to
satisfy the following: for any ergodic system
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$, any
$\varepsilon>0$, and any $A\in\mathcal{B}$, the set

$\left\\{g\in G:\mu\left(A\cap T_{\varphi(g)}^{-1}A\cap T_{\psi(g)}^{-1}A\cap
T_{\theta(g)}^{-1}A\right)>\mu(A)^{4}-\varepsilon\right\\}$ (21)

is syndetic in $G$?

Results about multiple recurrence in measure-preserving systems have
combinatorial implications. To formulate combinatorial consequences of the
above results, we need an ergodic version of the Furstenberg correspondence
principle. First, a definition: the _upper Banach density_ of a set
$E\subseteq G$ is the quantity

$\displaystyle d^{*}(E):=\sup\left\\{\limsup_{N\to\infty}{\frac{|E\cap
F_{N}|}{|F_{N}|}}:(F_{N})_{N\in\mathbb{N}}~{}\text{is a F{\o}lner sequence
in}~{}G\right\\}.$ (22)

######  Theorem 1.13 (Ergodic Furstenberg Correspondence Principle [BFe],
Theorem 2.8).

Let $G$ be a countable discrete abelian group. Let $E\subseteq G$ with
$d^{*}(E)>0$. Then there exists an ergodic measure-preserving system
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ and a set
$A\in\mathcal{B}$ with $\mu(A)=d^{*}(E)$ such that, for every $k\in\mathbb{N}$
and every $g_{1},\dots,g_{k}\in G$,

$\displaystyle
d^{*}\left(\bigcap_{i=1}^{k}{(E-g_{i})}\right)\geq\mu\left(\bigcap_{i=1}^{k}{T_{g_{i}}^{-1}A}\right).$
(23)

A routine application of Theorem 1.13 translates Theorems 1.10 and 1.11 into
the following two combinatorial results:777Rewriting the intersections, these
theorems show an abundance of combinatorial configurations of the form
$\\{x,x+\varphi(g),x+\psi(g)\\}$ and $\\{x,x+rg,x+sg,x+(r+s)g\\}$
respectively.

######  Theorem 1.14.

Let $G$ be a countable discrete abelian group. Let $\\{\varphi,\psi\\}$ be an
admissible pair of homomorphisms. For any $E\subseteq G$ with $d^{*}(E)>0$ and
any $\varepsilon>0$, the set

$\displaystyle\left\\{g\in
G:d^{*}\left(E\cap(E-\varphi(g))\cap(E-\psi(g))\right)>d^{*}(E)^{3}-\varepsilon\right\\}$
(24)

is syndetic.

######  Theorem 1.15.

Let $G$ be a countable discrete abelian group. Let $r,s\in\mathbb{Z}$ be
distinct nonzero integers such that $rG$, $sG$, and $(r\pm s)G$ have finite
index in $G$. For any $E\subseteq G$ with $d^{*}(E)>0$ and any
$\varepsilon>0$, the set

$\displaystyle\left\\{g\in
G:d^{*}\left(E\cap(E-rg)\cap(E-sg)\cap(E-(r+s)g)\right)>d^{*}(E)^{4}-\varepsilon\right\\}$
(25)

is syndetic.

Families of homomorphisms with the large intersections property produce an
endless variety of combinatorial configurations.

A natural question to ask is whether combinatorial results about large
intersections, such as Theorem 1.14 and Theorem 1.15, have finitary versions:

######  Question 1.16.

Fix a Følner sequence $(F_{N})_{N\in\mathbb{N}}$ in $G$. Suppose that
$\mathcal{F}=\\{\varphi_{1},\dots,\varphi_{k}\\}$ has the large intersections
property. Given $\delta>0$, $\varepsilon>0$, does there exist
$N_{0}=N_{0}(\delta,\varepsilon)$ such that if $N\geq N_{0}$ and $A\subseteq
F_{N}$ has cardinality $|A|\geq\delta|F_{N}|$, then there exists $g\neq 0$
such that

$\displaystyle\left|A\cap(A-\varphi_{1}(g))\cap\cdots\cap(A-\varphi_{k}(g))\right|>\left(\delta^{k+1}-\varepsilon\right)|F_{N}|?$
(26)

We will now briefly discuss a few examples and reformulate Question 1.16 for
these concrete situations.

When $G=\mathbb{Z}$ and $F_{N}=\\{1,\dots,N\\}$, Question 1.16 has a positive
answer:888In [Gre05, GreT10], these results are only stated for arithmetic
progressions (corresponding to the case $r=1,s=2$). However, the same method
extends to general triples or parallelogram configurations as stated here.

######  Theorem 1.17 ([Gre05], Theorem 1.10; [GreT10], Theorem 1.12).

Let $r,s\in\mathbb{Z}$ be distinct and nonzero. Suppose
$\delta,\varepsilon>0$. There exists
$N_{0}=N_{0}(r,s,\delta,\varepsilon)\in\mathbb{N}$ such that if $N\geq
N_{0}(\delta,\varepsilon)$ and $A\subseteq\\{1,\dots,N\\}$ has size
$|A|\geq\delta N$, then there exist $n,m\neq 0$ such that

$\displaystyle\left|A\cap(A-rn)\cap(A-sn)\right|$
$\displaystyle>\left(\delta^{3}-\varepsilon\right)N$ (27) and
$\displaystyle\left|A\cap(A-rm)\cap(A-sm)\cap(A-(r+s)m)\right|$
$\displaystyle>\left(\delta^{4}-\varepsilon\right)N.$ (28)

Now let us turn to $G=\mathbb{Z}^{2}$. We will show in Example 10.2 that the
pair of homomorphisms $\varphi(n,m)=(n,0)$, $\psi(n,m)=(0,n)$ fails to have
the large intersections property. The corresponding combinatorial
configurations, $\\{(a,b),(a+c,b),(a,b+c)\\}$, are known as _corners_.
Geometrically, we can view corners as isosceles right triangles in
$\mathbb{Z}^{2}$ with legs parallel to the axes. Sah, Sawhney, and Zhao have
shown a finitary analogue of the fact that this pattern is not good for large
intersections:

######  Theorem 1.18 ([SahSawZh], Theorem 1.4).

For any $l<4$, there exists $\delta>0$ such that for arbitrarily large $N$,
there exists a set $A\subseteq\\{1,\dots,N\\}$ with $|A|>\delta N$ such that

$\displaystyle\left|\left\\{(a,b)\in\mathbb{Z}:\\{(a,b),(a+c,b),(a,b+c)\\}\subseteq
A\right\\}\right|\leq\delta^{l}N$

for every $c\neq 0$.

We now show that a slight variant of corner configurations is good for large
intersections. Consider the homomorphisms $\varphi(n,m)=(n,m)$ and
$\psi(n,m)=(-m,n)$. A configuration produced by the pair $\\{\varphi,\psi\\}$
has the form $\\{(a,b),(a+n,b+m),(a-m,b+n)\\}$. This is an isosceles right
triangle (with legs of length $\sqrt{n^{2}+m^{2}}$), but we now have an
additional degree of freedom: varying the ratio $\frac{m}{n}$ changes the
angle between the legs of the triangle and the coordinate axes. When $m=0$,
this reduces to a corner with sides of length $n$. The pair
$\\{\varphi,\psi\\}$ is admissible, so by Theorem 1.14, there are syndetically
many pairs $(n,m)\in\mathbb{Z}^{2}$ such that

$\displaystyle
d^{*}\left(\left\\{(a,b)\in\mathbb{Z}^{2}:\\{(a,b),(a+n,b+m),(a-m,b+n)\\}\subseteq
E\right\\}\right)>d^{*}(E)^{3}-\varepsilon.$ (29)

Stated another way, out of all isosceles right triangle configurations in the
two-dimensional integer lattice, syndetically many of them appear up to a
shift with high frequency in $E$. A finitary version of this result also
holds, answering a question posed by us in an earlier version of this paper:

######  Theorem 1.19 ([Ko], Theorem 1; [BerSahSawTi], Theorem 1.1).

Let $\delta,\varepsilon>0$. There exists
$N_{0}=N_{0}(\delta,\varepsilon)\in\mathbb{N}$ such that if $N\geq
N_{0}(\delta,\varepsilon)$ and $A\subseteq\\{1,\dots,N\\}^{2}$ has size
$|A|\geq\delta N^{2}$, then there exists
$(n,m)\in\mathbb{Z}^{2}\setminus\\{(0,0)\\}$ such that

$\displaystyle\left|A\cap\left(A-(n,m)\right)\cap\left(A-(-m,n)\right)\right|>(\delta^{3}-\varepsilon)N^{2}.$
(30)

The structure of the paper is as follows. After a brief discussion of key
definitions and lemmas in Section 2, the paper is broken into three main
parts. First, in Sections 3–7, we establish characteristic factors for
families of admissible homomorphisms and prove limit formulae for families
$\\{\varphi,\psi\\}$ and $\\{r,s,r+s\\}$. This includes the bulk of the
difficult technical arguments, combining ideas from [CoLes84, FuWe96, Z07,
BTZ10] on characteristic factors and [FuWe96, HKr02] on computing multiple
ergodic averages as integrals. Then, in Sections 8 and 9, we apply the limit
formulae to prove Theorems 1.10 and 1.11 using a strategy from [Fr08, BTZ15].
The final four sections discuss each of the hypotheses that appear in Theorems
1.10 and 1.11: Section 10 deals with the issue of admissibility, Section 11
with ergodicity, Section 12 with larger families of homomorphisms, and Section
13 with the parallelogram condition.

##  2 Preliminaries

###  2.1 Standing assumptions

Let $G=(G,+)$ be a countable abelian group which acts on a probability space
$(X,\mathcal{B},\mu)$ by measure-preserving automorphisms $(T_{g})_{g\in G}$.
As usual, we assume the space $(X,\mathcal{B},\mu)$ is separable; that is, the
$\sigma$-algebra $\mathcal{B}$ is countably generated modulo null sets. We
refer to the quadruple $\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ as a
measure-preserving $G$-system, or $G$-system. Recall that a $G$-system
$\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ is ergodic if every set
$A\in\mathcal{B}$ satisfying $T_{g}^{-1}A=A$ for all $g\in G$ has
$\mu(A)\in\\{0,1\\}$.

Suppose $\mathbf{Y}=\left(Y,\mathcal{D},\nu,(S_{g})_{g\in G}\right)$ is a
factor of $\mathbf{X}$. Recall that we denote by
$\mathbb{E}\left[{f}\mid{Y}\right]$ the conditional expectation
$\mathbb{E}\left[{f}\mid{\pi^{-1}(\mathcal{D})}\right]$, where
$\pi:\mathbf{X}\to\mathbf{Y}$ is the factor map. We can also define the
_pullback map_ $\pi^{*}:L^{2}(Y)\to L^{2}(X)$ by $\pi^{*}f=f\circ\pi$ and the
_pushforward map_ $\pi_{*}:L^{2}(X)\to L^{2}(Y)$ as the adjoint of $\pi^{*}$.
In the case that the factor $\mathbf{Y}$ arises as a $(T_{g})_{g\in
G}$-invariant sub-$\sigma$-algebra of $\mathcal{B}$, we have the equality
$\pi_{*}f=\mathbb{E}\left[{f}\mid{Y}\right]$. In a standard abuse of notation,
we will therefore also use $\mathbb{E}\left[{f}\mid{Y}\right]$ to denote the
pushforward $\pi_{*}f$.

It is well known that $\mu$ can be disintegrated with respect to $\mathbf{Y}$
into a family of nonnegative Borel probability measures $(\mu_{y})_{y\in Y}$
on $X$ so that $\mu=\int_{Y}\mu_{y}d\nu(y)$. Note that
$\mathbb{E}\left[{f}\mid{Y}\right](y)=\int f\ d\mu_{y}$ for a.e. $y\in Y$ and
for every $f\in L^{1}(\mu)$ such that $f\in L^{1}(\mu_{y})$ for a.e. $y$. See
[Fu81, Theorem 5.8]. When $\mathbf{Y}$ is the factor associated to the
sub-$\sigma$-algebra of $(T_{g})_{g\in G}$-invariant sets, we obtain the
ergodic decomposition, which, should we need to write it, will be written
$\mu=\int\mu_{x}\ d\mu(x)$. See [Gl03, Theorem 3.22] or [V63].

For $1\leq p\leq\infty$, we write $L^{p}(\mu)$ or $L^{p}(X)$ for the Lebesgue
space $L^{p}(X,\mathcal{B},\mu)$ of complex-valued functions with finite
$p$-norm, where as usual, two functions in $L^{p}(\mu)$ are identified if they
agree $\mu$-a.e. This identification makes $L^{2}(\mu)$ into a separable
Hilbert space. We also denote by $L^{0}(\mu)$ the space of all measurable
complex-valued functions up to equivalence $\mu$-a.e.

###  2.2 Uniform Cesàro limits

We use uniform Cesàro limits extensively in this paper and present several
useful lemmas here. All of these results are standard, but we present them
here with proofs for the convenience of the reader. First, we show that the
uniform Cesàro limit is the same as the uniform limit of shifted Cesàro-type
averages:

######  Lemma 2.1.

Let $(F_{N})_{N\in\mathbb{N}}$ be a Følner sequence in $G$. For a sequence
$(u_{g})_{g\in G}$ in a Hilbert space $\mathcal{H}$,

$\displaystyle\text{UC-}\lim_{g\in G}{u_{g}}=u$

if and only if

$\displaystyle\frac{1}{|F_{N}|}\sum_{g\in F_{N}+h}{u_{g}}\to u$

uniformly in $h\in G$ as $N\to\infty$.

###### Proof.

Suppose $\text{UC-}\lim_{g\in G}{u_{g}}=u$. For each $N\in\mathbb{N}$, choose
$K_{N}\geq N$ and $h_{N}\in G$ such that

$\displaystyle\left\|\frac{1}{|F_{K_{N}}|}\sum_{g\in
F_{K_{N}}+h_{N}}{u_{g}}-u\right\|\geq\sup_{K\geq N}{\sup_{h\in
G}{\left\|\frac{1}{|F_{K}|}\sum_{g\in
F_{K}+h}{u_{g}}-u\right\|}}-\frac{1}{N}.$

Set $\Phi_{N}:=F_{K_{N}}+h_{N}$ for $N\in\mathbb{N}$. Since $K_{N}\to\infty$
and $(F_{N})_{N\in\mathbb{N}}$ is a Følner sequence,
$(\Phi_{N})_{N\in\mathbb{N}}$ is also a Følner sequence in $G$. Thus,

$\displaystyle\limsup_{N\to\infty}{\sup_{h\in
G}{\left\|\frac{1}{|F_{N}|}\sum_{g\in F_{N}+h}{u_{g}}-u\right\|}}$
$\displaystyle\leq\limsup_{N\to\infty}{\left\|\frac{1}{|F_{K_{N}}|}\sum_{g\in
F_{K_{N}}+h_{N}}{u_{g}}-u\right\|}$
$\displaystyle=\lim_{N\to\infty}{\left\|\frac{1}{|\Phi_{N}|}\sum_{g\in\Phi_{N}}{u_{g}}-u\right\|}=0.$

Conversely, suppose $\frac{1}{|F_{N}|}\sum_{g\in F_{N}+h}{u_{g}}$ converges to
$u$ uniformly in $h\in G$ as $N\to\infty$. Define a sequence of functions
$v_{N}:G\to\mathcal{H}$ by

$\displaystyle v_{N}(h):=\frac{1}{|F_{N}|}\sum_{g\in F_{N}+h}{u_{g}}.$

By assumption, $\sup_{h\in G}{\|v_{N}(h)-u\|}\to 0$ as $N\to\infty$. On the
other hand, since averages along Følner sequences are shift-invariant, we have
$\text{UC-}\lim_{g\in G}{\left(v_{N}(g)-u_{g}\right)}=0$ for each
$N\in\mathbb{N}$. Hence, $\text{UC-}\lim_{g\in G}{u_{g}}=u$. ∎

Using Lemma 2.1, we can prove a version of the van der Corput trick for
uniform Cesàro limits.

######  Lemma 2.2 (van der Corput Trick).

Let $G$ be a (countable discrete) abelian group, and let $(u_{g})_{g\in G}$ be
a bounded sequence in a Hilbert space $\mathcal{H}$. Suppose that for every
$h\in G$,

$\displaystyle\gamma_{h}:=\text{UC-}\lim_{g\in G}{\left\langle
u_{g+h},u_{g}\right\rangle}$

exists and

$\displaystyle\text{UC-}\lim_{h\in G}{\gamma_{h}}=0.$

Then

$\displaystyle\text{UC-}\lim_{g\in G}{u_{g}}=0.$

###### Proof.

Without loss of generality, assume $\|u_{g}\|\leq 1$ for every $g\in G$. Fix a
Følner sequence $(F_{N})_{N\in\mathbb{N}}$ in $G$, and let $\varepsilon>0$.
Since $\text{UC-}\lim_{h\in G}{\gamma_{h}}=0$, by Lemma 2.1, the sequence of
functions

$\displaystyle f_{M}(k):=\frac{1}{|F_{M}|}\sum_{h\in F_{M}-k}{\gamma_{h}}$

converges to $0$ uniformly in $k\in G$ as $M\to\infty$. Thus, we can find
$M\in\mathbb{N}$ such that $|f_{M}(k)|<\varepsilon$ for every $k\in G$.

Now, since $(F_{N})_{N\in\mathbb{N}}$ is a Følner sequence, there is an
$N_{1}\in\mathbb{N}$ such that if $N\geq N_{1}$, then $F_{N}$ is
$(F_{M},\varepsilon)$-invariant.999A set $F$ is called _$(K,\varepsilon)$
-invariant_ if $\frac{|(F+K)\triangle F|}{|F|}<\varepsilon$. It is easy to
check that $(F_{N})_{N\in\mathbb{N}}$ is a Følner sequence if and only if for
every finite set $K\subseteq G$ and every $\varepsilon>0$, there is an
$N_{0}\in\mathbb{N}$ so that $F_{N}$ is $(K,\varepsilon)$-invariant for every
$N\geq N_{0}$. In particular,

$\displaystyle\left\|\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}{u_{g}}-\frac{1}{|F_{M}|}\sum_{h\in F_{M}}{\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}{u_{g+h}}}\right\|<\varepsilon.$

Moreover, we can find $N_{2}\in\mathbb{N}$ such that if $N\geq N_{2}$, then

$\displaystyle\left|\frac{1}{|F_{N}|}\sum_{g\in F_{N}+k}{\left\langle
u_{g+h},u_{g}\right\rangle}-\gamma_{h}\right|<\varepsilon$

for $h\in F_{M}-F_{M}$, $k\in G$.

Set $N_{0}:=\max\\{N_{1},N_{2}\\}$. Then for $N\geq N_{0}$, we have

$\displaystyle\left\|\frac{1}{|F_{N}|}\sum_{g\in F_{N}}{u_{g}}\right\|^{2}$
$\displaystyle<\left\|\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}{\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}{u_{g+h}}}\right\|^{2}+\varepsilon(2+\varepsilon)$
$\displaystyle\leq\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}{\left\|\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}{u_{g+h}}\right\|^{2}}+\varepsilon(2+\varepsilon)$
$\displaystyle=\frac{1}{|F_{N}|}\sum_{g\in
F_{N}}{\frac{1}{|F_{M}|^{2}}\sum_{h_{1},h_{2}\in F_{M}}{\left\langle
u_{g+h_{1}},u_{g+h_{2}}\right\rangle}}+\varepsilon(2+\varepsilon)$
$\displaystyle=\frac{1}{|F_{M}|^{2}}\sum_{h_{1},h_{2}\in
F_{M}}{\frac{1}{|F_{N}|}\sum_{g\in F_{N}+h_{2}}{\left\langle
u_{g+h_{1}-h_{2}},u_{g}\right\rangle}}+\varepsilon(2+\varepsilon)$
$\displaystyle<\left|\frac{1}{|F_{M}|^{2}}\sum_{h_{1},h_{2}\in
F_{M}}{\gamma_{h_{1}-h_{2}}}\right|+\varepsilon(3+\varepsilon)$
$\displaystyle\leq\frac{1}{|F_{M}|}\sum_{k\in
F_{M}}{\left|\frac{1}{|F_{M}|}\sum_{h\in
F_{M}-k}{\gamma_{h}}\right|}+\varepsilon(3+\varepsilon)$
$\displaystyle<\varepsilon(4+\varepsilon).$

∎

The following lemma shows that positivity of uniform Cesàro limits implies
syndeticity of return times.

See 1.9

###### Proof.

Suppose $S$ is syndetic, and let $(F_{N})_{N\in\mathbb{N}}$ be a Følner
sequence. We will prove a stronger statement:
$\underline{d}_{(F_{N})}(S):=\liminf_{N\to\infty}{\frac{|S\cap
F_{N}|}{|F_{N}|}}>0$. Let $K\subseteq G$ be a finite set such that $S+K=G$. By
a standard averaging argument, for each $N\in\mathbb{N}$, there is a $k_{N}\in
K$ such that

$\displaystyle\frac{|S\cap(F_{N}-k_{N})|}{|F_{N}|}=\frac{|(S+k_{N})\cap
F_{N}|}{|F_{N}|}\geq\frac{1}{|K|}.$

Now by the Følner property, we have

$\displaystyle\limsup_{N\to\infty}{\max_{k\in K}{\frac{|(F_{N}-k)\triangle
F_{N}|}{|F_{N}|}}}=0.$

Thus,

$\displaystyle\liminf_{N\to\infty}{\frac{|S\cap F_{N}|}{|F_{N}|}}$
$\displaystyle\geq\liminf_{N\to\infty}{\frac{|S\cap(F_{N}-k_{N})\cap
F_{N}|}{|F_{N}|}}$
$\displaystyle\geq\liminf_{N\to\infty}{\frac{|S\cap(F_{N}-k_{N})|}{|F_{N}|}}-\limsup_{N\to\infty}{\frac{|(F_{N}-k_{N})\setminus
F_{N}|}{|F_{N}|}}$ $\displaystyle\geq\frac{1}{|K|}.$

Conversely, suppose $S$ is not syndetic. We want to construct a Følner
sequence entirely in $G\setminus S$. Let $(F_{N})_{N\in\mathbb{N}}$ be any
Følner sequence in $G$. Since $S$ is not syndetic, we have $S-F_{N}\neq G$ for
every $N\in\mathbb{N}$. Thus, we can find elements $g_{N}\in G$ such that
$g_{N}\notin S-F_{N}$. Equivalently, $S\cap(F_{N}+g_{N})=\emptyset$. Now
$\Phi_{N}:=F_{N}+g_{N}$ is a Følner sequence, and
$\bigcup_{N\in\mathbb{N}}{\Phi_{N}}\cap S=\emptyset$. ∎

###  2.3 Ergodic theorem

In our setting, we will need a general form of von Neumann’s mean ergodic
theorem. This result is a standard exercise using the invariant splitting of a
Hilbert space (see, e.g., [B00, Theorem 4.15] for a short proof). We record it
here for reference:

######  Theorem 2.3 (Ergodic theorem).

Let $(G,+)$ be a countable discrete abelian group. Let
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$, and let
$\mathcal{I}\subseteq\mathcal{B}$ be the $\sigma$-algebra of $T$-invariant
sets. Then for $f\in L^{2}(\mu)$,

$\displaystyle\text{UC-}\lim_{g\in
G}{T_{g}f}=\mathbb{E}\left[{f}\mid{\mathcal{I}}\right].$ (31)

In particular, if $\mathbf{X}$ is ergodic, then

$\displaystyle\text{UC-}\lim_{g\in G}{T_{g}f}=\int_{X}{f~{}d\mu}.$ (32)

In the language of characteristic factors, the ergodic theorem says that the
trivial factor is characteristic for the averages

$\displaystyle\text{UC-}\lim_{g\in G}{T_{g}f}$ (33)

in ergodic systems. If we apply an admissible homomorphism $\varphi:G\to G$,
then the action $\left(T_{\varphi(g)}\right)_{g\in G}$ will have finitely many
ergodic components, so $\text{UC-}\lim_{g\in G}{T_{\varphi(g)}f}$ is a
function with finitely many values (it is the conditional expectation with
respect to the finitely many ergodic components). Another way of describing
this behavior is to say that the characteristic factor for the average

$\displaystyle\text{UC-}\lim_{g\in G}{T_{\varphi(g)}f}$ (34)

is built from rotations on the finite group $G/\varphi(G)$.

###  2.4 Kronecker factor

Fix a countable discrete abelian group $(G,+)$. A system
$\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ is _compact_ if
there is a compact abelian group $Y$ and an action by group rotations
$S_{g}y=y+a_{g}$ such that $\mathbf{X}$ is isomorphic to
$\mathbf{Y}=\left(Y,\mathcal{D},\nu,(S_{g})_{g\in G}\right)$, where
$\mathcal{D}$ is the Borel $\sigma$-algebra on $Y$ and $\nu$ is the Haar
measure on $Y$. The _Kronecker factor_ of an ergodic system $\mathbf{X}$ is
the maximal compact factor of $\mathbf{X}$.

In our setting, the Kronecker factor can be described quite concretely.
Consider the Jacobs–de Leeuw–Glicksberg decomposition into compact and weakly
mixing functions:

$\displaystyle L^{2}(\mu)=L^{2}(\mu)_{c}\oplus L^{2}(\mu)_{wm},$

where

$\displaystyle L^{2}(\mu)_{c}$ $\displaystyle=\left\\{f\in
L^{2}(\mu):\overline{\\{T_{g}f:g\in G\\}}^{\|\cdot\|_{2}}~{}\text{is
compact}\right\\},$ $\displaystyle L^{2}(\mu)_{wm}$
$\displaystyle=\left\\{f\in L^{2}(\mu):\text{UC-}\lim_{g\in G}{|\left\langle
T_{g}f,f\right\rangle|}=0\right\\}.$

The compact functions, $L^{2}(\mu)_{c}$, are spanned by an orthonormal basis
of eigenfunctions, say $(f_{\lambda})_{\lambda\in\Lambda}$, where $\Lambda$ is
a countable subset of $\widehat{G}$, and $f_{\lambda}$ has the corresponding
“eigenvalue” $\lambda$, i.e. $T_{g}f_{\lambda}=\lambda(g)f_{\lambda}$ for all
$g\in G$ and $\lambda\in\Lambda$. We can assume, by rescaling if necessary,
that $|f_{\lambda}|=1$ and $f_{\lambda}f_{\mu}=f_{\lambda+\mu}$ for all
$\lambda,\mu\in\Lambda$ (the case $G=\mathbb{Z}$ is shown in [Wa82, Chapter
3], and this easily generalizes to our setting).

For $\lambda\in\Lambda$ and $\varphi:G\to G$, we can define
$\varphi\lambda:=\lambda\circ\varphi$. While this defines a homomorphism on
$G$ (so that $\varphi\lambda\in\widehat{G}$), it is not necessarily the case
that $\varphi\lambda\in\Lambda$.

Now let $Z=\widehat{\Lambda}$, where $\Lambda$ is treated as a discrete group
so that $Z$ is compact, and define an action

$\displaystyle\left(S_{g}(z)\right)(\lambda):=\lambda(g)z(\lambda).$

Equivalently, letting $\widehat{g}$ be the evaluation map
$\widehat{g}(\lambda)=\lambda(g)$ and writing $Z$ additively (since it is an
abelian group), we have $S_{g}(z)=z+\widehat{g}$. It is easy to check that
$L^{2}(Z)$ is spanned by the evaluation functions
$\\{e_{\lambda}:\lambda\in\Lambda\\}$, where $e_{\lambda}(z):=z(\lambda)$.
These functions are also the eigenfunctions for the action $(S_{g})_{g\in G}$.
The map $f_{\lambda}\mapsto e_{\lambda}$ then induces the factor map (see
[Wa82] for details).

We review a few facts about group rotations that will be helpful for analyzing
the behavior of the Kronecker factor. These facts are essentially the same as
in the special case of $\mathbb{Z}$-systems (c.f. [Wa82, Theorem 6.20]), but
we present them here with proofs in the general case for completeness. Recall
that a $G$-action by homeomorphisms $(T_{g})_{g\in G}$ on a compact Hausdorff
space $X$ is _minimal_ if the only closed $T$-invariant subsets of $X$ are
$\emptyset$ and $X$. Equivalently, every orbit $\\{T_{g}x:g\in G\\}$, $x\in
X$, is dense in $X$.

######  Lemma 2.4.

Let $G$ be an abelian group and $X$ a compact abelian group. Suppose $G$ acts
on $X$ by group rotations $S_{g}x=x+a_{g}$, where $g\mapsto a_{g}$ is a
(continuous) homomorphism $G\to X$. Then the following are equivalent:

  1. (i)

The orbit $\\{a_{g}:g\in G\\}$ is dense in $X$;

  2. (ii)

The action $(S_{g})_{g\in G}$ is minimal;

  3. (iii)

The action $(S_{g})_{g\in G}$ is uniquely ergodic, and the unique invariant
measure is the Haar measure on $X$.

###### Proof.

The implication (ii)$\implies$(i) is trivial. For (iii)$\implies$(ii), we note
that any closed invariant set supports an invariant measure, and Haar measure
always has full support.

Now we will show (i)$\implies$(iii). Assume that $\\{a_{g}:g\in G\\}$ is dense
in $X$. Let $\mu$ be a $G$-invariant measure on $X$, and let $f\in C(X)$. For
every $g\in G$, invariance of $\mu$ gives the identity

$\displaystyle\int_{X}{f(x+a_{g})~{}d\mu(x)}=\int_{X}{f(x)~{}d\mu(x)}.$

Now, given $y\in X$, condition (i) implies that
$y=\lim_{n\to\infty}{a_{g_{n}}}$ for some sequence $(g_{n})_{n\in\mathbb{N}}$
in $G$. Thus, by the dominated convergence theorem,

$\displaystyle\int_{X}{f(x+y)~{}d\mu(x)}=\lim_{n\to\infty}{\int_{X}{f(x+a_{g_{n}})~{}d\mu(x)}}=\int_{X}{f(x)~{}d\mu(x)}.$

This invariance property uniquely defines the Haar measure, so $\mu$ is the
Haar measure on $X$. ∎

##  3 Kronecker factor is characteristic for double recurrence

In this section, we show that the Kronecker factor is characteristic for the
averages

$\displaystyle\text{UC-}\lim_{g\in G}{T_{\varphi(g)}f_{1}\cdot
T_{\psi(g)}f_{2}}.$ (35)

For ease of notation, we will write $\widetilde{f}_{i}$ for the image of
$f_{i}$ under the factor map $\pi:X\to Z$. That is,
$\mathbb{E}\left[{f_{i}}\mid{Z}\right]=\widetilde{f}_{i}\circ\pi$.

######  Theorem 3.1.

Let $\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure-
preserving system with Kronecker factor $\mathbf{Z}$ and factor map
$\pi:\mathbf{X}\to\mathbf{Z}$. Let $\varphi,\psi:G\to G$ be homomorphisms such
that $\varphi$, $\psi$, and $\psi-\varphi$ have finite index image in $G$.
Then for $f_{1},f_{2}\in L^{\infty}(\mu)$, the limit

$\displaystyle\text{UC-}\lim_{g\in
G}{f_{1}(T_{\varphi(g)}x)f_{2}(T_{\psi(g)}x)}$

exists in $L^{2}(\mu)$ and is equal to

$\displaystyle\text{UC-}\lim_{g\in
G}{\widetilde{f}_{1}(z+\widehat{\varphi(g)})\widetilde{f}_{2}(z+\widehat{\psi(g)})}=\int_{Z^{2}}{\widetilde{f}_{1}(z+w_{1})\widetilde{f}_{2}(z+w_{2})~{}d\nu_{\varphi,\psi}(w_{1},w_{2})},$
(36)

where $z=\pi(x)$ and $\nu_{\varphi,\psi}$ is the Haar (probability) measure on
the subgroup
$Z_{\varphi,\psi}:=\overline{\\{(\widehat{\varphi(g)},\widehat{\psi(g)}):g\in
G\\}}$ of $Z^{2}$.

######  Remark 3.2.

If the spectrum of $\mathbf{X}$ has the additional property that
$\varphi\Lambda+\psi\Lambda\subseteq\Lambda$, then we can define $(\varphi
w)(\lambda):=w(\varphi\lambda)$ for $w\in Z$ and similarly for $\psi$, in
which case $Z_{\varphi,\psi}=\\{(\varphi w,\psi w):w\in Z\\}$, so the limit in
(36) simplifies to

$\displaystyle\int_{Z}{\widetilde{f}(z+\varphi w)\widetilde{g}(z+\psi
w)~{}dw}.$

A formula of this form appears in [FuWe96] for $\mathbb{Z}$-systems, where
$\varphi$ and $\psi$ are multiplication by $a$ and $b$ respectively.

###### Proof of Theorem 3.1.

First we will show that the Kronecker factor is characteristic. By linearity,
this is the same as showing

$\displaystyle\text{UC-}\lim_{g\in G}{T_{\varphi(g)}f_{1}\cdot
T_{\psi(g)}f_{2}}=0$ (37)

when either $\widetilde{f}_{1}=0$ or $\widetilde{f}_{2}=0$. Since the
expressions are symmetric in $f_{1}$ and $f_{2}$ (by interchanging $\varphi$
and $\psi$), we will assume $\widetilde{f}_{1}=0$.

Let $u_{g}=T_{\varphi(g)}f_{1}\cdot T_{\psi(g)}f_{2}$. We will use the van der
Corput trick (Lemma 2.2) to show (37). Since $T_{\varphi(g)}$ is
$\mu$-preserving, we have

$\displaystyle\left\langle
u_{g+h},u_{g}\right\rangle=\int{\left(\overline{f}_{1}T_{\varphi(h)}f_{1}\right)T_{(\psi-\varphi)(g)}\left(\overline{f}_{2}T_{\psi(h)}f_{2}\right)~{}d\mu}.$

Since $(\psi-\varphi)(G)\subseteq G$ has finite index, the action of this
subgroup $(T_{(\psi-\varphi)(g)})_{g\in G}$ has only finitely many components
in its ergodic decomposition. In fact, the ergodic decomposition is of the
form $\mu=\frac{1}{d}\sum_{j=1}^{d}{\mu_{j}}$, where $\mu_{j}$ is the
(normalized) restriction of $\mu$ to an invariant set of measure
$\frac{1}{d}$, and $d\leq[G:(\psi-\varphi)(G)]$. Now for each $h\in G$, the
ergodic theorem gives

$\displaystyle\gamma_{h}:=\text{UC-}\lim_{g\in G}{\left\langle
u_{g+h},u_{g}\right\rangle}=\frac{1}{d}\sum_{j=1}^{d}{\left(\int{\overline{f}_{1}T_{\varphi(h)}f_{1}~{}d\mu_{j}}\int{\overline{f}_{2}T_{\psi(h)}f_{2}~{}d\mu_{j}}\right)}.$

Applying the triangle inequality and the Cauchy–Schwarz inequality, we have
for any finite set $F\subseteq G$,

$\displaystyle\left|\frac{1}{|F|}\sum_{h\in F}{\gamma_{h}}\right|$
$\displaystyle\leq\frac{1}{d}\sum_{j=1}^{d}{\frac{1}{|F|}\sum_{h\in
F}{\left|\int{\overline{f}_{1}T_{\varphi(h)}f_{1}~{}d\mu_{j}\int{\overline{f}_{2}T_{\psi(h)}f_{2}~{}d\mu_{j}}}\right|}}$
$\displaystyle\leq\frac{1}{d}\sum_{j=1}^{d}{\left(\frac{1}{|F|}\sum_{h\in
F}{\left|\int{\overline{f}_{1}T_{\varphi(h)}f_{1}~{}d\mu_{j}}\right|^{2}}\frac{1}{|F|}\sum_{h\in
F}{\left|\int{\overline{f}_{2}T_{\psi(h)}f_{2}~{}d\mu_{j}}\right|^{2}}\right)^{1/2}}.$

We assumed $\widetilde{f}_{1}=0$, so $f_{1}\in L^{2}(\mu)_{wm}$. It follows
that $f_{1}$ is a weakly mixing function for the subaction along the finite
index subgroup101010 To see this, consider $(T_{g})_{g\in G}$ restricted to
$L^{2}(\mu)_{wm}$. We want to show that $(T_{\varphi(h)})_{h\in G}$ is a
weakly mixing action on this space. If not, then there is a nonzero function
$f\in L^{2}(\mu)_{wm}$ such that $\\{T_{\varphi(h)}f:h\in G\\}$ is pre-
compact. But $\varphi(G)$ has finite index, so every element of $G$ can be
expressed as $g=\varphi(h)+k$ for some $h\in G$ and some $k$ belonging to a
finite set $K$. Hence, $\\{T_{g}f:g\in G\\}=\bigcup_{k\in
K}{T_{k}\left(\\{T_{\varphi(h)}f:h\in G\\}\right)}$ is a finite union of pre-
compact sets and therefore pre-compact. But $f\in L^{2}(\mu)_{wm}$, so this is
a contradiction.  $\varphi(G)$, so for each $1\leq j\leq d$,

$\displaystyle\text{UC-}\lim_{h\in
G}{\left|\int{\overline{f}_{1}T_{\varphi(h)}f_{1}~{}d\mu_{j}}\right|^{2}}=0.$

Since $\left|\int{\overline{f}_{2}T_{\psi(h)}f_{2}~{}d\mu_{j}}\right|^{2}\leq
d^{2}\|f_{2}\|_{2}^{4}<\infty$, we have

$\displaystyle\text{UC-}\lim_{h\in G}{\text{UC-}\lim_{g\in G}{\left\langle
u_{g+h},u_{g}\right\rangle}}=0,$

so that (37) holds by Lemma 2.2. Thus, the Kronecker factor is characteristic:

$\displaystyle\text{UC-}\lim_{g\in
G}{f_{1}(T_{\varphi(g)}x)f_{2}(T_{\psi(g)}x)}=\text{UC-}\lim_{g\in
G}{\widetilde{f}_{1}(z+\widehat{\varphi(g)})\widetilde{f}_{2}(z+\widehat{\psi(g)})}.$

It remains to compute the limit as an integral. Consider the diagonal action
$(S_{g})_{g\in G}$ on $Z_{\varphi,\psi}$ given by
$S_{g}(w_{1},w_{2})=(w_{1}+\widehat{\varphi(g)},w_{2}+\widehat{\psi(g)})$. It
is easy to check that $S_{g}(Z_{\varphi,\psi})\subseteq Z_{\varphi,\psi}$, so
this action is well-defined. By Lemma 2.4, $(S_{g})_{g\in G}$ is uniquely
ergodic with unique invariant measure $\nu_{\varphi,\psi}$. Define
$f:Z_{\varphi,\psi}\to\mathbb{C}$ by

$\displaystyle
f(w_{1},w_{2}):=\widetilde{f}_{1}(z+w_{1})\widetilde{f}_{2}(z+w_{2}).$

If $\widetilde{f}_{1}$ and $\widetilde{f}_{2}$ are continuous, then $f$ is
continuous and unique ergodicity gives

$\displaystyle\text{UC-}\lim_{g\in
G}{\widetilde{f}_{1}(z+\widehat{\varphi(g)})\widetilde{f}_{2}(z+\widehat{\psi(g)})}$
$\displaystyle=\text{UC-}\lim_{g\in G}{f(S_{g}0)}$
$\displaystyle=\int_{Z_{\varphi,\psi}}{f(w)~{}dw}$
$\displaystyle=\int_{Z^{2}}{\widetilde{f}_{1}(z+w_{1})\widetilde{f}_{2}(z+w_{2})~{}d\nu_{\varphi,\psi}(w_{1},w_{2})}.$

In general, we can approximate $\widetilde{f}_{1}$ and $\widetilde{f}_{2}$ by
continuous functions (for example, with finite linear combinations of
characters) to get the desired convergence in $L^{2}(\mu)$. ∎

##  4 Compact extensions

The first step in studying characteristic factors for triple (and longer)
recurrence is to observe that they are formed as towers of _compact
extensions_.

For $k\in\mathbb{N}$, let $\mathbf{Z}_{k}$ be the minimal factor of
$\mathbf{X}$ that is characteristic for all $k$-element admissible families.
That is, for every admissible family $\\{\varphi_{1},\dots,\varphi_{k}\\}$ and
every $f_{1},\dots,f_{k}\in L^{\infty}(\mu)$,

$\text{UC-}\lim_{g\in G}{\left(T_{\varphi_{1}(g)}f_{1}\cdots
T_{\varphi_{k}(g)}f_{k}-T_{\varphi_{1}(g)}\mathbb{E}\left[{f_{1}}\mid{Z_{k}}\right]\cdots
T_{\varphi_{k}(g)}\mathbb{E}\left[{f_{k}}\mid{Z_{k}}\right]\right)}=0.$ (38)

Our goal is to prove the following:

######  Theorem 4.1.

For all $k\in\mathbb{N}$, $\mathbf{Z}_{k+1}$ is a compact extension of
$\mathbf{Z}_{k}$.

For $G=\mathbb{Z}$, this appears in [Z07], and our methodology here is
similar. There are two key ingredients in the proof. First, in the process of
using the van der Corput trick (Lemma 2.2), we will encounter averages of the
form

$\text{UC-}\lim_{g\in G}{\int_{X}{T_{\varphi_{1}(g)}f_{1}\cdots
T_{\varphi_{k+1}(g)}f_{k+1}~{}d\mu}}.$ (39)

Since $T_{\varphi_{1}(g)}$ is measure-preserving, this is equal to the average

$\text{UC-}\lim_{g\in
G}{\int_{X}{f_{1}T_{(\varphi_{2}-\varphi_{1})(g)}f_{2}\cdots
T_{(\varphi_{k+1}-\varphi_{1})(g)}f_{k+1}~{}d\mu}}.$

An important observation at this stage is that the family
$\\{\varphi_{2}-\varphi_{1},\dots,\varphi_{k+1}-\varphi_{1}\\}$ is admissible.
This follows easily from the definition of an admissible family, and it means
that the average (39) is controlled by the factor $\mathbf{Z}_{k}$.

Analyzing these averages involves a diagonal measure on $X^{k+1}$. We show
that this diagonal measure is sufficiently well behaved to conclude that the
projection onto the invariant $\sigma$-algebra for the diagonal action is
built from functions on the maximal compact extension of $\mathbf{Z}_{k}$. It
will then follow that the maximal compact extension of $\mathbf{Z}_{k}$ is a
characteristic factor for the averages (38), completing the proof.

First, we define our terms.

######  Definition 4.2.

Suppose $\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ is a measure-
preserving system with an ergodic factor
$\mathbf{Y}=(Y,\mathcal{D},\nu,(S_{g})_{g\in G})$ and associated factor map
$\alpha:X\to Y$.

  1. 1.

A closed subspace $M\subset L^{2}(X)$ is called a _$\mathbf{Y}$ -module_ if
for every $f\in M$ and $h\in L^{0}(Y)$, if $hf\in L^{2}(X)$, then $hf\in M$.

  2. 2.

If $M$ is a $\mathbf{Y}$-module, we say that a subset $L\subset M$ _spans_ $M$
if for every $f\in M$, there exist sequences of functions
$(f_{n})_{n=1}^{\infty}\subset L$ and $(c_{n})_{n=1}^{\infty}\subset L^{0}(Y)$
such that $f(x)=\sum_{n}c_{n}(\alpha(x))f_{n}(x)$.

  3. 3.

A function $f\in L^{2}(X)$ is called a _generalized eigenfunction_ or
_$\mathbf{Y}$ -eigenfunction_ if the $\mathbf{Y}$-module spanned by
$\\{T_{g}f:g\in G\\}$ is of finite rank.111111This is a technical assumption
which ensures, among other desirable consequences, that $Y$-module spanned by
$\\{T_{g}f:g\in G\\}$ has an orthonormal $Y$-basis.

  4. 4.

Denote by $\mathscr{E}(\mathbf{X}/\mathbf{Y})$ the closure in $L^{2}(X)$ of
the subspace of $\mathbf{Y}$-eigenfunctions. In words,
$\mathscr{E}(\mathbf{X}/\mathbf{Y})$ is called the _$\mathbf{Y}$
-eigenfunction space of $\mathbf{X}$_.

The $\mathbf{Y}$-eigenfunction space
$\mathscr{E}(\mathbf{X}/\mathbf{Y})\subset L^{2}(X)$ is a $\mathbf{Y}$-module
that contains not only $L^{2}(Y)$ but also the invariant space $\\{f\in
L^{2}(X):T_{g}f=f\text{ for all }g\in G\\}$. In general, if
$L^{2}(X)=\mathscr{E}(\mathbf{X}/\mathbf{Y})$, then $\mathbf{X}$ is called a
_compact extension_ of $\mathbf{Y}$. Moreover,
$\mathscr{E}(\mathbf{X}/\mathbf{Y})$ generates a sub-$\sigma$-algebra of
$\mathcal{B}$, and the resulting factor of $\mathbf{X}$ is called the _maximal
compact extension_ of $\mathbf{Y}$ in $\mathbf{X}$. See [Fu77] and [Gl03] for
references.

Here is a second round of definitions.

######  Definition 4.3.

Suppose two measure-preserving systems
$\mathbf{X}_{1}=\left(X_{1},\mathcal{B}_{1},\mu_{1},(T_{g}^{(1)})_{g\in
G}\right)$ and
$\mathbf{X}_{2}=\left(X_{2},\mathcal{B}_{2},\mu_{2},(T_{g}^{(2)})_{g\in
G}\right)$ share a common factor $\mathbf{Y}=(Y,\mathcal{D},\nu,(S_{g})_{g\in
G})$, and let $\alpha_{1}:\mathbf{X}_{1}\to\mathbf{Y}$ and
$\alpha_{2}:\mathbf{X}_{2}\to\mathbf{Y}$ be the associated factor maps. The
_relatively independent joining of $\mathbf{X}_{1}$ and $\mathbf{X}_{2}$ over
$\mathbf{Y}$_ is the measure-preserving system
$\mathbf{X}_{1}\times_{\mathbf{Y}}\mathbf{X}_{2}=(X_{1}\times_{\mathbf{Y}}X_{2},\mathcal{B}_{1}\times_{\mathbf{Y}}\mathcal{B}_{2},\mu_{1}\times_{\mathbf{Y}}\mu_{2},(T_{g}^{(1)}\times
T_{g}^{(2)})_{g\in G})$, where we set
$X_{1}\times_{\mathbf{Y}}X_{2}:=\\{(x_{1},x_{2})\in X_{1}\times
X_{2}:\alpha_{1}(x_{1})=\alpha_{2}(x_{2})\\}$, the $\sigma$-algebra
$\mathcal{B}_{1}\times_{\mathbf{Y}}\mathcal{B}_{2}$ is the restriction of
$\mathcal{B}_{1}\otimes\mathcal{B}_{2}$ to $X_{1}\times_{\mathbf{Y}}X_{2}$,
and $\mu_{1}\times_{\mathbf{Y}}\mu_{2}$ is the measure defined by setting, for
every $f_{1}\in L^{\infty}(X_{1})$ and $f_{2}\in L^{\infty}(X_{2})$,

$\int_{X_{1}\times_{\mathbf{Y}}X_{2}}f_{1}\otimes f_{2}\
d(\mu_{1}\times_{\mathbf{Y}}\mu_{2})\ =\
\int_{Y}\mathbb{E}\left[{f_{1}}\mid{\mathcal{D}}\right]\mathbb{E}\left[{f_{2}}\mid{\mathcal{D}}\right]\
d\nu,$ (40)

where $f_{1}\otimes f_{2}$ denotes the function $(x_{1},x_{2})\mapsto
f_{1}(x_{1})f_{2}(x_{2})$.

We will need to compute
$\mathscr{E}(\mathbf{X}_{1}\times_{\mathbf{Y}}\mathbf{X}_{2}/\mathbf{Y})$ in
terms of the $\mathbf{Y}$-eigenfunction spaces of $\mathbf{X}_{1}$ and of
$\mathbf{X}_{2}$. First, let us agree on one more piece of notation: If
$M_{1}\subset L^{2}(X_{1})$ and $M_{2}\subset L^{2}(X_{2})$ are
$\mathbf{Y}$-modules, denote by $M_{1}\otimes_{\mathbf{Y}}M_{2}$ the closed
$\mathbf{Y}$-module of $L^{2}(X_{1}\times_{\mathbf{Y}}X_{2})$ spanned by
products $f_{1}\otimes f_{2}$, where $f_{1}\in M_{1}$ and $f_{2}\in M_{2}$ are
bounded functions. After some effort, one can compute the
$\mathbf{Y}$-eigenfunction space of $X_{1}\times_{\mathbf{Y}}X_{2}$ as
follows.

######  Theorem 4.4.

Assuming the setup of Definition 4.3, we have

$\mathscr{E}(\mathbf{X}_{1}\times_{\mathbf{Y}}\mathbf{X}_{2}/\mathbf{Y})\ =\
\mathscr{E}(\mathbf{X}_{1}/\mathbf{Y})\otimes_{\mathbf{Y}}\mathscr{E}(\mathbf{X}_{2}/\mathbf{Y}).$
(41)

###### Proof.

See [Fu77, Theorem 7.1] for a proof in the restricted context of
$\mathbb{Z}$-systems, and see [Gl03, Theorem 9.21] for a direct quotation in
the context of $G$-systems. ∎

######  Theorem 4.5.

Assuming the setup of Definition 4.3, if moreover $\mathbf{X}_{2}$ is ergodic,
then

$\mathscr{E}(\mathbf{X}_{1}\times_{\mathbf{Y}}\mathbf{X}_{2}/\mathbf{X}_{2})\
=\ \mathscr{E}(\mathbf{X}_{1}/\mathbf{Y})\otimes L^{2}(X_{2}).$ (42)

###### Proof.

See [Fu77, Theorem 7.4] for a proof in the context of $\mathbb{Z}$-systems.
The argument in the context of $G$-systems is analogous. ∎

###  4.1 A theorem on conditional products

In this subsection, we repeatedly use the following set of assumptions:

  1. 1.

Fix a positive integer $k$.

  2. 2.

For each $i\in\\{1,\ldots,k\\}$, let
$\mathbf{X}_{i}=(X_{i},\mathcal{B}_{i},\mu_{i},(T^{(i)}_{g})_{g\in G})$ be a
measure-preserving system with a factor
$\mathbf{Y}_{i}=(Y_{i},\mathcal{D}_{i},\nu_{i},(S^{(i)}_{g})_{g\in G})$ and
associated factor map $\alpha_{i}:X_{i}\to Y_{i}$.

  3. 3.

Let $(X,\mathcal{B})$ denote the product
$(\prod_{{i}=1}^{{k}}X_{i},\otimes_{i=1}^{k}\mathcal{B}_{i})$, and let
$(Y,\mathcal{D})$ denote the product
$(\prod_{{i}=1}^{{k}}Y_{i},\otimes_{i=1}^{k}\mathcal{D}_{i})$.

  4. 4.

Let $\alpha:X\to Y$ be defined by $\alpha:=(\alpha_{1},\ldots,\alpha_{k})$.

######  Definition 4.6.

A measure $\mu$ on $(X,\mathcal{B})$ is to said to have _correct marginals_ if
its image on the $i$th coordinate is $\mu_{i}$.

######  Remark 4.7.

By definition, a _joining_ on $X$ has correct marginals and is invariant under
the diagonal action $T_{g}:=T^{(1)}_{g}\times\cdots\times T^{(k)}_{g}$, and
for some simpler lemmas here, we discuss joinings only when necessary. If
$\mu$ is a measure on $X$ with correct marginals, then the pushforward
$\alpha_{*}\mu=\mu\circ\alpha^{-1}$ is a measure on $Y$ with correct
marginals.

######  Definition 4.8.

Let $\mu$ be a measure on $(X,\mathcal{B})$ with correct marginals, and write
$\alpha_{*}\mu$ for its pushforward on $(Y,\mathcal{D})$. We say that $\mu$ is
a conditional product measure relative to $Y$ if

$\mu=\int_{Y}\mu_{1,y_{1}}\times\cdots\times\mu_{k,y_{k}}\
d\alpha_{*}\mu(y_{1},\ldots,y_{k}),$ (43)

where $\mu_{i}$ is disintegrated as $\mu_{i}=\int_{Y_{i}}\mu_{i,y_{i}}\
d\nu_{i}(y_{i})$.

######  Remark 4.9.

The definition of a conditional product measure is sensible since the set of
$(y_{1},\ldots,y_{k})$’s for which the integrand is defined has full
$\alpha_{*}\mu$-measure since $\alpha_{*}\mu$ has correct marginals. A
_conditional product joining_ relative to $Y$ is a conditional product measure
relative to $Y$ that is also invariant under the diagonal action.

Here is an equivalent characterization, proven by unfolding definitions.

######  Lemma 4.10.

A measure $\mu$ on $(X,\mathcal{B})$ with correct marginals is a conditional
product measure relative to $(Y,\mathcal{D})$ if and only if for all
$k$-tuples $(f_{i})_{i=1}^{k}$ with $f_{i}\in
L^{\infty}(X_{i},\mathcal{B}_{i},\mu_{i})$, we have

$\int_{X}\bigotimes_{i=1}^{k}f_{i}\ d\mu\ =\
\int_{Y}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{Y_{i}}\right]\
d\alpha_{*}\mu,$ (44)

where, as usual, we use the notation $\bigotimes_{i=1}^{k}f_{i}$ for the
function $(x_{1},\ldots,x_{k})\mapsto f_{1}(x_{1})\cdots f_{k}(x_{k})$.

We will need to change the factor with respect to which a measure is a
conditional product measure. The following lemma suffices for this purpose.

######  Lemma 4.11 ([Fu77], Lemma 9.2).

Suppose for each $i\in\\{1,\ldots,k\\}$, we have an intermediate factor
$(Z_{i},\mathcal{C}_{i})$ with $X_{i}\to Z_{i}\to Y_{i}$ and factor maps
$\beta_{i}:X_{i}\to Z_{i}$ and $\beta_{i}^{\prime}:Z_{i}\to Y_{i}$. Suppose
that $\mu$ is a measure on $X$ with correct marginals that is also a
conditional product measure relative to $Y$. Then $\mu$ is a conditional
product measure relative to $Z:=\prod_{{i}=1}^{{k}}Z_{i}$.

###### Proof.

For each $i$, set $\pi_{i}:=\beta_{i}^{\prime}\circ\beta_{i}$, and write
$\beta:=(\beta_{1},\ldots,\beta_{k})$ and similarly for $\pi$. We show the
lemma using Lemma 4.10. Fix a $k$-tuple $(f_{i})_{i=1}^{k}$ with $f_{i}\in
L^{\infty}(X_{i})$. We observe that

$\displaystyle\int_{Z}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{Z_{i}}\right]\
d\beta_{*}\mu\ $ $\displaystyle=\
\int_{X}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\mathcal{C}_{i}}\right]\
d\mu\ \overset{*}{=}\
\int_{Y}\bigotimes_{i=1}^{k}\mathbb{E}\left[{\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\mathcal{C}_{i}}\right]}\mid{Y_{i}}\right]\
d\pi_{*}\mu$ $\displaystyle=\
\int_{X}\bigotimes_{i=1}^{k}\mathbb{E}\left[{\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\mathcal{C}_{i}}\right]}\mid{\pi_{i}^{-1}\mathcal{D}_{i}}\right]\
d\mu\ \overset{**}{=}\
\int_{X}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\pi_{i}^{-1}\mathcal{D}_{i}}\right]\
d\mu$ $\displaystyle=\
\int_{Y}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{Y_{i}}\right]\
d\pi_{*}\mu\ \overset{*}{=}\ \int_{X}\bigotimes_{i=1}^{k}f_{i}\ d\mu,$

where the single-starred equalities hold since $X$ is a conditional product
measure relative to $Y$, and the double-starred equality holds since
$\pi_{i}^{-1}\mathcal{D}_{i}\subset\beta_{i}^{-1}\mathcal{C}_{i}$. ∎

According to the next lemma, the presence of a conditional product measure on
a product system $X$ ensures that $X$ is a relatively independent joining of
certain relatively independent joinings, a fact which we promise to use.

######  Lemma 4.12.

Let $\mathbf{X},\mathbf{Y},\mathbf{X}_{i}$, and $\mathbf{Y}_{i}$ be as above.
Put
$\widetilde{\mathbf{X}}:=(\mathbf{X}_{1}\times_{\mathbf{Y}_{1}}\mathbf{Y})\times_{\mathbf{Y}}(\mathbf{X}_{2}\times_{\mathbf{Y}_{2}}\mathbf{Y})\times_{\mathbf{Y}}\cdots\times_{\mathbf{Y}}(\mathbf{X}_{k}\times_{\mathbf{Y}_{k}}\mathbf{Y})$,
and denote by $\widetilde{\alpha}$ the function $X\to\widetilde{X}$ that maps
$x=(x_{1},x_{2},\ldots,x_{k})$ to
$\left((x_{1},\alpha(x)),(x_{2},\alpha(x)),\ldots,\right.$
$\left.(x_{k},\alpha(x))\right)$. If $\mu$ is a conditional product measure
relative to $Y$, then $\widetilde{\alpha}$ is a measurable isomorphism of
$\mathbf{X}$ with $\widetilde{\mathbf{X}}$.

###### Proof.

Since $\mathbf{X}$ and $\widetilde{\mathbf{X}}$ share the factor $\mathbf{Y}$,
it suffices to show that $\mu$ and the measure on $\widetilde{X}$ disintegrate
into the same measures with respect to the factor $\mathbf{Y}$. For reference,
see the proof in [Fu77] of Lemma 9.3. ∎

There is one more basic result we need.

######  Lemma 4.13.

If each $(X_{i},\mathcal{B}_{i},\mu_{i},(T^{(i)}_{g})_{g\in G})$ is ergodic
and $\mu$ is a joining on $X$, then almost every ergodic component of $\mu$
has correct marginals.

###### Proof.

Using the coordinate projection onto $X_{i}$, push the ergodic decomposition
of $\mu$ forward to get a decomposition of $\mu_{i}$ into invariant measures,
each of which is a pushforward of an ergodic component of $\mu$. As $\mu_{i}$
is already ergodic, these invariant measures are almost always $\mu_{i}$. We
know this for each $i$. ∎

The following is a generalized form of [Fu77, Theorem 9.4].

######  Theorem 4.14.

Assume each $(Y_{i},\mathcal{D}_{i},\nu_{i},(T^{(i)}_{g})_{g\in G})$ is
ergodic. For each $i$, let
$(\widehat{Y}_{i},\widehat{\mathcal{D}}_{i},\widehat{\nu}_{i},(T^{(i)}_{g})_{g\in
G})$ be the maximal compact extension of
$(Y_{i},\mathcal{D}_{i},\nu_{i},(S^{(i)}_{g})_{g\in G})$ in
$(X_{i},\mathcal{B}_{i},\mu_{i},(T^{(i)}_{g})_{g\in G})$. Let $G$ act on $X$
by $T_{g}:=T^{(1)}_{g}\times\cdots\times T^{(k)}_{g}$. If $\mu$ is a
conditional product joining on $X$ relative to $Y$, then almost all ergodic
components of $\mu$ are conditional product measures relative to
$\widehat{Y}:=\prod_{{i}=1}^{{k}}\widehat{Y}_{i}$.

###### Proof.

Let $\mu^{\prime}=\alpha_{*}\mu$. Without loss of generality, $\mu^{\prime}$
is ergodic. First, we need the following:

######  Claim 4.15.

The subspace of $T$-invariant functions $L^{2}(X,\mathcal{I}_{T},\mu)$ is
contained in $\bigotimes_{i=1}^{k}\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i})$.

###### Proof of Claim.

Applying Lemma 4.12, $\widetilde{\alpha}:X\to\widetilde{X}$ is an isomorphism
of measure-preserving systems, where
$\widetilde{X}:=(X_{1}\times_{Y_{1}}Y)\times_{Y}(X_{2}\times_{Y_{2}}Y)\times_{Y}\cdots\times_{Y}(X_{k}\times_{Y_{k}}Y)$.
For each $i$ the base space of $X_{i}\times_{Y_{i}}Y$ is a subset of
$X_{i}\times Y$; thus the base space of $\widetilde{X}$ is $X\times Y^{k}$.
Moreover, all of the coordinates that live in $Y$ are the same, so we can even
view $\widetilde{X}$ as a subset of $X\times Y$.

Now, the invariant subspace
$L^{2}(\widetilde{X},\widetilde{\mathcal{B}}_{T},\widetilde{\mu})$ is a
subspace of $\mathscr{E}(\widetilde{X}/Y)$. By Theorem 4.4,

$\mathscr{E}(\widetilde{\mathbf{X}}/\mathbf{Y})\ =\
\mathscr{E}(\mathbf{X}_{1}\times_{\mathbf{Y}_{1}}\mathbf{Y}/\mathbf{Y})\otimes_{Y}\cdots\otimes_{Y}\mathscr{E}(\mathbf{X}_{k}\times_{\mathbf{Y}_{k}}\mathbf{Y}/\mathbf{Y}).$
(45)

Then, for each $i$, by Theorem 4.5 (valid since $\mu^{\prime}$ is ergodic), we
have
$\mathscr{E}(\mathbf{X}_{i}\times_{\mathbf{Y}_{i}}\mathbf{Y}/\mathbf{Y})=\mathscr{E}(\mathbf{X}_{1}/\mathbf{Y}_{i})\otimes
L^{2}(Y)$, which implies that

$L^{2}(\widetilde{X},\widetilde{\mathcal{B}}_{T},\mu)\ \subset\
\left(\bigotimes_{i=1}^{k}\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i})\right)\otimes
L^{2}(Y).$ (46)

Viewing $\widetilde{X}$ as a subset of $X\times Y$, we pull back to transform
Equation (46) into the desired containment

$L^{2}(X,\mathcal{I}_{T},\mu)\ \subset\
\bigotimes_{i=1}^{k}\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i}).$ (47)

∎

Having proved the claim, we proceed as follows. We have the ergodic
decomposition $\mu=\int\mu_{x}d\mu(x)$; every $f\in L^{1}(X)$ satisfies
$\mathbb{E}\left[{f}\mid{\mathcal{I}_{T}}\right](x)=\int f\ d\mu_{x}$ for
$\mu$-a.e. $x\in X$. Since $\mu$ is a joining, Lemma 4.13 implies that almost
every $\mu_{x}$ has correct marginals.

Via the equivalent formulation in Lemma 4.10, we will show that almost every
component $\mu_{x}$ is a conditional product measure with respect to
$\widehat{Y}$. Thus, we need to show that for every $k$-tuple
$(f_{i})_{i=1}^{k}$ with $f_{i}\in L^{\infty}(X_{i},\mu_{i})$, the equation

$\int_{X}\bigotimes_{i=1}^{k}f_{i}\ d\mu_{x}\ =\
\int_{\widehat{Y}}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\widehat{Y}_{i}}\right]\
d\beta_{*}\mu_{x},$ (48)

holds for a.e. $x$, where $\beta:X\to\widehat{Y}$ is the composition
$\beta:=(\beta_{1},\ldots,\beta_{k})$ of the factor maps
$\beta_{i}:X_{i}\to\widehat{Y}_{i}$. This follows from an easy approximation
argument whose ingredients are the following four facts, the last of which we
prove.

######  Lemma 4.16.

For each $i$, let $f_{i},f_{i}^{\prime}\in L^{\infty}(X_{i},\mu_{i})$ be
functions, and let $M$ bound all of $||f_{1}||_{L^{\infty}(X_{1},\mu_{1})},\
\ldots,$ $\ ||f_{k}^{\prime}||_{L^{\infty}(X_{k},\mu_{k})}$. Then for any
measure $\theta$ on $X$ with correct marginals, we have

$\left|\left|\bigotimes_{i=1}^{k}f_{i}-\bigotimes_{i=1}^{k}f_{i}^{\prime}\right|\right|_{L^{1}(X,\theta)}\
\leq\ M^{k-1}\sum_{i=1}^{k}||f_{i}-f_{i}^{\prime}||_{L^{1}(X_{i},\mu_{i})}.$
(49)

######  Lemma 4.17.

The operator
$\mathbb{E}\left[{\cdot}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]$
is a contraction $L^{1}(X_{i},\mathcal{B}_{i},\mu_{i})\to
L^{1}(X_{i},\beta_{i}^{-1}\widehat{\mathcal{D}}_{i},\mu_{i})$.

######  Lemma 4.18.

There is a countable subset $\mathcal{F}_{i}\subset L^{\infty}(X_{i},\mu_{i})$
that is dense in $L^{1}(X_{i},\mu_{i})$.

######  Claim 4.19.

For a.e $x$, for all $k$-tuples $(f_{i})_{i=1}^{k}$ with
$f_{i}\in\mathcal{F}_{i}$, we have

$\int_{X}\bigotimes_{i=1}^{k}f_{i}\ d\mu_{x}\ =\
\int_{\widehat{Y}}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\widehat{Y}_{i}}\right]\
d\beta_{*}\mu_{x}.$ (50)

###### Proof of Claim.

Fix a $k$-tuple $(f_{i})_{i=1}^{k}$ with $f_{i}\in\mathcal{F}_{i}\subset
L^{\infty}(X_{i})$. By Lemma 4.11, since $\mu$ is a conditional product
measure relative to $Y$, it is also a conditional product measure relative to
$\widehat{Y}$. Thus, if $(f_{i}^{\prime})_{i=1}^{k}$ is a $k$-tuple of
functions with $f_{i}^{\prime}\in\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i})$,
it follows that

$\int_{X}\bigotimes_{i=1}^{k}f_{i}f_{i}^{\prime}\ d\mu\ =\
\int_{\widehat{Y}}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}f_{i}^{\prime}}\mid{\widehat{Y}_{i}}\right]\
d\beta_{*}\mu\ =\
\int_{X}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}f_{i}^{\prime}}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]\
d\mu\ =\
\int_{X}\bigotimes_{i=1}^{k}f_{i}^{\prime}\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]\
d\mu,$ (51)

which implies that for any
$f^{\prime}\in\bigotimes_{i=1}^{k}\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i})$,
we have

$\int_{X}f^{\prime}\cdot\bigotimes_{i=1}^{k}f_{i}\ d\mu\ =\
\int_{X}f^{\prime}\cdot\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]\
d\mu.$ (52)

Therefore, by the previous claim that
$\bigotimes_{i=1}^{k}\mathscr{E}(\mathbf{X}_{i}/\mathbf{Y}_{i})$ contains the
invariant subspace $L^{2}(X,\mathcal{I}_{T},\mu)$, we conclude

$\mathbb{E}\left.\left[\bigotimes_{i=1}^{k}f_{i}\ \right|\
\mathcal{I}_{T}\right]\ =\
\mathbb{E}\left.\left[\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]\
\right|\ \mathcal{I}_{T}\right].$ (53)

Finally, since every $f\in L^{2}(X)$ satisfies $\int f\
d\mu_{x}=\mathbb{E}\left[{f}\mid{\mathcal{I}_{T}}\right](x)$ for a.e. $x$, it
follows from Equation (53) that the equation

$\int_{X}\bigotimes_{i=1}^{k}f_{i}\ d\mu_{x}\ =\
\int_{X}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\beta_{i}^{-1}\widehat{\mathcal{D}}_{i}}\right]\
d\mu_{x}\ =\
\int_{\widehat{Y}}\bigotimes_{i=1}^{k}\mathbb{E}\left[{f_{i}}\mid{\widehat{Y}_{i}}\right]\
d\beta_{*}\mu_{x}$ (54)

holds for a.e. $x$. Since the collections $\mathcal{F}_{i}$ are countable, the
claim holds as stated. $\square$ ∎

∎

We can slightly strengthen the previous theorem to get the following theorem,
which generalizes [FuWe96, Theorem 5.1]. This theorem is stated with the
assumptions of this subsection made explicit, for ease of use elsewhere.

######  Theorem 4.20.

For $1\leq i\leq k$, let
$\mathbf{Y}_{i}=\left(Y_{i},\mathcal{D}_{i},\nu_{i},\left(S^{(i)}_{g}\right)_{g\in
G}\right)$ be a factor of
$\mathbf{X}_{i}=\left(X_{i},\mathcal{B}_{i},\mu_{i},\left(T^{(i)}_{g}\right)_{g\in
G}\right)$ with maximal compact extension $\widehat{\mathbf{Y}_{i}}$ in
$\mathbf{X}_{i}$. Suppose each $\mathbf{X}_{i}$ has finitely many ergodic
components. If $\mu$ is a conditional product joining on
$\prod_{i=1}^{k}{X_{i}}$ relative to $\prod_{i=1}^{k}{Y_{i}}$, then almost
every ergodic component of $\mu$ is a conditional product relative to
$\prod_{i=1}^{k}{\widehat{Y_{i}}}$.

###### Proof.

Since each $X_{i}$ has finitely many components, it follows that each $Y_{i}$
has finitely many components. The conclusion follows from the argument in
Theorem 4.14 in the same way that the proof of Theorem 9.5 follows from the
proof of Theorem 9.4 in [Fu77]. ∎

######  Remark 4.21.

The conclusion of this theorem can also be stated as follows: if a function
$F\in L^{2}(\prod_{{i}=1}^{{k}}X_{i})$ is invariant under
$T_{1}\times\cdots\times T_{k}$, then there is a function $D\in
L^{2}(\prod_{{i}=1}^{{k}}\widehat{Y_{i}})$ such that $F(x)=D(k)$, where
$x\mapsto k$ is the projection from $\prod_{{i}=1}^{{k}}X_{i}$ to
$\prod_{{i}=1}^{{k}}\widehat{Y_{i}}$.

###  4.2 A lemma for the diagonal measure

We now state and prove a lemma for the diagonal measure, which is needed to
apply Theorem 4.20 as a step in the proof of Theorem 4.1.

######  Lemma 4.22.

Let $\mathbf{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in G})$ be an ergodic measure-
preserving system, and suppose $(Z_{k},\nu)$ is characteristic for averages
(38). Let $\\{\varphi_{1},\dots,\varphi_{k+1}\\}$ be an admissible family. Let
$S_{g}:=T_{\varphi_{1}(g)}\times\cdots\times T_{\varphi_{k+1}(g)}$ be the
corresponding diagonal action on $X^{k+1}$. Then there are measures
$\widetilde{\mu}$ on $X^{k+1}$ and $\widetilde{\nu}$ on $Z_{k}^{k+1}$ such
that

  1. 1.

$\widetilde{\mu}$ is an $S$-invariant joining of $\mu$ with itself;

  2. 2.

$\widetilde{\nu}$ is an $S$-invariant joining of $\nu$ with itself;

  3. 3.

$\widetilde{\mu}$ is a conditional product joining relative to
$(Z_{k}^{k+1},\widetilde{\nu})$; and

  4. 4.

for all $f_{1},f_{2},\dots,f_{k+1}\in L^{\infty}(\mu)$,

$\displaystyle\text{UC-}\lim_{g\in
G}{\int_{X}{\prod_{i=1}^{k+1}{T_{\varphi_{i}(g)}f_{i}}~{}d\mu}}$
$\displaystyle=\int_{X^{k+1}}{\bigotimes_{i=1}^{k+1}{f_{i}}~{}d\widetilde{\mu}},$
$\displaystyle\text{UC-}\lim_{g\in
G}{\int_{Z_{k}}{\prod_{i=1}^{k+1}{T_{\varphi_{i}(g)}\pi_{k}f_{i}}~{}d\nu}}$
$\displaystyle=\int_{Z_{k}^{k+1}}{\bigotimes_{i=1}^{k+1}{\pi_{k}f_{i}}~{}d\widetilde{\nu}},$

where $\pi_{k}$ is the projection $\pi_{k}:L^{2}(X)\to L^{2}(Z_{k})$.

###### Proof.

Define a functional $\Phi:C(X^{k+1})\to\mathbb{C}$ by

$\displaystyle\Phi(F):=\text{UC-}\lim_{g\in
G}{\int_{X^{k+1}}{S_{g}F~{}d\sigma}},$ (55)

where $\sigma$ is the diagonal measure

$\displaystyle\int_{X^{k+1}}{F(x_{1},x_{2},\dots,x_{k+1})~{}d\sigma(x_{1},x_{2},\dots,x_{k+1})}:=\int_{X}{F(x,x,\dots,x)~{}d\mu(x)}.$
(56)

The limit in (55) exists (see [Au16, Z-K16]), and it is controlled by the
factor $\mathbf{Z}_{k}$.121212This is because $T$ is measure-preserving, so
$\Phi(F)=\text{UC-}\lim_{g\in
G}{\int_{X}{F(x,T_{(\varphi_{2}-\varphi_{1})(g)}x,\dots,T_{(\varphi_{k+1}-\varphi_{1})(g)}x)~{}d\mu(x)}}$.
Approximating $F$ by linear combinations of simple tensors
$\otimes_{i=1}^{k+1}{f_{i}}$ shows that this limit is controlled by the
characteristic factor for the family
$\\{\varphi_{2}-\varphi_{1},\dots,\varphi_{k+1}-\varphi_{1}\\}$, which is an
admissible family with $k$ elements. Observe that $\Phi$ is a positive linear
functional and $|\Phi(F)|\leq\|F\|_{\sup}$. Thus, by the Riesz representation
theorem, there is a (positive) Borel measure $\widetilde{\mu}$ on $X^{k+1}$
such that $\Phi(F)=\int_{X^{k+1}}{F~{}d\widetilde{\mu}}$. Moreover, since
$\Phi(\mathbbm{1})=1$, $\widetilde{\mu}$ is a probability measure.

Note that $\widetilde{\mu}$ is invariant with respect to the diagonal action
$S$ by the Følner property: for any Følner sequence
$(F_{N})_{N\in\mathbb{N}}$, we have

$\displaystyle\left|\int_{X^{k+1}}{S_{g}F~{}d\widetilde{\mu}}-\int_{X^{k+1}}{F~{}d\widetilde{\mu}}\right|=$
$\displaystyle\left|\lim_{M\to\infty}{\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}{\int_{X}{S_{g+h}F~{}d\sigma}}}-\lim_{M\to\infty}{\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}{\int_{X}{S_{h}F~{}d\sigma}}}\right|$ $\displaystyle=$
$\displaystyle\left|\lim_{M\to\infty}{\frac{1}{|F_{M}|}\sum_{h\in
F_{M}+g}{\int_{X}{S_{h}F~{}d\sigma}}}-\lim_{M\to\infty}{\frac{1}{|F_{M}|}\sum_{h\in
F_{M}}{\int_{X}{S_{h}F~{}d\sigma}}}\right|$ $\displaystyle\leq$
$\displaystyle\lim_{M\to\infty}{\frac{1}{|F_{M}|}\sum_{h\in(F_{M}+g)\triangle
F_{M}}{\int_{X}{|S_{h}F|~{}d\sigma}}}$ $\displaystyle\leq$
$\displaystyle\lim_{M\to\infty}{\frac{\left|(F_{M}+g)\triangle
F_{M}\right|}{|F_{M}|}}\|F\|_{\sup}=0.$

Moreover, it is easy to check from the definition that $\widetilde{\mu}$ is a
joining of $\mu$ with itself, since $\mu$ is $T$-invariant.

By repeating this construction on the factor $\mathbf{Z}_{k}$, we may define
an invariant probability measure $\widetilde{\nu}$ on $Z_{k}^{k+1}$ satisfying

$\displaystyle\int_{Z_{k}^{k+1}}{F~{}d\widetilde{\nu}}=\text{UC-}\lim_{g\in
G}{\int_{Z_{k}}{F\left(S_{g}(z,z,\dots,z)\right)~{}d\nu(z)}}.$

Now, we claim that $\widetilde{\mu}$ is a conditional product joining relative
to the measure $\widetilde{\nu}$ on $Z_{k}^{k+1}$. Let $\pi_{k}:L^{2}(X)\to
L^{2}(Z_{k})$ be the projection onto the factor $\mathbf{Z}_{k}$. Since $\mu$
is $T$-invariant and $Z_{k}$ is a $T$-invariant sub-$\sigma$-algebra, we can
use the fact that $\mathbf{Z}_{k}$ is characteristic for length $k$ averages
to compute the integral with respect to $\widetilde{\mu}$:

$\displaystyle\int_{X^{k+1}}{\bigotimes_{i=1}^{k+1}{f_{i}}~{}d\widetilde{\mu}}$
$\displaystyle=\text{UC-}\lim_{g\in
G}{\int_{X}{\prod_{i=1}^{k+1}{f_{i}(T_{\varphi_{i}(g)}x)}~{}d\mu(x)}}$
$\displaystyle=\text{UC-}\lim_{g\in
G}{\int_{X}{f_{1}(x)\prod_{i=2}^{k+1}{f_{i}(T_{(\varphi_{i}-\varphi_{1})(g)}x)}~{}d\mu(x)}}$
$\displaystyle=\text{UC-}\lim_{g\in
G}{\int_{X}{f_{1}(x)\prod_{i=2}^{k+1}{\mathbb{E}\left[{f_{i}}\mid{Z_{k}}\right](T_{(\varphi_{i}-\varphi_{1})(g)}x)}~{}d\mu(x)}}$
$\displaystyle=\text{UC-}\lim_{g\in
G}{\int_{X}{f_{1}(T_{\varphi_{1}(g)}x)\prod_{i=2}^{k+1}{\mathbb{E}\left[{f_{i}}\mid{Z_{k}}\right](T_{\varphi_{i}(g)}x)}~{}d\mu(x)}}$
$\displaystyle=\text{UC-}\lim_{g\in
G}{\int_{Z_{k}}{\prod_{i=1}^{k+1}{\pi_{k}f_{i}(T_{\varphi_{i}(g)}z)}~{}d\nu(z)}}$
$\displaystyle=\int_{Z_{k}^{k+1}}{\bigotimes_{i=1}^{k+1}{\pi_{k}f_{i}}~{}d\widetilde{\nu}}.$

∎

######  Remark 4.23.

When $k=2$, so that $\mathbf{Z}_{2}=Z$ is the Kronecker factor, we can see the
measure $\widetilde{\nu}$ as the Haar measure on the subgroup
$W_{\varphi,\psi,\theta}:=\overline{\\{(z+\widehat{\varphi(g)},z+\widehat{\psi(g)},z+\widehat{\theta(g)}):z\in
Z,g\in G\\}}\subseteq Z^{3}$. If
$\varphi\Lambda+\psi\Lambda+\theta\Lambda\subseteq\Lambda$, then
$W_{\varphi,\psi,\theta}$ has the simpler description
$W_{\varphi,\psi,\theta}=\\{(z+\varphi w,z+\psi w,z+\theta w):z,w\in
Z\\}=Z_{id,id,id}+Z_{\varphi,\psi,\theta}$.

###  4.3 Proof of Theorem 4.1

With all of this machinery, we now prove Theorem 4.1. Let
$\widehat{\mathbf{Z}_{k}}$ be the maximal compact extension of
$\mathbf{Z}_{k}$. Let $u_{g}:=\prod_{i=1}^{k+1}T_{\varphi_{i}(g)}f_{i}$. By
symmetry, it suffices to show that

$\displaystyle\text{UC-}\lim_{g\in G}{u_{g}}=0$ (57)

when $\mathbb{E}\left[{f_{1}}\mid{\widehat{Z_{k}}}\right]=0$.

We will assume $\mathbb{E}\left[{f_{1}}\mid{\widehat{Z_{k}}}\right]=0$ and
show that (57) holds using the van der Corput trick (Lemma 2.2). We can use
the diagonal measure from Lemma 4.22 to compute the limit

$\displaystyle\gamma_{h}$ $\displaystyle:=\text{UC-}\lim_{g\in G}{\left\langle
u_{g+h},u_{g}\right\rangle}$
$\displaystyle=\int_{X^{k+1}}{\bigotimes_{i=1}^{k+1}{\left(\overline{f}_{i}T_{\varphi_{i}(h)}f_{i}\right)}~{}d\widetilde{\mu}}$
$\displaystyle=\int_{X^{k+1}}{\left(\bigotimes_{i=1}^{k+1}{\overline{f}_{i}}\right)\left(\bigotimes_{i=1}^{k+1}{T_{\varphi_{i}(h)}f_{i}}\right)~{}d\widetilde{\mu}}.$

Thus, letting $F=\bigotimes_{i=1}^{k+1}{f_{i}}$ and
$S_{h}=T_{\varphi_{1}(h)}\times T_{\varphi_{2}(h)}\times\cdots\times
T_{\varphi_{k+1}(h)}$ the diagonal action, we have by the ergodic theorem that

$\displaystyle\text{UC-}\lim_{h\in G}{\gamma_{h}}$
$\displaystyle=\text{UC-}\lim_{h\in
G}{\int_{X^{k+1}}{\overline{F}~{}S_{h}F~{}d\widetilde{\mu}}}.$
$\displaystyle=\int_{X^{k+1}}{\overline{F}~{}\mathbb{E}\left[{F}\mid{\mathcal{I}_{S}}\right]~{}d\widetilde{\mu}},$

where $\mathcal{I}_{S}$ is the $\sigma$-algebra of $S$-invariant sets. Now,
Lemma 4.22 showed that the measure $\widetilde{\mu}$ is a conditional product
joining. Furthermore, since $\\{\varphi_{1},\dots,\varphi_{k+1}\\}$ is an
admissible family, the actions $\left(Z_{k},(T_{\varphi_{i}(h)})_{h\in
G}\right)$ each have finitely many ergodic components. So Theorem 4.20
applies, and we can write
$\mathbb{E}\left[{F}\mid{\mathcal{I}_{S}}\right](x)=D(k)$ for some function
$D\in L^{2}(\widehat{Z_{k}}^{k+1})$, where $x\mapsto k$ is the projection
$X^{k+1}\to\widehat{Z_{k}}^{k+1}$. Thus,

$\displaystyle\text{UC-}\lim_{h\in
G}{\gamma_{h}}=\int_{X^{k+1}}{\overline{F}D~{}d\widetilde{\mu}}.$ (58)

It remains to show that the quantity in (58) is in fact equal to zero.

To do so, we define a functional
$\Psi:L^{2}(\widehat{Z_{k}}^{k+1})\to\mathbb{C}$ by

$\displaystyle\Psi(H):=\int_{X^{k+1}}{\overline{F}H~{}d\widetilde{\mu}}$

That is, $\Psi=\left\langle\cdot,F\right\rangle_{\widetilde{\mu}}$. We want to
show that $\Psi\equiv 0$. Since $\Psi$ is continuous and linear and
$L^{2}(\widehat{Z_{k}}^{k+1})=\bigotimes_{i=1}^{k+1}L^{2}(\widehat{Z_{k}})$,
it suffices to show that $\Psi(H)=0$ for $H$ of the form
$H=\bigotimes_{i=1}^{k+1}{h_{i}}$ with $h_{i}\in L^{2}(\widehat{Z_{k}})$. By
Lemma 4.22, $\widetilde{\mu}$ is a conditional product joining with respect to
the measure $\widetilde{\nu}$ on $Z_{k}^{k+1}$, so

$\displaystyle\Psi\left(\bigotimes_{i=1}^{k+1}{h_{i}}\right)$
$\displaystyle=\int_{X^{k+1}}{\bigotimes_{i=1}^{k+1}{\left(\overline{f}_{i}h_{i}\right)}~{}d\widetilde{\mu}}$
$\displaystyle=\int_{Z_{k}^{k+1}}{\bigotimes_{i=1}^{k+1}{\mathbb{E}\left[{\overline{f}_{i}h_{i}}\mid{Z_{k}}\right]}~{}d\widetilde{\nu}}.$

Now, since $\widehat{\mathbf{Z}_{k}}$ is an extension of $\mathbf{Z}_{k}$,

$\displaystyle\mathbb{E}\left[{\overline{f}_{1}h_{1}}\mid{Z_{k}}\right]=\mathbb{E}\left[{\mathbb{E}\left[{\overline{f}_{1}h_{1}}\mid{\widehat{Z_{k}}}\right]}\mid{Z_{k}}\right]=\mathbb{E}\left[{h_{1}\overline{\mathbb{E}\left[{f_{1}}\mid{\widehat{Z_{k}}}\right]}}\mid{Z_{k}}\right]=0,$

since we assumed that $\mathbb{E}\left[{f_{1}}\mid{\widehat{Z_{k}}}\right]=0$.
Thus, $\Psi\left(\bigotimes_{i=1}^{k+1}{h_{i}}\right)=0$. This proves that
$\Psi\equiv 0$, so in particular $\Psi(D)=0$. Therefore, by Lemma 2.2,
equation (57) holds. We have shown that $\widehat{\mathbf{Z}_{k}}$ is an
extension of $\mathbf{Z}_{k+1}$.

Since $\widehat{\mathbf{Z}_{k}}$ is an extension of $\mathbf{Z}_{k+1}$, which
is an extension of $\mathbf{Z}_{k}$, and $\widehat{\mathbf{Z}_{k}}$ is a
compact extension of $\mathbf{Z}_{k}$, it follows by, e.g., [Gl03, Lemma 9.12]
that $\mathbf{Z}_{k+1}$ is a compact extension of $\mathbf{Z}_{k}$. ∎

##  5 From compact extensions to abelian group extensions

We now move to analyzing the multiple ergodic averages

$\displaystyle\text{UC-}\lim_{g\in G}{T_{\varphi(g)}f_{1}\cdot
T_{\psi(g)}f_{2}\cdot T_{\theta(g)}f_{3}}.$ (59)

Later, we will restrict to the case $\theta=\varphi+\psi$. By Theorem 4.1, the
average (59) is controlled by a compact extension of the Kronecker factor.
Utilizing a technique from [CoLes84] and [FuWe96], we can obtain a more
refined description of this factor.

First, a general compact extension of the Kronecker factor is a skew product
of the form $Z\times_{\rho}K/L$, where $K$ is a compact group, $L$ a closed
subgroup, and $\rho_{g}:Z\to K$ a cocycle (see [Gl03, Theorem 9.14]). That is,
$(\rho_{g})_{g\in G}$ satisfies the cocycle equation

$\displaystyle\rho_{g+h}(z)=\rho_{g}(z+\widehat{h})\rho_{h}(z),$ (60)

and $T$ acts on $Z\times K/L$ by
$T_{g}(z,kL)=\left(z+\widehat{g},\rho_{g}(z)kL\right)$.

It is easiest to deal with the case that $L$ is the trivial group, so that the
compact extension is just a group extension. Following [FuWe96], we call a
system $\mathbf{X}$ _normal_ if the maximal compact extension of the Kronecker
factor is a group extension. Fortunately, it is enough to study normal
systems:

######  Proposition 5.1 ([FuWe96], Theorem 8.8).

Every ergodic system is a factor of a normal ergodic system.

Furstenberg and Weiss prove this in the case $G=\mathbb{Z}$, but their proof
relies only on general facts from group theory and categorical arguments with
commutative diagrams that can be interpreted in the category of $G$-systems
with no added difficulty (see [FuWe96, Sections 7 and 8]). The one exception
is [FuWe96, Lemma 8.3], which states that a homomorphism
$\theta:\mathbf{X}\to\mathbf{X}^{\prime}$ between measure-preserving systems
induces a homomorphism of pairs
$(\mathbf{X},\mathbf{Z})\to(\mathbf{X},\mathbf{Z}^{\prime})$, where
$\mathbf{Z}$ and $\mathbf{Z}^{\prime}$ are the Kronecker factors of
$\mathbf{X}$ and $\mathbf{X}^{\prime}$ respectively. To prove this in the
setting of general countable abelian groups requires a few modifications. For
completeness, we provide a proof of this fact in our setting:

######  Lemma 5.2 (c.f. [FuWe96], Lemma 8.3).

Let $\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})\right)$ be an ergodic
$G$-system with Kronecker factor $\mathbf{Z}$. Let $\mathbf{X}^{\prime}$ be
another $G$-system with Kronecker factor $\mathbf{Z}^{\prime}$. Suppose
$\theta:\mathbf{X}\to\mathbf{X}^{\prime}$ is a homomorphism. Then $\theta$
defines a homomorphism of pairs131313This means that $\theta$ induces a
homomorphism $\mathbf{Z}\to\mathbf{Z}^{\prime}$.
$(\mathbf{X},\mathbf{Z})\to(\mathbf{X}^{\prime},\mathbf{Z}^{\prime})$.

###### Proof.

Since all of the systems involved here are homomorphic images of $\mathbf{X}$,
we deal with the space $L^{2}(X)$ and view the other systems as
sub-$\sigma$-algebras of $\mathcal{B}$ (or equivalently, as $L^{2}$-closures
of sub-algebras of $L^{\infty}(X)\subseteq L^{2}(X)$). Let $f\in
L^{2}(X^{\prime})$. We want to show $\mathbb{E}\left[{f}\mid{Z}\right]\in
L^{2}(Z^{\prime})$.

We know $Z$ is a compact abelian group and $L^{2}(Z)$ is spanned by characters
on $Z$. Let $\Lambda=\widehat{Z}$. We may think of $\Lambda$ as a subgroup of
$\widehat{G}$. Hence,
$\mathbb{E}\left[{f}\mid{Z}\right]=\sum_{\lambda\in\Lambda}{\left\langle
f,e_{\lambda}\right\rangle e_{\lambda}}$. Suppose $\left\langle
f,e_{\lambda}\right\rangle\neq 0$. Then

$\displaystyle\left\langle\overline{\lambda(g)}T_{g}f,e_{\lambda}\right\rangle=\left\langle
T_{g}f,T_{g}e_{\lambda}\right\rangle=\left\langle
f,e_{\lambda}\right\rangle\neq 0.$

Thus, $h:=\text{UC-}\lim_{g\in G}{\overline{\lambda}(g)T_{g}f}\neq 0$. But $h$
is an eigenfunction in $L^{2}(X^{\prime})$ with eigenvalue $\lambda$.
Therefore, $h\in L^{2}(Z^{\prime})$. It follows that
$\mathbb{E}\left[{f}\mid{Z}\right]\in L^{2}(Z^{\prime})$ as desired. ∎

The lemmas in [FuWe96, Section 9] are again of a general nature and allow for
further reduction to abelian group extensions (assuming $\mathbf{X}$ is
normal).

######  Theorem 5.3.

Let $\mathbf{X}$ be a normal ergodic system. There is a factor $\mathbf{Y}$ of
$\mathbf{X}$ that is an abelian group extension of the Kronecker factor,
$Y=Z\times_{\rho}H$, such that $\mathbf{Y}$ is characteristic for the averages
(59).

##  6 Conze–Lesigne factors

Now we restrict to the case $\theta=\varphi+\psi$ in (59). Instead of handling
the cocycle $\rho_{g}:Z\to H$ appearing in Theorem 5.3 directly, we will deal
with the family of cocycles $\sigma_{g}=\chi\circ\rho_{g}:Z\to S^{1}$, where
$\chi\in\widehat{H}$ is a character.

The next step in [FuWe96] is to use a clever argument involving the notion of
Mackey groups to reduce to an identity for the resulting cocycles
$\sigma_{g}:Z\to S^{1}$:

$\displaystyle\sigma_{\varphi(g)}(z_{1})p_{g}(z_{2})q_{g}(z_{3})=\frac{F\left(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)},z_{3}+\widehat{(\varphi+\psi)(g)}\right)}{F(z_{1},z_{2},z_{3})},$
(61)

where $p_{g},q_{g}:Z\to S^{1}$, $z_{1},z_{2},z_{3}\in
W_{\varphi,\psi,\varphi+\psi}$, and $F:W_{\varphi,\psi,\varphi+\psi}\to
S^{1}$. This argument is contained in [FuWe96, Section 10] for
$\mathbb{Z}$-systems. The required technical results are all of a general
group-theoretic nature, so the same argument works for our setting. We proceed
from this identity following the approach in [FuWe96]. Several modifications
are required to push the results from $\mathbb{Z}$-systems to general
$G$-systems, but the skeleton of the argument is the same.

For $\varphi,\psi:G\to G$, recall
$Z_{\varphi,\psi}:=\overline{\left\\{\left(\widehat{\varphi(g)},\widehat{\psi(g)}\right):g\in
G\right\\}}$. Let $\Delta=Z_{id,id}\cap Z_{\varphi,\psi}$ and
$W_{\varphi,\psi}=Z_{id,id}+Z_{\varphi,\psi}$. By the second isomorphism
theorem, $W_{\varphi,\psi}/Z_{\varphi,\psi}\cong Z_{id,id}/\Delta$.

Let $\lambda_{0}:Z_{id,id}/\Delta\to Z_{id,id}$ be a measurable cross-section
of the quotient homomorphism. Using $W_{\varphi,\psi}/Z_{\varphi,\psi}\cong
Z_{id,id}/\Delta$ and $Z_{id,id}\cong Z$, lift this to a measurable map
$\lambda:W_{\varphi,\psi}\to Z$. Since this comes from the cross-section, we
have $\left(z_{1}-\lambda(z_{1},z_{2}),z_{2}-\lambda(z_{1},z_{2})\right)\in
Z_{\varphi,\psi}$ for every $(z_{1},z_{2})\in W_{\varphi,\psi}$, and
$\lambda(z_{1}+u,z_{2}+v)=\lambda(z_{1},z_{2})$ for $(u,v)\in
Z_{\varphi,\psi}$.

For $(z_{1},z_{2})\in W_{\varphi,\psi}$, define
$u(z_{1},z_{2}):=z_{1}+z_{2}-\lambda(z_{1},z_{2})$.

######  Lemma 6.1.

For $(z_{1},z_{2})\in W_{\varphi,\psi}$, $(z_{1},z_{2},u(z_{1},z_{2}))\in
W_{\varphi,\psi,\theta}$.

###### Proof.

Since $\left(z_{1}-\lambda(z_{1},z_{2}),z_{2}-\lambda(z_{1},z_{2})\right)\in
Z_{\varphi,\psi}$, we can find a sequence $(g_{k})_{k\in\mathbb{N}}$ in $G$ so
that

$\displaystyle
z_{1}-\lambda(z_{1},z_{2})=\lim_{k\to\infty}{\widehat{\varphi(g_{k})}};$
$\displaystyle
z_{2}-\lambda(z_{1},z_{2})=\lim_{k\to\infty}{\widehat{\psi(g_{k})}}.$

Thus, since $\varphi+\psi=\theta$, we have

$\displaystyle u(z_{1},z_{2})-\lambda(z_{1},z_{2})$
$\displaystyle=\left(z_{1}-\lambda(z_{1},z_{2})\right)+\left(z_{2}-\lambda(z_{1},z_{2})\right)$
$\displaystyle=\lim_{k\to\infty}{\widehat{\varphi(g_{k})}}+\lim_{k\to\infty}{\widehat{\psi(g_{k})}}$
$\displaystyle=\lim_{k\to\infty}{\left(\widehat{\varphi(g_{k})}+\widehat{\psi(g_{k})}\right)}$
$\displaystyle=\lim_{k\to\infty}{\widehat{\theta(g_{k})}}.$

Therefore,

$\displaystyle(z_{1},z_{2},u(z_{1},z_{2}))=(\lambda(z_{1},z_{2}),\lambda(z_{1},z_{2}),\lambda(z_{1},z_{2}))+\lim_{k\to\infty}{\left(\widehat{\varphi(g_{k})},\widehat{\psi(g_{k})},\widehat{\theta(g_{k})}\right)}\in
W_{\varphi,\psi,\theta}.$

∎

Now let $\Delta^{\prime}:=\\{\delta\in Z:(\delta,\delta)\in\Delta\\}$. For
$\delta\in\Delta^{\prime}$, the map
$\lambda_{\delta}(z_{1},z_{2}):=\lambda(z_{1},z_{2})+\delta$ is another cross-
section, so Lemma 6.1 also holds for
$u_{\delta}(z_{1},z_{2}):=z_{1}+z_{2}-\lambda_{\delta}(z_{1},z_{2})$.

Now define a map $\widetilde{u}:W_{\varphi,\psi}\times\Delta^{\prime}\to
W_{\varphi,\psi,\theta}$ by
$\widetilde{u}(z_{1},z_{2},\delta):=\left(z_{1},z_{2},u_{\delta}(z_{1},z_{2})\right)$.

######  Lemma 6.2.

The map $\widetilde{u}:W_{\varphi,\psi}\times\Delta^{\prime}\to
W_{\varphi,\psi,\theta}$ is onto and preserves the Haar measure.

###### Proof.

Suppose $(z_{1},z_{2},z_{3})\in W_{\varphi,\psi,\theta}$. Then there is a
sequence $(g_{k})_{k\in\mathbb{N}}$ in $G$ and an element $z\in Z$ such that

$\displaystyle z_{1}=z+\lim_{k\to\infty}{\widehat{\varphi(g_{k})}};$
$\displaystyle z_{2}=z+\lim_{k\to\infty}{\widehat{\psi(g_{k})}};$
$\displaystyle z_{3}=z+\lim_{k\to\infty}{\widehat{\theta(g_{k})}}.$

Let $\delta:=z-\lambda(z,z)\in\Delta^{\prime}$. We have:

$\displaystyle u_{\delta}(z_{1},z_{2})$
$\displaystyle=z_{1}+z_{2}-\lambda(z_{1},z_{2})-\delta$
$\displaystyle=2z+\lim_{k\to\infty}{\left(\widehat{\varphi(g_{k})}+\widehat{\psi(g_{k})}\right)}-\lambda(z,z)-\delta$
$\displaystyle=2z+(z_{3}-z)-\lambda(z,z)-\left(z-\lambda(z,z)\right)$
$\displaystyle=z_{3}.$

Hence $\widetilde{u}(z_{1},z_{2},\delta)=(z_{1},z_{2},z_{3})$, so
$\widetilde{u}$ is onto.

Moreover, for each $\delta\in\Delta^{\prime}$, the map
$(z_{1},z_{2})\mapsto\left(z_{1},z_{2},u_{\delta}(z_{1},z_{2})\right)$ is a
cross-section of the homomorphism $W_{\varphi,\psi,\theta}\to
W_{\varphi,\psi}$, so $\widetilde{u}$ preserves the Haar measure. ∎

######  Lemma 6.3.

For $(z_{1},z_{2})\in W_{\varphi,\psi}$, $g\in G$, and
$\delta\in\Delta^{\prime}$:

$\displaystyle
u_{\delta}(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)})=u_{\delta}(z_{1},z_{2})+\widehat{\theta(g)}.$

###### Proof.

$\displaystyle u_{\delta}(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)})$
$\displaystyle=z_{1}+\widehat{\varphi(g)}+z_{2}+\widehat{\psi(g)}-\lambda_{\delta}(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)})$
$\displaystyle=z_{1}+z_{2}+\widehat{\theta(g)}-\lambda_{\delta}(z_{1},z_{2})$
$\displaystyle=u_{\delta}(z_{1},z_{2})+\widehat{\theta(g)}.$

∎

Recall equation (61):

$\displaystyle\sigma_{\varphi(g)}(z_{1})p_{g}(z_{2})q_{g}(z_{3})=\frac{F\left(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)},z_{3}+\widehat{(\varphi+\psi)(g)}\right)}{F(z_{1},z_{2},z_{3})}.$

We now wish to eliminate the variable $z_{3}$.

First, for $(u,v,w)\in Z_{\varphi,\psi,\theta}$, we replace
$(z_{1},z_{2},z_{3})$ by

$\displaystyle(z_{1}+(u-w),z_{2}+(v-w),z_{3})=(z_{1}+u,z_{2}+v,z_{3}+w)-(w,w,w)\in
W_{\varphi,\psi,\theta}.$

Substituting into (61), we get

$\displaystyle\sigma_{\varphi(g)}(z_{1}+(u-w))p_{g}(z_{2}+(v-w))q_{g}(z_{3})=\frac{F\left(z_{1}+(u-w)+\widehat{\varphi(g)},z_{2}+(v-w)+\widehat{\psi(g)},z_{3}+\widehat{\theta(g)}\right)}{F(z_{1}+(u-w),z_{2}+(v-w),z_{3})}.$

Dividing by the original expression gives

$\displaystyle\frac{\sigma_{\varphi(g)}(z_{1}+(u-w))}{\sigma_{\varphi(g)}(z_{1})}\frac{p_{g}(z_{2}+(v-w))}{p_{g}(z_{2})}=\frac{F_{u,v,w}\left(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)},z_{3}+\widehat{\theta(g)}\right)}{F_{u,v,w}(z_{1},z_{2},z_{3})},$
(62)

where

$\displaystyle
F_{u,v,w}(z_{1},z_{2},z_{3}):=\frac{F\left(z_{1}+u-w,z_{2}+v-w,z_{3}\right)}{F(z_{1},z_{2},z_{3})}.$

We now want to replace $z_{3}$ by $u_{\delta}(z_{1},z_{2})$. The equation (62)
holds for almost every $(z_{1},z_{2},z_{3})\in W_{\varphi,\psi,\theta}$ and
$(u,v,w)\in Z_{\varphi,\psi,\theta}$. By Lemma 5.2, we can decompose
$W_{\varphi,\psi,\theta}$ (up to isomorphism) as the Cartesian product
$W_{\varphi,\psi}\times\Delta^{\prime}$. By Fubini’s theorem, it follows that
(62) holds on almost every slice
$(z_{1},z_{2},z_{3})\in\widetilde{u}\left(W_{\varphi,\psi}\times\\{\delta\\}\right)$.
That is, for almost every $\delta\in\Delta^{\prime}$, (62) holds almost
everywhere upon replacing $z_{3}$ by $u_{\delta}(z_{1},z_{2})$. Fix any such
$\delta$, and let
$H_{u,v,w}(z_{1},z_{2}):=F_{u,v,w}\left(z_{1},z_{2},u_{\delta}(z_{1},z_{2})\right)$.
Applying Lemma 6.3, we now have

$\displaystyle\frac{\sigma_{\varphi(g)}(z_{1}+(u-w))}{\sigma_{\varphi(g)}(z_{1})}\frac{p_{g}(z_{2}+(v-w))}{p_{g}(z_{2})}=\frac{H_{u,v,w}\left(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)}\right)}{H_{u,v,w}(z_{1},z_{2})}.$
(63)

We want to use this identity to get a similar identity for the cocycle
$\sigma$ on its own. To do so, we need the following lemma:

######  Lemma 6.4.

Let $\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ and
$\mathbf{Y}=\left(Y,\mathcal{D},\nu,(S_{g})_{g\in G}\right)$ be ergodic
systems. Suppose $(\alpha_{g})_{g\in G}$ and $(\beta_{g})_{g\in G}$ are
cocycles for $\mathbf{X}$ and $\mathbf{Y}$ respectively, each taking values in
$S^{1}$. Suppose $L:X\times Y\to\mathbb{C}$ is not 0 a.e. and satisfies for
all $g\in G$

$\displaystyle\alpha_{g}(x)\beta_{g}(y)L(x,y)=L(T_{g}x,S_{g}y).$ (64)

Then there are functions $M:X\to S^{1}$ and $N:Y\to S^{1}$ and characters
$c^{\prime},c^{\prime\prime}\in\widehat{G}$ so that

$\displaystyle\alpha_{g}(x)=c^{\prime}(g)\frac{M(T_{g}x)}{M(x)},\quad\beta_{g}(y)=c^{\prime\prime}(g)\frac{N(S_{g}y)}{N(y)}.$
(65)

###### Proof.

Form $S^{1}$-extensions by defining

$\displaystyle\widetilde{T}_{g}(x,\zeta)=\left(T_{g}x,\alpha_{g}(x)^{-1}\zeta\right),\quad\widetilde{S}_{g}(y,\eta)=\left(S_{g}y,\beta_{g}(y)^{-1}\eta\right).$

These are actions because $\alpha$ and $\beta$ are cocycles for $T$ and $S$
respectively. Denote by $\widetilde{X}$ and $\widetilde{Y}$ the product spaces
$\widetilde{X}=X\times S^{1}$ and $\widetilde{Y}=Y\times S^{1}$. Define
$\widetilde{L}:\widetilde{X}\times\widetilde{Y}\to\mathbb{C}$ by

$\displaystyle\widetilde{L}(x,\zeta;y,\eta):=\zeta\eta L(x,y).$

By construction, $\widetilde{L}$ is $\widetilde{T}\times\widetilde{S}$
invariant:

$\displaystyle\left(\widetilde{T}_{g}\times\widetilde{S}_{g}\right)\widetilde{L}(x,\zeta;y,\eta)$
$\displaystyle=\left(\alpha_{g}(x)^{-1}\zeta\right)\left(\beta_{g}(y)^{-1}\eta\right)L(T_{g}x,S_{g}y)$
$\displaystyle=\alpha_{g}(x)^{-1}\zeta\beta_{g}(y)^{-1}\eta\alpha_{g}(x)\beta_{g}(y)L(x,y)$
$\displaystyle=\zeta\eta L(x,y)$
$\displaystyle=\widetilde{L}(x,\zeta;y,\eta).$

It follows that $\widetilde{L}$ comes from the product of the Kronecker
factors for $\widetilde{T}$ and $\widetilde{S}$, so we can write
$\widetilde{L}$ as a sum of products of eigenfunctions:

$\displaystyle\zeta\eta L(x,y)=\sum_{i}{H_{i}(x,\zeta)K_{i}(y,\eta)},$ (66)

where $\widetilde{T}_{g}H_{i}=c^{\prime}_{i}(g)H_{i}$ and
$\widetilde{S}_{g}K_{i}=c^{\prime}_{i}(g)^{-1}K_{i}$ for some character
$c^{\prime}_{i}\in\widehat{G}$. Now we can write

$\displaystyle H_{i}(x,\zeta)=\sum_{j}{h_{ij}(x)\zeta^{j}},\quad
K_{i}(y,\eta)=\sum_{j}{k_{ij}(y)\eta^{j}}.$

Substituting back into (66) gives

$\displaystyle\zeta\eta
L(x,y)=\sum_{i,j,j^{\prime}}{\zeta^{j}\eta^{j^{\prime}}h_{ij}(x)k_{ij^{\prime}}(y)}$

But $L$ does not depend on $\zeta$ or $\eta$, so only the $j=j^{\prime}=1$
term contributes, and evaluating at $\zeta=\eta=1$, we have

$\displaystyle L(x,y)=\sum_{i}{h_{i1}(x)k_{i1}(y)}$

Therefore, for some $i$, $h_{i1}\not\equiv 0$ and $k_{i1}\not\equiv 0$.

But

$\displaystyle\widetilde{T}_{g}\left(\sum_{j}{h_{ij}(x)\zeta^{j}}\right)=c^{\prime}_{i}(g)\sum_{j}{h_{ij}(x)\zeta^{j}},$

so

$\displaystyle\sum_{j}{h_{ij}(T_{g}x)\alpha_{g}(x)^{-j}\zeta^{j}}=c^{\prime}_{i}(g)\sum_{j}{h_{ij}(x)\zeta^{j}}.$

Matching powers of $\zeta$, we get
$h_{i1}(T_{g}x)\alpha_{g}(x)^{-1}=c^{\prime}_{i}(g)h_{i1}(x)$. But $T$ is
ergodic, so taking the absolute value of both sides, we find that $|h_{i1}|$
is constant, say $|h_{i1}|=\lambda\neq 0$. Let $M(x)=\lambda^{-1}h_{i1}(x)$
and $c^{\prime}(g)=c^{\prime}_{i}(g)^{-1}$. Then $\alpha_{g}(x)$ has the
desired form (65). This process can be repeated to show that $\beta_{g}(y)$
also has this form. ∎

For $(u,v,w)\in Z_{\varphi,\psi,\theta}$, we can define a cocycle

$\displaystyle\alpha_{g}(x):=\frac{\sigma_{\varphi(g)}(x+(u-w))}{\sigma_{\varphi(g)}(x)}.$

However, Lemma 6.4 does not immediately apply, since the maps $z_{1}\mapsto
z_{1}+\widehat{\varphi(g)}$ and $z_{2}\mapsto z_{2}+\widehat{\psi(g)}$ may not
be ergodic, and (63) only holds on $W_{\varphi,\psi}\subseteq Z\times Z$. We
solve the first problem by taking ergodic components $\widehat{j}+Z_{\varphi}$
and $\widehat{j}+Z_{\psi}$. For the latter issue, we need the following lemma
(note that $\xi$ will have finite index image, since $(\varphi,\psi)$ is an
admissible pair):

######  Lemma 6.5.

Let $\xi=\psi-\varphi$. Then $Z_{\xi}\times Z_{\xi}\subseteq
W_{\varphi,\psi}$.

###### Proof.

It suffices to show $(\widehat{\xi(g)},\widehat{\xi(h)})\in W_{\varphi,\psi}$
for all $g,h\in G$, since $W_{\varphi,\psi}$ is a closed subgroup of $Z\times
Z$. Suppose $g,h\in G$. Then set $k:=h-g$ and
$z:=\widehat{\psi(g)}-\widehat{\varphi(h)}$. Then

$\displaystyle(\widehat{\xi(g)},\widehat{\xi(h)})$
$\displaystyle=(\widehat{\psi(g)}-\widehat{\varphi(g)},\widehat{\psi(h)}-\widehat{\varphi(h)})$
$\displaystyle=(\widehat{\psi(g)}-\widehat{\varphi(h)}+\widehat{\varphi(h)}-\widehat{\varphi(g)},\widehat{\psi(h)}-\widehat{\psi(g)}+\widehat{\psi(g)}-\widehat{\varphi(h)})$
$\displaystyle=(z+\widehat{\varphi(k)},z+\widehat{\psi(k)})\in
W_{\varphi,\psi}.$

∎

From this lemma, we can conclude that

$\displaystyle\left(\widehat{j}+Z_{\xi\circ\varphi}\right)\times\left(\widehat{j}+Z_{\xi\circ\psi}\right)\subseteq
W_{\varphi,\psi}\cap\left(\left(\widehat{j}+Z_{\varphi}\right)\times\left(\widehat{j}+Z_{\psi}\right)\right).$

But $Z_{\xi\circ\varphi}$ and $Z_{\xi\circ\psi}$ have finite index in $Z$, so
$W_{\varphi,\psi}$ has positive measure intersection with the set
$\left(\widehat{j}+Z_{\varphi}\right)\times\left(\widehat{j}+Z_{\psi}\right)$.
Thus, if we define $H^{\prime}_{u,v,w}$ on
$\left(\widehat{j}+Z_{\varphi}\right)\times\left(\widehat{j}+Z_{\psi}\right)$
by

$\displaystyle
H^{\prime}_{u,v,w}(z_{1},z_{2})=\begin{cases}H_{u,v,w}(z_{1},z_{2}),&(z_{1},z_{2})\in
W_{\varphi,\psi};\\\ 0,&(z_{1},z_{2})\notin W_{\varphi,\psi},\end{cases}$

then $H^{\prime}_{u,v,w}$ is not zero almost everywhere. Moreover,

$\displaystyle
H^{\prime}_{u,v,w}(z_{1}+\widehat{\varphi(g)},z_{2}+\widehat{\psi(g)})=\alpha_{g}(z_{1})\beta_{g}(z_{2})H^{\prime}_{u,v,w}(z_{1},z_{2})$

for
$(z_{1},z_{2})\in\left(\widehat{j}+Z_{\varphi}\right)\times\left(\widehat{j}+Z_{\psi}\right)$,
where $\alpha$ and $\beta$ are the cocycles given by

$\displaystyle\alpha_{g}(x):=\frac{\sigma_{\varphi(g)}(x+(u-w))}{\sigma_{\varphi(g)}(x)},\quad\beta_{g}(x):=\frac{p_{g}(x+(v-w))}{p_{g}(x)}.$

Hence, applying Lemma 6.4, we have

$\displaystyle\frac{\sigma_{\varphi(g)}(z+(u-w))}{\sigma_{\varphi(g)}(z)}=c_{j}(g)\frac{K_{j}(u,v,w;z+\widehat{\varphi(g)})}{K_{j}(u,v,w;z)}$

for each $j\in G$ and $z\in\widehat{j}+Z_{\varphi}$. Observe that the left-
hand side is parametrized by $u-w$ with $u\in Z_{\varphi}$ and $w\in
Z_{\theta}$. Since $\theta=\varphi+\psi$, we have
$Z_{\varphi}-Z_{\theta}\supseteq Z_{-\psi}=Z_{\psi}$. Thus, we can restrict to
$u\in Z_{\psi}$, and choose the functions $K_{j}$ so that

$\displaystyle\frac{\sigma_{\varphi(g)}(z+u)}{\sigma_{\varphi(g)}(z)}=c_{j}(g)\frac{K_{j}(u,z+\widehat{\varphi(g)})}{K_{j}(u,z)}$

for $g\in G$, $z\in\widehat{j}+Z_{\varphi}$, and $u\in Z_{\psi}$. Now since
cosets are disjoint and cover all of $Z$, we can patch together $c_{j}$ and
$K_{j}$ for different values of $j$ to get a new expression

$\displaystyle\frac{\sigma_{\varphi(g)}(z+u)}{\sigma_{\varphi(g)}(z)}=\Lambda_{u}(z+Z_{\varphi})(g)\frac{K_{u}(z+\widehat{\varphi(g)})}{K_{u}(z)}$
(67)

that holds for $z\in Z$, $u\in Z_{\psi}$, and $g\in G$. The functions
$\Lambda_{u}$ and $K_{u}$ can be chosen to depend measurably on $u$ (see
[Les93, Proposition 2] and [FuWe96, Proposition 10.5]).

The foregoing discussion motivates the following definition (see [CoLes84] and
[FuWe96] for the case $G=\mathbb{Z}$):

######  Definition 6.6.

Let $\mathbf{X}$ be an ergodic system with Kronecker factor $Z$. A cocycle
$\rho_{g}:Z\to S^{1}$ is a _$(\varphi,\psi)$ -Conze–Lesigne cocycle_
($(\varphi,\psi)$-CL cocycle for short) if there are measurable functions
$\Lambda:Z_{\psi}\times\left(Z/Z_{\varphi}\right)\to\widehat{G}$ and
$K:Z_{\psi}\times Z\to S^{1}$ such that

$\displaystyle\frac{\rho_{\varphi(g)}(z+u)}{\rho_{\varphi(g)}(z)}=\Lambda_{u}(z+Z_{\varphi})(g)\frac{K_{u}(z+\widehat{\varphi(g)})}{K_{u}(z)}$
(68)

for almost all $z\in Z$, $u\in Z_{\psi}$, and all $g\in G$. The set of all
$(\varphi,\psi)$-CL cocycles is denoted by $CL_{\mathbf{X}}(\varphi,\psi)$.

If $\varphi(g)=rg$ and $\psi(g)=sg$ with $r,s\in\mathbb{Z}$, we will denote
$CL_{\mathbf{X}}(\varphi,\psi)$ by $CL_{\mathbf{X}}(r,s)$.

######  Definition 6.7.

Let $\mathbf{X}$ be an ergodic system with Kronecker factor $Z$. We call a
function $f:X\to S^{1}$ a _$(\varphi,\psi)$ -CL function_ if there is a
cocycle $\rho\in CL_{\mathbf{X}}(\varphi,\psi)$ such that

$\displaystyle f(T_{g}x)=\rho_{g}(z)f(x),$

where $z=\pi(x)$ for $\pi:X\to Z$ the projection onto the Kronecker factor.

Since $CL_{\mathbf{X}}(\varphi,\psi)$ is a group, it follows immediately that
the span of $(\varphi,\psi)$-CL functions forms a subalgebra of
$L^{\infty}(\mu)$. We call the corresponding factor the _$(\varphi,\psi)$ -CL
factor_, denoted by $\mathcal{B}_{CL(\varphi,\psi)}$. By the computations in
this section, we have reduced the general cocycle in Theorem 5.3 to cocycles
satisfying the Conze–Lesigne equation (68). We can therefore summarize the
results of this section by the following theorem:

######  Theorem 6.8.

Let $\mathbf{X}$ be a normal ergodic system, and let $\\{\varphi,\psi\\}$ be
an admissible pair of homomorphisms. Then

$\displaystyle\text{UC-}\lim_{g\in G}$ $\displaystyle\
{T_{\varphi(g)}f_{1}\cdot T_{\psi(g)}f_{2}\cdot T_{(\varphi+\psi)(g)}f_{3}}$
$\displaystyle=$ $\displaystyle~{}\text{UC-}\lim_{g\in
G}{T_{\varphi(g)}\mathbb{E}\left[{f_{1}}\mid{\mathcal{B}_{CL(\varphi,\psi)}}\right]\cdot
T_{\psi(g)}\mathbb{E}\left[{f_{2}}\mid{\mathcal{B}_{CL(\varphi,\psi)}}\right]\cdot
T_{(\varphi+\psi)(g)}\mathbb{E}\left[{f_{3}}\mid{\mathcal{B}_{CL(\varphi,\psi)}}\right]}$

in $L^{2}(\mu)$.

##  7 Limit formula for triple averages

The goal of this section is to prove a formula for the limit of the multiple
ergodic averages

$\displaystyle\text{UC-}\lim_{g\in
G}{f_{1}(T_{rg}x)f_{2}(T_{sg}x)f_{3}(T_{(r+s)g}x)},$

where $T$ is an ergodic action and $\\{r,s,r+s\\}$ is an admissible triple.
Our approach follows closely the method used by Host and Kra in [HKr02] in
establishing a formula for $\mathbb{Z}$-actions (see [HKr02, Theorem 12]). In
order to extend this method to our more general setting of countable discrete
abelian groups, we need to restrict our attention to a class of systems called
_quasi-affine systems_ (defined in Section 7.4).

######  Theorem 7.1.

Let $G$ be a countable discrete abelian group. Let $r,s\in\mathbb{Z}$ such
that $rG$, $sG$, and $(r\pm s)G$ have finite index in $G$. Let
$k^{\prime}_{1}=-rs(r+s)$, $k^{\prime}_{2}=rs(r+s)$, and
$k^{\prime}_{3}=-rs(s-r)$. Set
$D:=\gcd(k^{\prime}_{1},k^{\prime}_{2},k^{\prime}_{3})=rs\gcd(r+s,s-r)$ and
$k_{i}=\frac{k^{\prime}_{i}}{D}$. Let $b_{1},b_{2},b_{3}\in\mathbb{Z}$ so that
$\sum_{i=1}^{3}{k_{i}b_{i}}=1$.

Let $\mathbf{X}=\mathbf{Z}\times_{\sigma}H$ be an ergodic quasi-affine system.
There is a function $\psi:Z\times Z\to H$ such that $\psi(0,z)=0$,
$t\mapsto\psi(t,\cdot)$ is a continuous map from $Z$ to
$\mathcal{M}(Z,H)$,141414We denote by $\mathcal{M}(Z,H)$ the space of
measurable functions $Z\to H$ in the topology of convergence in measure. and
for every $f_{1},f_{2},f_{3}\in L^{\infty}(\mu)$,

$\displaystyle\text{UC-}\lim_{g\in
G}{f_{1}(T_{rg}x)f_{2}(T_{sg}x)f_{3}(T_{(r+s)g}x)}=\int_{Z\times
H^{2}}{\prod_{i=1}^{3}{f_{i}(z+a_{i}t,h+a_{i}u+a_{i}^{2}v+b_{i}\psi(t,z))}~{}du~{}dv~{}dt},$
(69)

in $L^{2}(\mu)$, where $x=(z,h)\in Z\times H$, and
$a_{1}=r,a_{2}=s,a_{3}=r+s$.

###  7.1 Cohomology for abelian group extensions

To enable our usage of cocycles, we introduce cohomology for abelian group
extensions.

######  Definition 7.2.

Let $\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ be a measure-
preserving system and $(H,+)$ a compact abelian group.

  1. 1.

Denote by $Z^{1}_{G}(\mathbf{X},H)$ the set of _cocycles_ , i.e. measurable
functions $\sigma:G\times X\to H$ satisfying the cocycle equation

$\displaystyle\sigma_{g+h}(x)=\sigma_{g}(T_{h}x)+\sigma_{h}(x).$

  2. 2.

A cocycle $\sigma\in Z^{1}_{G}(\mathbf{X},H)$ is a _coboundary_ if there is a
measurable function $F:X\to H$ such that

$\displaystyle\sigma_{g}(x)=F\left(T_{g}x\right)-F(x).$

We denote the set of all coboundaries by $B^{1}_{G}(\mathbf{X},H)$.

  3. 3.

Two cocycles $\sigma$ and $\sigma^{\prime}$ are _cohomologous_ , denoted
$\sigma\sim\sigma^{\prime}$, if $\sigma-\sigma^{\prime}\in
B^{1}_{G}(\mathbf{X},H)$.

Recall that, given a cocycle $\sigma\in Z^{1}_{G}(\mathbf{X},H)$, we denote by
$\mathbf{X}\times_{\sigma}H$ the $G$-system

$\displaystyle T^{\sigma}_{g}(x,y)=(T_{g}x,y+\sigma_{g}(x))$

for $(x,y)\in X\times H$ and $g\in G$.

The set of cocycles $Z^{1}_{G}(\mathbf{X},H)$ and the set of coboundaries
$B^{1}_{G}(\mathbf{X},H)$ both form groups under pointwise addition
$(\sigma+\sigma^{\prime})(g,x)=\sigma(g,x)+\sigma^{\prime}(g,x)$. Moreover,
$B^{1}_{G}(\mathbf{X},H)$ is a subgroup of $Z^{1}_{G}(\mathbf{X},H)$, and if
$\sigma\sim\sigma^{\prime}$, then
$\mathbf{X}\times_{\sigma}H\cong\mathbf{X}\times_{\sigma^{\prime}}H$. As a
result, the isomorphism class of the extension $\mathbf{X}\times_{\sigma}H$
depends only on the congruence class of $\sigma$ in the _cohomology group_
$H^{1}_{G}(\mathbf{X},H):=Z^{1}_{G}(\mathbf{X},H)/B^{1}_{G}(\mathbf{X},H)$.

Since we are ultimately interested in studying the Conze–Lesigne factor, which
is an extension of the Kronecker factor by a compact abelian group, we need
the following notions for cocycles defined on the Kronecker factor:

######  Definition 7.3.

Suppose $\mathbf{Z}$ is an ergodic Kronecker system.151515By this, we mean
that $\mathbf{Z}$ is measurably isomorphic to an ergodic action of $G$ by
rotations on a compact abelian group. Let $H$ be a compact abelian group, and
let $\sigma:G\times Z\to H$ be a cocycle.

  1. 1.

$\sigma$ is _ergodic_ if $\mathbf{Z}\times_{\sigma}H$ is ergodic.

  2. 2.

$\sigma$ is _weakly mixing_ if the extension by $H$ over $\sigma$ is
relatively weakly mixing. That is, $\mathbf{Z}\times_{\sigma}H$ is ergodic
with Kronecker factor equal to $\mathbf{Z}$.

It is helpful to study cocycles $\sigma\in Z^{1}_{G}(\mathbf{X},H)$ by
considering the family of cocycles $\chi\circ\sigma:G\times X\to S^{1}$ for
$\chi\in\widehat{H}$. Here we will use multiplicative notation, rather than
the additive notation above. For $S^{1}$-valued cocycles, we introduce an
additional definition:

######  Definition 7.4.

Let $\mathbf{X}=\left(X,\mathcal{B},\mu,(T_{g})_{g\in G}\right)$ be a measure-
preserving system. A cocycle $\rho$ is _cohomologous to a character_ if there
is a character $\gamma\in\widehat{G}$ such that $\rho_{g}(x)\sim\gamma(g)$.
That is, for some measurable function $F:X\to S^{1}$,

$\displaystyle\rho_{g}(x)=\gamma(g)\frac{F\left(T_{g}x\right)}{F(x)}$

for every $g\in G$ and almost every $x\in X$.

The following proposition characterizes dynamical properties of a cocycle
$\sigma:G\times Z\to H$ in terms of the behavior of the cocycles
$\chi\circ\sigma$ for $\chi\in\widehat{H}$:

######  Proposition 7.5.

Suppose $\mathbf{Z}$ is an ergodic Kronecker system. Let $H$ be a compact
abelian group, and let $\sigma:G\times Z\to H$ be a cocycle.

  1. (1)

$\sigma$ is ergodic if and only if, for every
$\chi\in\widehat{H}\setminus\\{1\\}$, $\chi\circ\sigma$ is not a coboundary.

  2. (2)

$\sigma$ is weakly mixing if and only if, for every
$\chi\in\widehat{H}\setminus\\{1\\}$, $\chi\circ\sigma$ is not cohomologous to
a character.

###### Proof.

(1). Suppose $\sigma$ is ergodic and $\chi\circ\sigma$ is a coboundary for
some $\chi\in\widehat{H}$. We want to show $\chi=1$. Write

$\displaystyle\chi\left(\sigma_{g}(z)\right)=\frac{F\left(z+\widehat{g}\right)}{F(z)}$
(70)

for some function $F:Z\to S^{1}$. Define $f:Z\times H\to S^{1}$ by
$f(z,x):=\overline{F}(z)\chi(x)$. The coboundary identity (70) implies that
$f$ is invariant for $\mathbf{Z}\times_{\sigma}H$. Since $\sigma$ is ergodic,
it follows that $f$ is constant. Therefore, $\chi=1$.

Conversely, suppose that for every $\chi\in\widehat{H}\setminus\\{1\\}$,
$\chi\circ\sigma$ is not a coboundary. Suppose $f$ is invariant for
$\mathbf{Z}\times_{\sigma}H$. We will show $f$ is constant. Write
$f(z,x)=\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}$. Invariance means

$\displaystyle\sum_{\chi\in\widehat{H}}{c_{\chi}\left(z+\widehat{g}\right)\chi(\sigma_{g}(z))\chi(x)}=\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}.$

So, for every $\chi\in\widehat{H}$,

$\displaystyle
c_{\chi}\left(z+\widehat{g}\right)\chi\left(\sigma_{g}(z)\right)=c_{\chi}(z).$

By ergodicity of $\mathbf{Z}$, $|c_{\chi}|$ is constant, say
$|c_{\chi}|=C_{\chi}$. Thus, if $c_{\chi}\neq 0$, then

$\displaystyle\chi\left(\sigma_{g}(z)\right)=\frac{C_{\chi}^{-1}\overline{c_{\chi}}\left(z+\widehat{g}\right)}{C_{\chi}^{-1}\overline{c_{\chi}}(z)},$

so $\chi\circ\sigma_{g}$ is a coboundary. It follows that $c_{\chi}=0$ for
$\chi\neq 1$, so $f(z,x)=c_{1}(z)$. But $\mathbf{Z}$ is ergodic, so $f$ is
constant.

(2). Suppose $\sigma$ is weakly mixing. Let $\chi\in\widehat{H}$, and assume
$\chi\circ\sigma$ is cohomologous to a character. We will show $\chi=1$. Let
$F:Z\to S^{1}$ and $\gamma\in\widehat{G}$ so that

$\displaystyle\chi\left(\sigma_{g}(z)\right)=\gamma(g)\frac{F\left(z+\widehat{g}\right)}{F(z)}.$

Then $f(z,x):=\overline{F}(z)\chi(x)$ is an eigenfunction of
$\mathbf{Z}\times_{\sigma}H$ with eigenvalue $\gamma$. Since $\sigma$ is
weakly mixing, it follows that $f$ is measurable over $\mathbf{Z}$. Therefore,
$\chi$ is constant, so $\chi=1$.

Conversely, suppose that for every $\chi\in\widehat{H}\setminus\\{1\\}$,
$\chi\circ\sigma$ is not cohomologous to a character. Suppose $f$ is an
eigenfunction of $\mathbf{Z}\times_{\sigma}H$ with eigenvalue
$\gamma\in\widehat{G}$. We will show that $f$ is measurable over $Z$. Write
$f(z,x)=\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}$. Then

$\displaystyle\sum_{\chi\in\widehat{H}}{c_{\chi}\left(z+\widehat{g}\right)\chi(\sigma_{g}(z))\chi(x)}=\gamma(g)\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}.$

Hence, for every $\chi\in\widehat{H}$,

$\displaystyle
c_{\chi}\left(z+\widehat{g}\right)\chi\left(\sigma_{g}(z)\right)=\gamma(g)c_{\chi}(z),$

so $\chi\circ\sigma\sim\gamma$ for any $\chi\in\widehat{H}$ with $c_{\chi}\neq
0$. It follows that $f(z,x)=c_{1}(z)$. ∎

The characterization of ergodicity in Proposition 7.5 motivates the following
definition:

######  Definition 7.6.

Let $\mathbf{Z}$ be an ergodic Kronecker system, $H$ a compact abelian group,
and $\sigma:G\times Z\to H$ a cocycle. The _Mackey group associated to
$\sigma$_ is the subgroup

$\displaystyle M:=\left\\{x\in H:\chi(x)=1~{}\text{for
all}~{}\chi\in\widehat{H}~{}\text{such that}~{}\chi\circ\sigma~{}\text{is a
coboundary}\right\\}$

Note that the annihilator161616The _annihilator_ of a set $A$ in a group $H$
is the set $A^{\perp}:=\left\\{\chi\in\widehat{H}:\chi(a)=1~{}\text{for
every}~{}a\in A\right\\}$. of $M$ is given by
$M^{\perp}=\left\\{\chi\in\widehat{H}:\chi\circ\sigma~{}\text{is a
coboundary}\right\\}$. Using the isomorphism $M^{\perp}=\widehat{H/M}$,
Proposition 7.5 says that $\sigma$ is ergodic if and only if $M=H$ if and only
if $M^{\perp}=\\{1\\}$. The following proposition refines this criterion to
describe the splitting of $L^{2}(Z\times H)$ into invariant and ergodic
functions:

######  Proposition 7.7.

Let $\mathbf{Z}$ be an ergodic Kronecker system, $H$ a compact abelian group,
and $\sigma:G\times Z\to H$ a cocycle. Let $M$ be the Mackey group associated
to $\sigma$. Let $f\in L^{2}(Z\times H)$. Suppose for every $\chi\in
M^{\perp}$,

$\displaystyle\int_{H}{f(z,x)\overline{\chi(x)}~{}dx}=0$

for almost every $z\in Z$. Then

$\displaystyle\text{UC-}\lim_{g\in
G}{f\left(z+\widehat{g},x+\sigma_{g}(z)\right)}=0$

in $L^{2}(Z\times H)$.

###### Proof.

Let $T^{\sigma}_{g}(z,x):=\left(z+\widehat{g},x+\sigma_{g}(z)\right)$. Write
$f=f_{1}+f_{2}$, where $f_{1}$ is $T^{\sigma}$-invariant and $f_{2}$ is in the
ergodic subspace for $T^{\sigma}$. We want to show $f_{1}=0$. We can expand
$f_{1}(z,x)=\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}$. Using
$T^{\sigma}$-invariance, we have

$\displaystyle\sum_{\chi\in\widehat{H}}{c_{\chi}\left(z+\widehat{g}\right)\chi\left(\sigma_{g}(z)\right)\chi(x)}=\sum_{\chi\in\widehat{H}}{c_{\chi}(z)\chi(x)}.$

Thus,

$\displaystyle
c_{\chi}\left(z+\widehat{g}\right)\chi\left(\sigma_{g}(z)\right)=c_{\chi}(z)$

for every $\chi\in\widehat{H}$. If $c_{\chi}\neq 0$, this implies
$\chi\circ\sigma$ is a coboundary, so $\chi\in M^{\perp}$. But for $\chi\in
M^{\perp}$,

$\displaystyle c_{\chi}(z)=\int_{H}{f(z,x)\overline{\chi(x)}~{}dx}=0.$

Therefore, $f_{1}=0$. ∎

###  7.2 Mackey groups for admissible triples

When dealing with multiple recurrence, one needs a modified version of the
Mackey group. First we define several notations. Let $\mathbf{Z}$ be an
ergodic Kronecker system and $H$ a compact abelian group. Fix an admissible
triple $\\{a_{1},a_{2},a_{3}\\}\subseteq\mathbb{Z}$ and a cocycle
$\sigma:G\times Z\to H$. Let $W\subseteq Z^{3}$ be the subgroup

$\displaystyle
W:=W_{a_{1},a_{2},a_{3}}=\left\\{(z+a_{1}t,z+a_{2}t,z+a_{3}t):z,t\in
Z\right\\}.$

For $g\in G$, let
$\alpha_{g}:=(a_{1}\widehat{g},a_{2}\widehat{g},a_{3}\widehat{g})\in W$, and
define $\widetilde{S}_{g}:W\to W$ by $\widetilde{S}_{g}(w)=w+\alpha_{g}$. Let
$\widetilde{\sigma}:G\times W\to H^{3}$ be the cocycle

$\displaystyle\widetilde{\sigma}_{g}(w_{1},w_{2},w_{3})=\left(\sigma_{a_{1}g}(w_{1}),\sigma_{a_{2}g}(w_{2}),\sigma_{a_{3}g}(w_{3})\right).$

Finally, let $M=M(a_{1},a_{2},a_{3})\subseteq H^{3}$ be the Mackey group
associated to the cocycle $\widetilde{\sigma}$.

The action $\widetilde{S}$ is not ergodic, but its ergodic decomposition is
easy to describe. For each $z\in Z$, let
$W_{z}:=\left\\{(z+a_{1}t,z+a_{2}t,z+a_{3}t):t\in Z\right\\}$. Note that
$W_{0}$ is an $\widetilde{S}$-invariant subgroup of $Z^{3}$, and
$(W_{0},\widetilde{S})$ is uniquely ergodic by Lemma 2.4. Now, for each $z\in
Z$, $(W_{z},\widetilde{S})$ is an isomorphic topological dynamical system
supporting a unique invariant measure $m_{z}$, which is a shift of the Haar
measure $m_{0}$ on $W_{0}$. We claim that the Haar measure $m_{W}$ on $W$ can
be decomposed as

$\displaystyle m_{W}=\int_{Z}{m_{z}~{}dz}.$

To see this, let $f:W\to\mathbb{C}$ be a continuous function and
$w_{0}=(u+a_{1}v,u+a_{2}v,u+a_{3}v)\in W$. Then

$\displaystyle\int_{Z}{\int_{W}{f(w+w_{0})~{}dm_{z}(w)}~{}dz}$
$\displaystyle=\int_{Z}{\int_{W}{f\left(w+(u,u,u)\right)~{}dm_{z}(w)}~{}dz}$
$\displaystyle(m_{z}~{}\text{is}~{}W_{0}\text{-invariant})$
$\displaystyle=\int_{Z}{\int_{W}{f(w)~{}dm_{z+u}(w)}~{}dz}$
$\displaystyle(W_{z}+(u,u,u)=W_{z+u})$
$\displaystyle=\int_{Z}{\int_{W}{f(w)~{}dm_{z}(w)}~{}dz}.$
$\displaystyle(\text{Haar measure on}~{}Z~{}\text{is shift-invariant})$

Thus, the measure $\int_{Z}{m_{z}~{}dz}$ is shift-invariant on $W$. By the
uniqueness of Haar measure, $m_{W}=\int_{Z}{m_{z}~{}dz}$ as claimed.

######  Remark 7.8.

The decomposition given here is not the standard ergodic decomposition, since
the cosets $W_{z}$ may coincide for different values of $z\in Z$. However,
this form will be more convenient for our purposes.

For each $z\in Z$, $\widetilde{\sigma}$ is defined $m_{z}$-a.e., so we can
define the Mackey groups

$\displaystyle
M_{z}^{\perp}:=\left\\{\widetilde{\chi}\in\widehat{H}^{3}:\widetilde{\chi}\circ\widetilde{\sigma}~{}\text{is
a coboundary for}~{}(W_{z},m_{z},\widetilde{S})\right\\}.$

Ergodicity of the base system $\mathbf{Z}$ allows us to prove the following:

######  Proposition 7.9.

For almost every $z\in Z$, $M_{z}=M$.

###### Proof.

Let
$\stackrel{{\scriptstyle\triangle}}{{g}}=\left(\widehat{g},\widehat{g},\widehat{g}\right)$,
and let
$\stackrel{{\scriptstyle\triangle}}{{S}}_{g}(w)=w+\stackrel{{\scriptstyle\triangle}}{{g}}$.
Then for each $g\in G$, $\stackrel{{\scriptstyle\triangle}}{{S}}_{g}$ gives an
isomorphism
$(W_{z},m_{z},\widetilde{S})\cong(W_{z+\widehat{g}},m_{z+\widehat{g}},\widetilde{S})$.
Moreover, by the cocycle equation,

$\displaystyle\sigma_{a_{i}g}\left(w_{i}+\widehat{h}\right)=\sigma_{a_{i}g}(w_{i})+\sigma_{h}\left(w_{i}+a_{i}\widehat{g}\right)-\sigma_{h}(w_{i}),$

so

$\displaystyle\widetilde{\sigma}_{g}\left(\stackrel{{\scriptstyle\triangle}}{{S}}_{h}w\right)=\widetilde{\sigma}_{g}(w)+\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}\left(\widetilde{S}_{g}w\right)-\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}(w),$
(71)

where
$\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}(w)=\left(\sigma_{h}(w_{1}),\sigma_{h}(w_{2}),\sigma_{h}(w_{3})\right)$.

Now suppose $\widetilde{\chi}\in M_{z}^{\perp}$. Then there is a function
$F:W_{z}\to S^{1}$ such that

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}(w)\right)=\frac{F\left(\widetilde{S}_{g}w\right)}{F(w)}$

for every $g\in G$ and almost every $w\in W_{z}$. Then by (71), we have

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}\left(\stackrel{{\scriptstyle\triangle}}{{S}}_{h}w\right)\right)$
$\displaystyle=\frac{F\left(\widetilde{S}_{g}w\right)}{F(w)}\frac{\chi\left(\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}\left(\widetilde{S}_{g}w\right)\right)}{\chi\left(\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}(w)\right)}.$

Letting
$\Phi(w):=F(w)\chi\left(\stackrel{{\scriptstyle\triangle}}{{\sigma}}_{h}(w)\right)$,
this simplifies to

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}\left(\stackrel{{\scriptstyle\triangle}}{{S}}_{h}w\right)\right)=\frac{\Phi\left(\widetilde{S}_{g}w\right)}{\Phi(w)},$

so $\widetilde{\chi}\in M_{z+\widehat{h}}^{\perp}$. We can repeat this
argument to show that $M_{z+\widehat{h}}^{\perp}\subseteq
M_{z+\widehat{h}-\widehat{h}}^{\perp}=M_{z}^{\perp}$. Thus,
$M_{z}=M_{z+\widehat{h}}$ for every $h\in G$.

Now, for each $\widetilde{\chi}\in\widehat{H}^{3}$, $\\{z\in
Z:\widetilde{\chi}\in M_{z}^{\perp}\\}$ is measurable (see [Les93, Proposition
2]) and therefore has measure either $0$ or $1$ by ergodicity of $\mathbf{Z}$.
Since $\widehat{H}^{3}$ is countable, it follows that $M_{z}$ is constant
a.e., say $M_{z}=N$. It remains to check $M=N$.

Let $\widetilde{\chi}\in N^{\perp}$. Then
$\widetilde{\chi}\circ\widetilde{\sigma}$ is a coboundary for
$(W_{z},m_{z},\widetilde{S})$ for almost every $z\in Z$. That is, for almost
every $z\in Z$, there is a function $F_{z}:W_{z}\to S^{1}$ such that

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}(w)\right)=\frac{F_{z}\left(\widetilde{S}_{g}w\right)}{F_{z}(w)}$

for $m_{z}$-almost every $w\in W_{z}$. Define
$F\left(z+a_{1}t,z+a_{2}t,z+a_{3}t\right):=F_{z}(z+a_{1}t,z+a_{2}t,z+a_{3}t)$.
We can ensure this is well-defined by taking $F_{z}=F_{z^{\prime}}$ whenever
$W_{z}=W_{z^{\prime}}$. Moreover, we can ensure that the map $z\mapsto F_{z}$
is measurable (see [Les93, Proposition 2]) so that $F$ is a measurable
function. Then

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}(w)\right)=\frac{F\left(\widetilde{S}_{g}w\right)}{F(w)}$

for almost every $w\in W$. Hence, $\widetilde{\chi}\in M^{\perp}$.

Now suppose $\widetilde{\chi}\in M^{\perp}$. Then there is a function $F:W\to
S^{1}$ such that

$\displaystyle\widetilde{\chi}\left(\widetilde{\sigma}_{g}(w)\right)=\frac{F\left(\widetilde{S}_{g}w\right)}{F(w)}$

for almost every $w\in W$. By Fubini’s theorem, it follows that this equation
holds for $m_{z}$-almost every $w\in W_{z}$ for almost every $z\in Z$. That
is, $\widetilde{\chi}\in M_{z}^{\perp}$ a.e., so $\widetilde{\chi}\in
N^{\perp}$. ∎

We use this to prove a version of Proposition 7.7 for $M(a_{1},a_{2},a_{3})$:

######  Proposition 7.10.

Assume $\sigma$ is weakly mixing. Let $M=M(a_{1},a_{2},a_{3})$ be the Mackey
group associated to $\widetilde{\sigma}$. Let $f_{1},f_{2},f_{3}\in
L^{\infty}(Z\times H)$. Suppose for every $\widetilde{\chi}\in M^{\perp}$,

$\displaystyle\int_{H^{3}}{\prod_{i=1}^{3}{f_{i}(w_{i},x_{i})}\overline{\widetilde{\chi}(x)}~{}dx}=0$
(72)

for almost every $w=(w_{1},w_{2},w_{3})\in W$. Then

$\displaystyle\text{UC-}\lim_{g\in
G}{\prod_{i=1}^{3}{f_{i}\left(z+a_{i}\widehat{g},x+\sigma_{a_{i}g}(z)\right)}}=0$

in $L^{2}(Z\times H)$.

###### Proof.

Let $T^{\sigma}_{g}(z,h):=\left(z+\widehat{g},h+\sigma_{g}(z)\right)$, and let
$\widetilde{T}_{g}(w,x)=\left(\widetilde{S}_{g}w,x+\widetilde{\sigma}_{g}(w)\right)$.
Let $F(w,x):=\prod_{i=1}^{3}{f_{i}(w_{i},x_{i})}$. First, we will use
Propositions 7.7 and 7.9 together to show that $F$ is in the ergodic subspace
of $L^{2}(W\times H^{3})$ for $\widetilde{T}$.

Let $E_{1}:=\\{z\in Z:M_{z}=M\\}$. Let $E_{2}:=\left\\{z\in Z:\text{\eqref{eq:
orthogonal to annihilators} holds for}~{}m_{z}\text{-a.e.}~{}w\in
W_{z}\right\\}$. Finally, let $E_{3}:=\left\\{z\in Z:F\in
L^{\infty}(W_{z}\times H^{3})\right\\}$. By Proposition 7.9, $E_{1}$ has full
measure in $Z$. By Fubini’s theorem and the hypothesis, $E_{2}$ has full
measure. Also by Fubini’s theorem, $E_{3}$ has full measure. Hence, the set
$E:=E_{1}\cap E_{2}\cap E_{3}\subseteq Z$ has full measure.

Now let $z\in E$. Since $z\in E_{3}$, one has $F\in L^{\infty}(W_{z}\times
H^{3})\subseteq L^{2}(W_{z}\times H^{3})$. Let $\widetilde{\chi}\in
M_{z}^{\perp}$. Since $z\in E_{1}$, we have $\widetilde{\chi}\in M^{\perp}$.
Therefore, since $z\in E_{2}$, (72) holds for $\widetilde{\chi}$ and
$m_{z}$-a.e. $w\in W_{z}$. Thus, $F$ satisfies the hypotheses of Proposition
7.7, so

$\displaystyle\text{UC-}\lim_{g\in G}{\widetilde{T}_{g}F}=0$ (73)

in $L^{2}(W_{z}\times H^{3})$.

Fix a Følner sequence $(F_{N})_{N\in\mathbb{N}}$ in $G$, and set
$A_{N}:=\frac{1}{|F_{N}|}\sum_{g\in F_{N}}{\widetilde{T}_{g}F}\in
L^{2}(W\times H^{3})$. We want to show $\left\|A_{N}\right\|_{L^{2}(W\times
H^{3})}\to 0$. By decomposing $m_{W}=\int_{Z}{m_{z}~{}dz}$, we have

$\displaystyle\left\|A_{N}\right\|_{L^{2}(W\times
H^{3})}^{2}=\int_{W}{\int_{H^{3}}{\left|A_{N}(w,x)\right|^{2}~{}dx}~{}dw}=\int_{Z}{\int_{W_{z}}{\int_{H^{3}}{\left|A_{N}(w,x)\right|^{2}~{}dx}~{}dm_{z}(w)}~{}dz}.$

Now by (73),

$\displaystyle\int_{W_{z}}{\int_{H^{3}}{\left|A_{N}(w,x)\right|^{2}~{}dx}~{}dm_{z}(w)}\to
0$

for every $z\in E$ (so for $m_{Z}$-a.e. $z\in Z$). By the dominated
convergence theorem, it follows that $\left\|A_{N}\right\|_{L^{2}(W\times
H^{3})}\to 0$ as desired.

We will now use the van der Corput trick (Lemma 2.2). Let
$u_{g}:=\prod_{i=1}^{3}{T^{\sigma}_{a_{i}g}f_{i}}$. Observe that
$\mathbf{Z}\times_{\sigma}H$ has Kronecker factor $\mathbf{Z}$ because
$\sigma$ is weakly mixing. Therefore, we can use Lemma 4.22 (see also Remark
4.23) to compute

$\displaystyle\gamma_{h}:=\text{UC-}\lim_{g\in G}{\left\langle
u_{g+h},u_{g}\right\rangle}=\int_{W}{\prod_{i=1}^{3}{\mathbb{E}\left[{\overline{f_{i}}\cdot
T^{\sigma}_{a_{i}h}f_{i}}\mid{Z}\right](w_{i})}~{}dm_{W}(w_{1},w_{2},w_{3})}$

Now, by Fubini’s theorem, we have

$\displaystyle\mathbb{E}\left[{\overline{F}\cdot\widetilde{T}_{h}F}\mid{W}\right](w)$
$\displaystyle=\int_{H^{3}}{\left(\overline{F}\cdot\widetilde{T}_{h}F\right)(w,x)~{}dx}=\int_{H^{3}}{\prod_{i=1}^{3}{\left(\overline{f_{i}}\cdot
T^{\sigma}_{a_{i}h}f_{i}\right)(w_{i},x_{i})}~{}dx}$
$\displaystyle=\prod_{i=1}^{3}{\left(\int_{H}{\left(\overline{f_{i}}\cdot
T^{\sigma}_{a_{i}h}f_{i}\right)(w_{i},x_{i})~{}dx_{i}}\right)}=\prod_{i=1}^{3}{\mathbb{E}\left[{\overline{f_{i}}\cdot
T^{\sigma}_{a_{i}h}f_{i}}\mid{Z}\right]}(w_{i}),$

so

$\displaystyle\gamma_{h}=\int_{W}{\mathbb{E}\left[{\overline{F}\cdot\widetilde{T}_{h}F}\mid{W}\right]~{}dm_{W}}=\int_{W\times
H^{3}}{\overline{F}\cdot\widetilde{T}_{h}F~{}dm_{W\times H^{3}}}=\left\langle
F,\widetilde{T}_{h}F\right\rangle.$

But we showed that $F$ is in the ergodic subspace of $L^{2}(W\times H^{3})$,
so $\text{UC-}\lim_{h\in G}{\widetilde{T}_{h}F}=0$ in $L^{2}(W\times H^{3})$.
Thus, $\text{UC-}\lim_{h\in G}{\gamma_{h}}=0$. By Lemma 2.2,
$\text{UC-}\lim_{g\in G}{u_{g}}=0$ in $L^{2}(Z\times H)$ as claimed. ∎

###  7.3 Analytic characterizations of cocycles

In this section, we characterize classes of cocycles in terms of various
analytic properties. These characterizations will be useful in proving the
limit formula in Theorem 7.1. As usual, throughout this section we let $G$ be
a countable discrete abelian group. For a Kronecker system
$\mathbf{Z}=\left(Z,\mathcal{B},m_{Z},(T_{g})_{g\in G}\right)$, we let
$\widehat{g}\in Z$ denote the group element such that $T_{g}z=z+\widehat{g}$.
The $G=\mathbb{Z}$ case of the results in the section are proved in [HKr02].
We follow their approach, making several modifications in order to extend the
results to our more general setting.

######  Lemma 7.11.

Let $\mathbf{Z}$ be a Kronecker system and $\rho:G\times Z\to S^{1}$ a
cocycle. Suppose $(g_{n})_{n\in\mathbb{N}}$ is a sequence in $G$ such that
$\widehat{g}_{n}\to 0$ in $Z$. Then for every $h\in G$,

$\displaystyle\left\|\rho_{g_{n}}\left(z+\widehat{h}\right)-\rho_{g_{n}}(z)\right\|_{L^{2}(Z)}\to
0.$

###### Proof.

Expand the cocycle equation two ways:

$\displaystyle\rho_{g_{n}+h}(z)$
$\displaystyle=\rho_{g_{n}}(z)\rho_{h}\left(z+\widehat{g}_{n}\right),$ and
$\displaystyle\rho_{g_{n}+h}(z)$
$\displaystyle=\rho_{h}(z)\rho_{g_{n}}\left(z+\widehat{h}\right).$

Dividing, we get

$\displaystyle\frac{\rho_{g_{n}}\left(z+\widehat{h}\right)}{\rho_{g_{n}}(z)}=\frac{\rho_{h}\left(z+\widehat{g}_{n}\right)}{\rho_{h}(z)}.$

Since $\widehat{g}_{n}\to 0$ in $Z$ and the operators $U_{t}f(z)=f(z+t)$
define a continuous action of $Z$ on $L^{2}(Z)$, we have

$\displaystyle\frac{\rho_{h}\left(z+\widehat{g}_{n}\right)}{\rho_{h}(z)}\to 1$

in $L^{2}(Z)$. The result follows immediately. ∎

######  Proposition 7.12.

Let $\mathbf{Z}$ be a Kronecker system and $\rho:G\times Z\to S^{1}$ a