2.1 Uniform Cesàro limits

The large intersection property of a family $\{\varphi _1, \dots , \varphi _k\}$ is related to the limit behaviour of the multiple ergodic averages

(4) $$ \begin{align} \frac{1}{|\Phi_N|} \sum_{g \in \Phi_N} \prod_{i=1}^k{T_{\varphi_i(g)}f_i}, \end{align} $$

where $(\Phi _N)_{N \in \mathbb N}$ is a Følner sequenceFootnote ² in G and $f_1, \dots , f_k \in L^{\infty }(\mu )$ . By [Reference Austin3] and [Reference Zorin-Kranich32], the quantity in (4) converges in $L^2(\mu )$ as $N \to \infty $ , and the limit is independent of the choice of Følner sequence $(\Phi _N)_{N \in \mathbb N}$ . For more concise notation, we define the uniform Cesàro limit $x = \text {UC -}\lim _{g \in G}{x_g}$ if $\frac {1}{|\Phi _N|} \sum _{g \in \Phi _N}{x_g} \to x$ for every Følner sequence $(\Phi _N)_{N\in \mathbb N}$ in G.

One crucial tool for handling uniform Cesàro limits is the following version of the van der Corput differencing trick:

Lemma 2.1 (van der Corput lemma, cf. [Reference Ackelsberg, Bergelson and Best2], Lemma 2.2)

Let $\mathcal {H}$ be a Hilbert space and G a countable amenable group. Let $(u_g)_{g\in G}$ be a bounded sequence in $\mathcal {H}$ . If $\text {UC -}\lim _{g\in G} \left\langle u_{g+h}, u_g \right\rangle $ exists for every $h\in G$ , and

$$ \begin{align*} \text{UC -}\lim_{h\in G} \text{UC -}\lim_{g\in G} \left\langle u_{g+h}, u_g \right\rangle=0 \end{align*} $$

then,

$$ \begin{align*}\text{UC -}\lim_{g\in G} u_g=0 \end{align*} $$

strongly.

Another useful tool for computing uniform Cesàro limits is the following ‘Fubini’ trick, which we use extensively in Section 7:

Lemma 2.2 ([Reference Bergelson and Leibman9], special case of Lemma 1.1)

Let G and H be countable discrete amenable groups, and let $(v_{h,g})_{(h,g) \in H \times G}$ be a bounded sequence. Suppose

$$ \begin{align*}\text{UC -}\lim_{(h,g) \in H \times G}{v_{h,g}}\end{align*} $$

exists, and for every $g \in G$ ,

$$ \begin{align*}\text{UC -}\lim_{h \in H}{v_{h,g}}\end{align*} $$

exists. Then

$$ \begin{align*} \text{UC -}\lim_{g \in G} \text{UC -}\lim_{h \in H} v_{h,g} = \text{UC -}\lim_{(h,g) \in H \times G} v_{h,g}. \end{align*} $$

2.2 Factors

A factor of $\textbf {X}$ is a G-system $\textbf {Y}=(Y,\mathcal {Y},\nu ,(S_g)_{g\in G})$ together with a measurable map $\pi :X\to Y$ , such that $\pi _*\mu = \nu $ and $\pi \circ T_g = S_g\circ \pi $ for all $g\in G$ . There is a natural one-to-one correspondence between factors and $(T_g)_{g \in G}$ -invariant sub- $\sigma $ -algebras of $\mathcal {X}$ . Throughout the paper, we freely move between the system $\textbf {Y}$ and the $\sigma $ -algebra $\pi ^{-1}(\mathcal {Y})$ and refer to both of them as factors of $\textbf {X}$ . Given $f\in L^2(\mu )$ , we denote by $E(f|\mathcal {Y})$ the conditional expectation of f with respect to the $\sigma $ -algebra $\pi ^{-1}(\mathcal {Y})$ . We say that f is measurable with respect to $\mathcal {Y}$ if $f=E(f|\mathcal {Y})$ .

2.3 Factor of invariant sets

Let $\textbf {X}=(X,\mathcal {X},\mu , (T_g)_{g\in G})$ be a G-system. We write $\mathcal {I}_G(X)$ for the sub- $\sigma $ -algebra of G-invariant sets. We say that $\textbf {X}$ is ergodic if $\mathcal {I}_G(X)$ is the $\sigma $ -algebra comprised of null and conull subsets of $(X,\mathcal {X},\mu )$ . For a subgroup $H \le G$ , we denote by $\mathcal {I}_H(X)$ the sub- $\sigma $ -algebra of H-invariant sets. Given a homomorphism $\varphi :G\rightarrow G$ , it is convenient to denote by $\mathcal {I}_{\varphi }(X)$ the $\sigma $ -algebra $\mathcal {I}_{\varphi (G)}(X)$ .

2.4 Host–Kra factors

The Gowers–Host–Kra seminorms are an ergodic-theoretic version of the uniformity norms introduced by Gowers in [Reference Gowers22]. These seminorms were first introduced by Host and Kra in [Reference Host and Kra23] in the case of ergodic $\mathbb {Z}$ -systems, and then generalised by Chu, Frantzikinakis and Host to $\mathbb {Z}$ -systems that are not necessarily ergodic in [Reference Chu, Frantzikinakis and Host14]. In [Reference Bergelson, Tao and Ziegler10, Appendix A], a general theory of Gower–Host–Kra seminorms is developed for (not necesssarily ergodic) G-systems, where G is an arbitrary countable discrete abelian group.

Definition 2.3. Let G be a countable discrete abelian group, and let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be a G-system. Let $f \in L^{\infty } (X)$ , and let $k\geq 1$ be an integer. The Gowers–Host–Kra seminorm $\|f\|_{U^k(G)}$ of order k of f is defined recursively by the formula

$$\begin{align*}\|f\|_{U^1(G)}:= \|E(\phi|\mathcal{I}_G(X))\|_{L^2} \end{align*}$$

for $k=1$ , and

$$\begin{align*}\|f\|_{U^k(G)}:=\text{UC -}\lim_{g\in G}\left(\|\Delta_gf\|_{U^{k-1}}^{2^{k-1}}\right)^{1/2^k} \end{align*}$$

for $k>1$ , where .

In [Reference Bergelson, Tao and Ziegler10, Appendix A], it is shown that the Gower–Host–Kra seminorms for general G-systems are indeed seminorms. Moreover, these seminorms correspond to factors of $\textbf {X}$ .

The factors $\mathbf {Z}^k$ guaranteed by Proposition 2.4 are called the Host–Kra factors of $\textbf {X}$ .

Let $\textbf {X}=\left(X,\mathcal {X},\mu ,(T_g)_{g \in G} \right)$ be a G-system. Then, $\mathcal {Z}^0(X)$ is the same as the $\sigma $ -algebra $\mathcal {I}_G(X)$ . In particular, if $\textbf {X}$ is ergodic, then $\mathcal {Z}^0(X)$ is trivial. In the literature, $\mathbf {Z}^1(X)$ is often called the Kronecker factor, and $\mathbf {Z}^2(X)$ the Conze–Lesigne or quasi-affine factor of $\textbf {X}$ .

We summarise some basic results about the Host–Kra factors.

2.5 Joins and meets of factors

Let G be a countable discrete abelian group, let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be a G-system and let $\varphi ,\psi :G\rightarrow G$ be arbitrary homomorphisms.

1. 1. Let $\mathcal {Z}^1_{\varphi }(X)$ , or just $\mathcal {Z}_{\varphi }(X)$ , denote the $\sigma $ -algebra of the Kronecker factor of X with respect to the action of $\varphi (G)$ , that is, the $\sigma $ -algebra of the factor $\mathbf {Z}^1_{\varphi }(X)$ obtained by applying Proposition 2.4 for the G-system $(X,\mathcal {X},\mu ,(T_{\varphi (g)})_{g\in G})$ and $k=1$ . More generally, let H be a subgroup of G and $k\geq 1$ , we let $\mathcal {Z}^k_H(X)$ denote the $\sigma $ -algebra of the k-th Host–Kra factor $\mathbf {Z}^k_H(X)$ with respect to the action of H.

2. 2. Let $\mathcal {A}$ , $\mathcal {A}_1,\mathcal {A}_2$ be $\sigma $ -algebras on X. Then,

   • ○ We write $\mathcal {A}\preceq \mathcal {X}$ if the $\sigma $ -algebra $\mathcal {A}$ is a sub- $\sigma $ -algebra of $\mathcal {X}$ .

   • ○ We let $\mathcal {A}_1\lor \mathcal {A}_2$ denote the join of $\mathcal {A}_1$ and $\mathcal {A}_2$ , that is, the $\sigma $ -algebra generated by $\mathcal {A}_1$ and $\mathcal {A}_2$ in $\mathcal {X}$ .

   • ○ We let $\mathcal {A}_1\land \mathcal {A}_2$ denote the meet of $\mathcal {A}_1$ and $\mathcal {A}_2$ , that is, the maximal $\sigma $ -algebra which is also a sub- $\sigma $ -algebra of $\mathcal {A}_1$ and $\mathcal {A}_2$ .

   • ○ We say that $\mathcal {A}_1$ and $\mathcal {A}_2$ are $\mu $ -independent if their meet is trivial modulo $\mu $ -null sets.

   • ○ More generally, we say that $\mathcal {A}_1$ and $\mathcal {A}_2$ are relatively independent over the $\sigma $ -algebra $\mathcal {A}$ if $\mathcal {A}_1\land \mathcal {A}_2\preceq \mathcal {A}$ .

3. 3. We let $\mathcal {I}_{\varphi ,\psi }(X)$ denote the meet of $\mathcal {I}_{\varphi }(X)$ and $\mathcal {I}_{\psi }(X)$ and $\mathcal {Z}_{\varphi ,\psi }(X)$ the meet of $\mathcal {Z}_{\varphi }(X)$ and $\mathcal {Z}_{\psi }(X)$ . We let $\mathbf {Z}_{\varphi ,\psi }(X)$ denote the factor of $\textbf {X}$ which corresponds to the $\sigma $ -algebra $\mathcal {Z}_{\varphi ,\psi }(X)$ .

The next two lemmas give convenient alternative descriptions of independent and relatively independent $\sigma $ -algebras. These results are classical and can be found, for example, in [Reference Zimmer31, Proposition 1.4]; we provide short proofs for the convenience of the reader.

Proof. The first definition of independence above is clearly equivalent to (i). We prove the equivalence between (i) and (ii).

(i) $\Rightarrow $ (ii).

$$ \begin{align*} \int_X f\cdot g ~d\mu = \int_X E(f|\mathcal{A}_2)\cdot g ~d\mu = \int_X E(f|\mathcal{A}_2) ~d\mu \cdot \int_X g ~d\mu = \int_X f ~d\mu \cdot \int_X g ~d\mu, \end{align*} $$

where the second equality holds since $E(f|\mathcal {A}_2)$ is a constant $\mu $ -a.e. by (i).

For (ii) $\Rightarrow $ (i), let $\widetilde {f} = f-\int f d\mu $ . Then,

$$ \begin{align*} \|\widetilde{f}\|_{L^2(\mu)}^2 = \int |\widetilde{f}|^2 ~d\mu = \left|\int_X \widetilde{f} ~d\mu\right| ^2=0. \end{align*} $$

We conclude that $f=\int fd\mu $ .

Proof. Condition $(i)$ is equivalent to the definition of relative independence above. Therefore, it is enough to prove the equivalence of $(i)$ and $(ii)$ .

(i) $\Rightarrow $ (ii). We have $E(fg|\mathcal {A}_1) = f\cdot E(g|\mathcal {A}_1) = f\cdot E(g|\mathcal {A})$ , where the last equality follows from $(i)$ . Now, by taking the conditional expectation over $\mathcal {A}$ , we have

$$ \begin{align*} E(fg|\mathcal{A})=E(f|\mathcal{A})\cdot E(g|\mathcal{A}). \end{align*} $$

(ii) $\Rightarrow $ (i). Let $\widetilde {f} = f-E(f|\mathcal {A})$ . Then $E(|\widetilde {f}|^2|\mathcal {A})=E(\widetilde {f}|\mathcal {A})^2 = 0$ . In particular, $\int |\widetilde {f}|^2 d\mu = 0$ , thus, $f=E(f|\mathcal {A})$ .

2.6 Characteristic factors

Let $\textbf {X}=(X,\mathcal {X},\mu ,T)$ be an invertible ergodic measure preserving system and $f_1,...,f_k\in L^{\infty }(X)$ , $k\geq 0$ . The convergence of the multiple ergodic averages

(5) $$ \begin{align} \frac{1}{N}\sum_{n=0}^{N-1} \prod_{i=1}^k T^{in} f_i \end{align} $$

in $L^2(\mu )$ for general k was established by Host and Kra [Reference Host and Kra23] and independently, though somewhat later, by Ziegler [Reference Ziegler30].

Host and Kra proved convergence by showing that the averages in (5) are controlled by the Gowers–Host–Kra seminorms defined above. This reduces the general convergence problem to convergence under the additional assumption that each function $f_i$ is measurable with respect to the Host–Kra factor.

Ziegler, on the other hand, studied the universal (minimal) characteristic factors for the multiple ergodic averages

$$ \begin{align*} \frac{1}{N}\sum_{n=0}^{N-1} \prod_{i=1}^k T^{a_in} f_i, \end{align*} $$

where $a_1,...,a_k\in \mathbb {Z}$ are distinct and nonzero. These are the minimal factors $\mathcal {Z}_{k-1}(X)$ , such that

$$ \begin{align*} \lim_{N\rightarrow \infty} \frac{1}{N}\sum_{n=0}^{N-1} \prod_{i=1}^k T^{a_in} f_i=0 \end{align*} $$

whenever $E(f_i|\mathcal {Z}_{k-1}(X))=0$ for some i.

In [Reference Bergelson5, Appendix A], Leibman proved that, for ${\mathbb Z}$ -systems, the factors studied by Host and Kra coincide with the factors studied by Ziegler, thus giving these factors the name Host–Kra–Ziegler factors. Using Følner sequences in order to define averages, one can generalise the above to arbitrary countable discrete abelian groups (or even more generally, to amenable groups). However, in the setting of general abelian groups, Host–Kra factors may no longer coincide with the characteristic factors for averages of the form

$$ \begin{align*} \text{UC -}\lim_{g \in G} \prod_{i=1}^k T_{a_ig} f_i. \end{align*} $$

We give a very simple example. Let p be a prime number and $\mathbb {F}_p$ be the group with p elements. We denote by $\mathbb {F}_p^{\infty }$ the direct sum of countably many copies of $\mathbb {F}_p$ . In [Reference Bergelson, Tao and Ziegler10], it is shown that there are many nontrivial ergodic $\mathbb {F}_p^{\infty }$ -systems with nontrivial Host–Kra factors $\mathcal {Z}^k(X)$ for any $k \ge 0$ . However, the only characteristic factor for the average

$$ \begin{align*} \text{UC -}\lim_{g \in G} T_g f_1\cdot...\cdot T_{pg} f_p \end{align*} $$

is $\mathcal {X}$ . Indeed, since $T_{pg}=Id$ , the average is nonzero for every $f_p\not =0$ , assuming that $f_1=...=f_{p-1}=1$ (say). To overcome this technicality, one may restrict to the case where $k<p$ , but the situation is not that simple for arbitrary countable discrete abelian groups, and, in general, Host–Kra factors may not coincide with the universal characteristic factors.

This phenomenon was not studied previously in the literature, but it plays an important role in this paper. More specifically, we study how the Host–Kra factor $\mathcal {Z}^1(X)$ , which coincides with the classical Kronecker factor, is related to the the universal characteristic factor, $\mathcal {Z}_1(X)$ , for the average

$$ \begin{align*} \text{UC -}\lim_{g \in G} T_g f_1 T_{2g} f_2, \end{align*} $$

where $f_1,f_2\in L^{\infty }(\mu )$ , in the setting of actions of countable discrete abelian groups. One of our main tools is a result which asserts, roughly speaking, that by adding eigenfunctions to the system $\textbf {X}$ , one has that the characteristic factor $\mathcal {Z}_1(X)$ is generated by the Host–Kra factor $\mathcal {Z}^1(X)$ and the $\sigma $ -algebra of $2G$ -invariant functions. We also give an example that illustrates the necessity of adding eigenfunctions to the system (see Example 4.1).

2.7 Seminorms for multiplicative configurations

We now give a brief explanation of where our methods come up short of fully answering Questions 1.19 and 1.20. As discussed above, our approach to the large intersections property is to study families of seminorms and their corresponding characteristic factors. However, in the case of Questions 1.19 and 1.20, these seminorms have somewhat exotic behaviour.

For example, Question 1.19 is related to the averages

(6) $$ \begin{align} \text{UC -}\lim_{q\in\mathbb{Q}_{>0}} f_1(T_{q^n} x) f_2(T_{q^m}x) \end{align} $$

for some ergodic $(\mathbb {Q}_{>0}, \cdot )$ -system. An application of the van der Corput lemma (Lemma 2.1) shows that (6) is equal to zero if

$$ \begin{align*} \text{UC -}\lim_{q\in \mathbb{Q}_{>0}} \left|\int \Delta_{q^m} f_1 \cdot E(\Delta_{q^m}f_2|\mathcal{I}_{q^{n-m}}(X))~d\mu \right|=0. \end{align*} $$

If the action of $T_{q^{n-m}}$ , $q\in (\mathbb {Q}_{>0}, \cdot )$ , were ergodic (e.g. if $n=m+1$ ), then the above expression is manageable as we will see in this paper. Presumably, if n and m are coprime, then this expression may also be manageable, but we do not see how.

Question 1.20 is related to the average

(7) $$ \begin{align} \text{UC -}\lim_{q\in\mathbb{Q}_{>0}} T_qf_1 T_{q^2} f_2 T_{q^3} f_3. \end{align} $$

Using the van der Corput lemma, the Cauchy–Schwarz inequality and then the van der Corput lemma again, we see that the average in (7) is zero if

$$ \begin{align*} \text{UC -}\lim_{q_1\in\mathbb{Q}_{>0}} \left| \text{UC -}\lim_{q_2\in\mathbb{Q}_{>0}} \int \Delta_{q_1^2} \Delta_{q_2^3} f_3~d\mu \right|=0. \end{align*} $$

If in the expression above we had $q_1^2,q_2^2$ , or $q_1^3,q_2^3$ , then this expression would be related to the Gowers–Host–Kra seminorm of $f_3$ with respect to the action of all squares or cubes of $(\mathbb {Q}_{>0}, \cdot )$ . The above quantity is therefore some combination of the two. Again, presumably, the fact that $2$ and $3$ are coprime may be useful to analyse these seminorms. Studying the structure of these new peculiar seminorms is an interesting problem that we do not pursue in this paper.

3.1 Characteristic factors

We start with a definition of characteristic factors (cf. [Reference Furstenberg and Weiss21, Section 3]).

Definition 3.3. Let G be a countable discrete abelian group, let $\varphi ,\psi :G\rightarrow G$ be arbitrary homomorphisms and let $X=(X,\mathcal {X},\mu , (T_g)_{g\in G})$ be a G-system. A factor $\textbf {Y}=(Y,\mathcal {Y},\nu , (S_g)_{g\in G})$ of $\textbf {X}$ is called a partial characteristic factor for the pair $(\varphi ,\psi )$ with respect to $\varphi $ if

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_{\varphi(g)}f_1 T_{\psi(g)} f_2 = \text{UC -}\lim_{g\in G} T_{\varphi(g)} E(f_1|\mathcal{Y}) T_{\psi(g)} f_2 \end{align*} $$

for every $f_1,f_2\in L^{\infty }(X)$ . We define a partial characteristic factor with respect to $\psi $ similarly and say that $\textbf {Y}$ is a characteristic factor if it is a partial characteristic factor with respect to both $\varphi $ and $\psi $ , that is

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_{\varphi(g)}f_1 T_{\psi(g)} f_2 = \text{UC -}\lim_{g\in G} T_{\varphi(g)} E(f_1|\mathcal{Y}) T_{\psi(g)} E(f_2|\mathcal{Y}) \end{align*} $$

for every $f_1,f_2\in L^{\infty }(X)$ .

In other words, a factor of a measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ is a characteristic factor for a certain multiple ergodic average, if the study of the limit behaviour of the average can be reduced to this factor. The following easy lemma is related to the well-known result of Furstenberg, which asserts that a system $\textbf {X}=(X,\mathcal {X},\mu ,T)$ is weakly mixing if and only if the Kronecker factor, $\mathcal {Z}^1(X)$ , is trivial.

Lemma 3.4. Let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g \in G})$ be a G-system, let $\varphi :G\to G$ be a homomorphism and let $f\in L^2(X)$ . If $E(f|\mathcal {Z}_{\varphi }(X))=0$ , then for every $h\in L^2(X)$ , we have

$$ \begin{align*}\text{UC -}\lim_{g\in G}\left| \int_X T_{\varphi(g)} f \cdot h~d\mu \right|=0.\end{align*} $$

Proof. Assume $E(f\mid \mathcal {Z}_{\varphi }(X)) = 0$ . Then by Proposition 2.4, $\|f\|_{U^2(\varphi (G))}=0$ , that is,

$$ \begin{align*} \text{UC -}\lim_{g\in G} \left|\int_X \Delta_{\varphi(g)} f d\mu \right| = 0. \end{align*} $$

Since $\text {UC -}\lim _{g\in G} |a_g|=0 \iff \text {UC -}\lim _{g\in G} a_g^2 = 0$ for every bounded complex-valued sequence $g\mapsto a_g$ , we have

The mean ergodic theorem implies that

and

. Therefore, for every $h\in L^2(X)$ , we have,

which implies that

$$ \begin{align*} \text{UC -}\lim_{g\in G} \left|\int_X T_{\varphi(g)}f \cdot h~ d\mu\right|=0 \end{align*} $$

as required.

Using the van der Corput lemma (Lemma 2.1), we show that $\mathcal {Z}_{\varphi }(X)$ and $\mathcal {Z}_{\psi }(X)$ are partial characteristic factors for the pair $(\varphi ,\psi )$ with respect to $\varphi $ and $\psi $ , respectively.

Proposition 3.5. Let $\textbf {X} = (X, \mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system. Let $\varphi , \psi : G \rightarrow G$ be homomorphisms, such that $(\psi -\varphi )(G)$ has finite index in G. Then, for any $f_1,f_2\in L^{\infty }(\mu )$ , one has

$$ \begin{align*}\text{UC -}\lim_{g\in G} T_{\varphi(g)} f_1\cdot T_{\psi(g)} f_2 = \text{UC -}\lim_{g\in G} T_{\varphi(g)} E(f_1|\mathcal {Z}_{\varphi}(X)) \cdot T_{\psi(g)} E(f_2|\mathcal {Z}_{\psi}(X)) \end{align*} $$

in $L^2(\mu )$ .

Proof. We follow the argument of Furstenberg and Weiss [Reference Furstenberg and Weiss21]. By linearity and symmetry, it is enough to show that

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_{\varphi(g)} f_1\cdot T_{\psi(g)} f_2 = 0 \end{align*} $$

whenever $E(f_1|\mathcal {Z}_{\varphi }(X))=0$ . Dividing through by a constant, we may assume that $\|f_i\|_{\infty } \le 1$ for $i=1,2$ .

We use the van der Corput lemma with $u_g = T_{\varphi (g)} f_1\cdot T_{\psi (g)} f_2$ . For every $g,h\in G$ , we have

(10)

Since the measure $\mu $ is $T_{\varphi (g)}$ -invariant, (10) is equal to

Hence, by the mean ergodic theorem, we have

Since $H:=(\psi -\varphi )(G)$ has finite index in G and the action of G on X is ergodic, we can find a partition $X=\bigcup _{i=1}^{l} A_i$ to H-invariant sets, where l is at most the index of H in G. Since $f_2$ is bounded by $1$ ,

Now, since $E(f_1|Z_{\varphi }(X))=0$ , Lemma 3.4 implies that

, for every $1\leq i \leq k$ . The van der Corput lemma (Lemma 2.1) then implies that

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_{\varphi(g)} f_1\cdot T_{\psi(g)} f_2 = 0, \end{align*} $$

and this completes the proof.

In [Reference Bergelson5, Appendix A], Leibman proved the following result in the special case where $G=\mathbb {Z}$ . For the sake of completeness, we give a proof for arbitrary countable discrete abelian G in Appendix A.

In particular, if $\varphi (G)$ has finite index in G, then the factor $\mathcal {Z}_{\varphi }(X)$ coincides with $\mathcal {Z}(X)$ .

Corollary 3.7. Let G be a countable discrete abelian group, let $X=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be a G-system and let $\varphi ,\psi :G\rightarrow G$ be arbitrary homomorphisms, such that $\varphi (G)$ and $(\psi -\varphi )(G)$ have finite index in G. Then, for any bounded functions $f_1,f_2\in L^{\infty }(X)$ ,

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_{\varphi(g)} f_1\cdot T_{\psi(g)} f_2 = \text{UC -}\lim_{g\in G} T_{\varphi(g)} E(f_1|\mathcal {Z}(X)) \cdot T_{\psi(g)} E(f_2|\mathcal {Z}_{\psi}(X)). \end{align*} $$

Let G be a countable discrete abelian group and $\textbf {X} = (X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system. By Theorem 2.5(ii), the Kronecker factor of $\textbf {X}$ , $\mathbf {Z}^1(X)$ is isomorphic to an ergodic rotation. Therefore, it is convenient to identify the Kronecker factor with the system $\textbf {Z}=(Z,\alpha )$ , where Z is a compact abelian group and $\alpha :G\rightarrow Z$ is a homomorphism, such that $T^1_g z = z + \alpha _g$ , where $T^1$ is the G-action on Z. The following corollary of Proposition 3.5 will be useful later on in this paper.

Proposition 3.8. Let $\textbf {X} = \left(X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ be an ergodic G-system with Kronecker factor $\textbf {Z}=(Z, \alpha )$ . Let $\varphi , \psi : G \to G$ be homomorphisms, such that $(\psi - \varphi )(G)$ has finite index in G. Then, for any $f_0, f_1, f_2 \in L^{\infty }(\mu )$ and any continuous function $\eta : Z^2 \to \mathbb {C}$ , we have

$$ \begin{align*} \text{UC -}\lim_{g \in G}&~{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\int_X{f_0 \cdot T_{\varphi(g)}f_1 \cdot T_{\psi(g)}f_2~d\mu}} \\ & = \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\int_X{f_0 \cdot T_{\varphi(g)}E(f_1|\mathcal{Z}_{\varphi}(X)) \cdot T_{\psi(g)}E(f_2|\mathcal{Z}_{\psi}(X))~d\mu}}. \end{align*} $$

Proof. By the Stone–Weierstrass theorem and linearity, we may assume $\eta (u,v) = \lambda _1(u)\lambda _2(v)$ for some characters $\lambda _1, \lambda _2 \in \widehat {Z}$ . Let $\pi : X \to Z$ be the factor map, and let $\chi _i := \lambda _i \circ \pi $ . Note that $T_g\chi _i = \lambda _i(\alpha _g)\chi _i$ , so $\chi _i$ is a G-eigenfunction with eigenvalue $\lambda _i \circ \alpha $ .

Now, set

Since $\chi _1$ and $\chi _2$ are measurable with respect to the Kronecker factor $\mathcal {Z}(X)$ , which is a sub- $\sigma $ -algebra of $\mathcal {Z}_{\varphi }(X)$ and $\mathcal {Z}_{\psi }(X)$ , we have the identities

$$ \begin{align*} E(h_1|\mathcal {Z}_{\varphi}(X)) & = \chi_1 \cdot E({f_1}|{\mathcal {Z}_{\varphi}(X)}), \\[6pt] E(h_2|\mathcal {Z}_{\varphi}(X)) & = \chi_2 \cdot E({f_2}|{\mathcal {Z}_{\varphi}(X)}). \end{align*} $$

Thus, applying Proposition 3.5 for the functions $h_1, h_2$ and integrating against $h_0$ , we have

$$ \begin{align*} \text{UC -}\lim_{g \in G}&~{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\int_X{f_0 \cdot T_{\varphi(g)}f_1 \cdot T_{\psi(g)}f_2~d\mu}} \\& = \text{UC -}\lim_{g \in G}{\int_X{h_0 \cdot T_{\varphi(g)}h_1 \cdot T_{\psi(g)}h_2~d\mu}} \\& = \text{UC -}\lim_{g \in G}{\int_X{h_0 \cdot T_{\varphi(g)} E(h_1|\mathcal {Z}_{\varphi}(X)) \cdot T_{\psi(g)} E(h_2|\mathcal {Z}_{\psi}(X))~d\mu}} \\& = \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\int_X{f_0 \cdot T_{\varphi(g)} E(f_1|\mathcal {Z}_{\varphi}(X)) \cdot T_{\psi(g)} E(f_2|\mathcal {Z}_{\psi}(X))~d\mu}}.\\[-42pt] \end{align*} $$

In the next section, we will study the factor $\mathcal {Z}_{\psi }(X)$ further.

3.2 Relative orthonormal basis

Let G be a countable discrete abelian group, and let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be a G-system. Under the assumption that the system is ergodic, it is well known that the factor $\mathcal {Z}^1(X)$ admits an orthonormal basis of eigenfunctions. The following example demonstrates that this may fail for nonergodic systems.

Example 3.9. Let $S^1 = \{z\in \mathbb {C} : |z|=1\}$ . Consider $X=S^1\times S^1$ equipped with the Borel $\sigma $ -algebra, the Haar probability measure $\mu $ and the measure-preserving transformation $T(x,y) = (x,y\cdot x)$ . Any function $f\in L^2(X)$ takes the form

$$ \begin{align*} f(x,y) = \sum_{n,m\in\mathbb{N}} a_{n,m} x^n y^m \end{align*} $$

for some $a_{n,m}\in \mathbb {C}$ with

(11) $$ \begin{align} \sum_{n,m\in\mathbb{N}} |a_{n,m}|^2 <\infty. \end{align} $$

Now suppose that there exists some constant $c\in S^1$ , such that $Tf(x,y) = c\cdot f(x,y)$ for $\mu $ -a.e. $(x,y)\in S^1\times S^1$ . By the uniqueness of the Fourier series, we deduce that

$$ \begin{align*} a_{n+m,m} = c\cdot a_{n,m} \end{align*} $$

for every $n,m\in \mathbb {N}.$ If $m\not = 0$ , this is a contradiction to (11) unless $a_{n,m}=0$ . We conclude that f is an eigenfunction if and only if it is independent of the y coordinate. In particular, $L^2(X)$ is not generated by the eigenfunctions of X.

On the other hand, the functions $\{x^n\}_{n\in \mathbb {N}}$ are invariant and therefore measurable with respect to $\mathcal {Z}^1(X)$ . Moreover, the functions $\{y^m\}_{n\in \mathbb {N}}$ satisfy $\Delta _n (y^m) = T^n (y^m) \cdot y^{-m} = x^{n\cdot m}$ , which is an invariant function. Hence, $y^m$ is also measurable with respect to $\mathcal {Z}^1(X)$ . We thus conclude that X coincides with $\mathcal {Z}^1(X)$ .

In order to handle nonergodic systems, Frantzikinakis and Host [Reference Frantzikinakis and Host19] came up with the following definition.

We also give a definition of eigenfunctions that applies to nonergodic systems.

Note that under the assumption that the H-action is ergodic, this definition coincides with the standard definition of an eigenfunction. Observe, moreover, that the functions $\{y^m\}_{m\in \mathbb {N}}$ from Example 3.9 are eigenfunctions according to this definition.

Frantzikinakis and Host proved the following result:

The proof of Theorem 3.12 is given for $\mathbb {Z}$ -actions in [Reference Frantzikinakis and Host19], but the same argument can be easily generalised for arbitrary group actions.

3.3 Proof of Theorem 1.11

In this subsection, we prove Theorem 1.11. Example 3.1 is a good example to have in mind while reading this section.

Let $\textbf {X} = \left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ be an ergodic G-system, and let $\textbf {Z}=(Z,\alpha )$ be the Kronecker factor of $\textbf {X}$ . Let $A \in \mathcal {X}$ and $f = \mathbf {1}_A$ . We can write

$$ \begin{align*} f_c:=E(f|\mathcal{Z}(X)) = \sum_{i\in\mathbb{N}}a_i\zeta_i, \end{align*} $$

where $\{\zeta _i\}_{i\in \mathbb {N}}$ is an orthonormal basis of eigenfunctions and $a_i\in \mathbb {C}$ . Moreover, using Theorem 3.12,

$$ \begin{align*} f_{\psi} := E(f_{\psi}|\mathcal{Z}_{\psi}(X)) = \sum_{i\in\mathbb{N}} b_i \xi_i, \end{align*} $$

where $\{\xi _i\}_{i\in \mathbb {N}}$ is a relative orthonormal basis of $\psi (G)$ -eigenfunctions and are $\psi (G)$ -invariant functions.

Choose $N_1 \in \mathbb {N}$ sufficiently large so that

$$ \begin{align*} \left\| f_c - \sum_{i=1}^{N_1}{a_i\zeta_i} \right\|_{2} & < \frac{\varepsilon}{8} \end{align*} $$

and

$$ \begin{align*} \left\| f_{\psi} - \left( \sum_{i=1}^{N_1}{b_i\xi_i}\right) \right\|_{2} & < \frac{\varepsilon}{8}. \end{align*} $$

For each $j \in \mathbb {N}$ , the function $\xi _j$ is a $\psi (G)$ -eigenfunction, so we can write $\xi _j \left( T_{\psi (g)}x \right) = \mu _j(x, \psi (g)) \xi _j(x)$ for some $\psi (G)$ -invariant function $\mu _j : X \to \widehat {\psi (G)}$ . The group Z is compact, so $\widehat Z$ is countable and we can write $\widehat {Z} = \bigcup _{n \in \mathbb {N}}{F_n}$ , where $F_1 \subseteq F_2 \subseteq \cdots $ are finite sets. Let

$$ \begin{align*} C_n := \left\{ g \mapsto \chi_1(\alpha_{\varphi(g)}) \chi_2(\alpha_{\psi(g)}) : \chi_1, \chi_2 \in F_n \right\}, \end{align*} $$

and let $C = \bigcup _{n \in \mathbb {N}}{C_n}.$ Finally, let

$$ \begin{align*} E_{j,n} := \left\{ x \in X : \mu_j(x, \cdot) \in C_n \cup \left( \widehat{G} \setminus C \right) \right\}. \end{align*} $$

Note that the complement of $E_{j,n}$ consists of all $x\in X$ , such that $\mu _j(x,\cdot )$ belongs to a finite set. Since $\mu _j$ is measurable, we conclude that so is the complement of $E_{j,n}$ . Hence, $E_{j,n}$ are measurable. Since $\bigcup _{n=1}^{\infty }{E_{j,n}} = X$ for every $j \in \mathbb {N}$ , there exists sufficiently large $N_2 \in \mathbb {N}$ , such that

$$ \begin{align*} \left( \int_{X \setminus E_{j,N_2}}{|b_j \xi_j|^2~d\mu} \right)^{1/2} < \frac{\varepsilon}{16N_1} \end{align*} $$

for $j = 1, \dots , N_1$ . Then, let $N \ge \max \{N_1,N_2\}$ , such that: if $T_g\zeta _i = \chi (\alpha _g)\zeta _i$ for some $i = 1, \dots , N_1$ , then $\chi \in F_N$ .

Now, let $B_0 \in Z$ be a small neighborhood of $0$ in Z, such that if $z \in B_0$ and $\chi \in F_N$ , then

$$ \begin{align*} |\chi(z) - 1| < \frac{\varepsilon}{16N}. \end{align*} $$

Let $\eta _0 : Z \to [0, \infty )$ be a continuous function supported on $B_0$ normalised so that

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\eta_0(\alpha_{\varphi(g)}) \eta_0(\alpha_{\psi(g)})} = 1. \end{align*} $$

Put $\eta (u,v) := \eta _0(u) \eta _0(v)$ . Then, by Proposition 3.8, we have

$$ \begin{align*} \text{UC -}\lim_{g \in G}&~{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right)} \\ & = \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\int_X{f \cdot T_{\varphi(g)}f_c \cdot T_{\psi(g)}f_{\psi}~d\mu}} \\ & = \int_X{f \cdot \text{UC -}\lim_{g \in G}{\eta_0(\alpha_{\varphi(g)}) T_{\varphi(g)} f_c \cdot \eta_0(\alpha_{\psi(g)}) T_{\psi(g)} f_{\psi}}~d\mu}. \end{align*} $$

From the definition of $B_0$ , if $\alpha _{\varphi (g)} \in B_0$ , then $\left\| T_{\varphi (g)}\zeta _i - \zeta _i \right\|_{\infty } < \frac {\varepsilon }{16N}$ for $i = 1, \dots , N_1$ . Hence, for every $g \in G$ , since $\eta _0$ is supported on $B_0$ , we have

$$ \begin{align*} \left\| \eta_0(\alpha_{\varphi(g)}) T_{\varphi(g)} f_c - \eta_0(\alpha_{\varphi(g)})f_c \right\|_{2} \le &~\left\| \eta_0(\alpha_{\varphi(g)}) \left( T_{\varphi(g)} f_c - \sum_{i=1}^{N_1}{a_i T_{\varphi(g)} \zeta_i} \right) \right\|_{2} \\ & + \left\| \eta_0(\alpha_{\varphi(g)}) \left( \sum_{i=1}^{N_1}{a_i T_{\varphi(g)} \zeta_i} - \sum_{i=1}^{N_1}{a_i\zeta_i} \right) \right\|_{2} \\ & + \left\| \eta_0(\alpha_{\varphi(g)}) \left( \sum_{i=1}^{N_1}{a_i\zeta_i} - f_c \right) \right\|_{2} \\ \le &~\eta_0(\alpha_{\varphi(g)}) \left( \left\| f_c - \sum_{i=1}^{N_1}{a_i\zeta_i} \right\|_{2} + N_1 \frac{\varepsilon}{16N} + \left\| f_c - \sum_{i=1}^{N_1}{a_i\zeta_i} \right\|_{2} \right) \\ < &~\eta_0(\alpha_{\varphi(g)}) \left( \frac{\varepsilon}{8} + \frac{\varepsilon}{16} + \frac{\varepsilon}{8} \right) = \frac{5\varepsilon}{16} \eta_0(\alpha_{\varphi(g)}). \end{align*} $$

Therefore,

$$ \begin{align*} \left| \int_X{f \cdot \eta_0(\alpha_{\varphi(g)}) T_{\varphi(g)} f_c} \right. & \left. {\cdot~\eta_0(\alpha_{\psi(g)}) T_{\psi(g)} f_{\psi}~d\mu} - \int_X{f_c \cdot f \cdot \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) T_{\psi(g)} f_{\psi}~d\mu} \right| \\& = \left| \int_X{f \cdot \eta_0(\alpha_{\psi(g)}) T_{\psi(g)}f_{\psi} \cdot \left( \eta_0(\alpha_{\varphi(g)}) T_{\varphi(g)} f_c - \eta_0(\alpha_{\varphi(g)})f_c \right)~d\mu} \right| \\& \le \eta_0(\alpha_{\psi(g)}) \left\| \eta_0(\alpha_{\varphi(g)}) T_{\varphi(g)} f_c - \eta_0(\alpha_{\varphi(g)})f_c \right\|_{1} \\& < \frac{5\varepsilon}{16} \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right). \end{align*} $$

Taking a Cesàro average, we have the inequality

(12) $$ \begin{align} \text{UC -}\lim_{g \in G}&~{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right)} \nonumber \\ &> \int_X{f_c \cdot f \cdot \text{UC -}\lim_{g \in G}{ \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} f_{\psi}}~d\mu} - \frac{5\varepsilon}{16}. \end{align} $$

Now, we estimate the average

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} f_{\psi}}. \end{align*} $$

First, for each $i = 1, \dots , N_1$ , we have

$$ \begin{align*} \left\| \eta_0(\alpha_{\psi(g)}) \left( T_{\psi(g)}(b_i\xi_i) - b_i\xi_i \right) \right\|_{\infty} = \left\| b_i \cdot \eta_0(\alpha_{\psi(g)}) \left( T_{\psi(g)}\xi_i - \xi_i \right) \right\|_{\infty} < \frac{\varepsilon}{16N} \eta_0(\alpha_{\psi(g)}). \end{align*} $$

Next, let $1 \le j \le N_1$ . Write $T_{\psi (g)}(b_j\xi _j) = b_j \mu _j(x, \psi (g)) \psi _j$ . If $\mu _j(x, \cdot ) \notin C$ , then for any $\chi _1, \chi _2 \in \widehat {Z}$ , the character $g \mapsto \chi _1(\alpha _{\varphi (g)}) \chi _2(\alpha _{\psi (g)}) \mu _j(x, \psi (g))$ is nontrivial, so

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\chi_1(\alpha_{\varphi(g)}) \chi_2(\alpha_{\psi(g)}) \mu_j(x, \psi(g))} = 0. \end{align*} $$

Hence, by the Stone–Weierstrass theorem,

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \mu_j(x, \psi(g))} = 0. \end{align*} $$

Therefore,

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} f_{\psi}} = \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} \widetilde{f}_{\psi}}, \end{align*} $$

where $\widetilde {f}_{\psi } = E(f|\widetilde {\mathcal {Z}}_{\psi }(X))$ and $\widetilde {\mathcal {Z}}_{\psi }(X)$ is the factor generated by $\psi (G)$ -eigenfunctions whose eigenvalues come from C. Note that

$$ \begin{align*} \widetilde{f}_{\psi} = \sum_{i \in \mathbb{N}}b_i\widetilde{\xi}_i , \end{align*} $$

where

$$ \begin{align*} \widetilde{\xi}_j(x) = \begin{cases} \xi_j(x), & \mu_j(x, \cdot) \in C; \\ 0, & \mu_j(x, \cdot) \notin C. \end{cases} \end{align*} $$

We note that since C is at most countable, $\widetilde {\chi }_j$ is measurable. Moreover,

$$ \begin{align*} \widetilde{f}_{\psi} - \sum_{i=1}^{N_1}b_i\widetilde{\xi}_i = E(f - \sum_{i=1}^{N_1} b_i \xi_i |\widetilde{\mathcal {Z}}_{\psi}(X)), \end{align*} $$

so

$$ \begin{align*} \left\| \widetilde{f}_{\psi} - \sum_{i=1}^{N_1}{b_i\widetilde{\xi}_i} \right\|_{2} < \frac{\varepsilon}{8}. \end{align*} $$

If $x \in E_{j,N}$ , then we must have $\mu _j(x, \cdot ) \in C_N$ . That is, $\mu _j(x, \psi (g)) = \chi _1(\alpha _{\varphi (g)}) \chi _2(\alpha _{\psi (g)})$ for some $\chi _1, \chi _2 \in F_N$ . Thus,

$$ \begin{align*} \left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \mu_j(x, \psi(g)) - 1 \right) \right| = &~\left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \chi_1(\alpha_{\varphi(g)}) \chi_2(\alpha_{\psi(g)}) - 1 \right) \right| \\\leq &~\left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \chi_1(\alpha_{\varphi(g)}) \chi_2(\alpha_{\psi(g)}) - \chi_2(\alpha_{\psi(g)}) \right) \right| \\[6pt]& + \left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \chi_2(\alpha_{\psi(g)}) - 1 \right) \right| \\[4pt]\leq &~\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \frac{\varepsilon}{16N} + \frac{\varepsilon}{16N} \right) = \frac{\varepsilon}{8N} \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right). \end{align*} $$

Therefore,

$$ \begin{align*} \bigg\| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) & \left( T_{\psi(g)}(b_j\widetilde{\xi}_j) - b_j\widetilde{\xi}_j \right) \bigg\|_2^2 \\ = &~\int_X{\left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( T_{\psi(g)}(b_j\widetilde{\xi}_j) - b_j\widetilde{\xi}_j \right) \right|^2~d\mu} \\ = &~\int_X{\left| b_j(x) \widetilde{\xi}_j(x) \right|^2 \left| \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left( \mu_j(x, \psi(g)) - 1 \right) \right|^2~d\mu(x)} \\ \le &~\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)^2 \left( \int_{E_{j,N}}{\left( \frac{\varepsilon}{8N} \right)^2 \left| b_j \widetilde{\xi}_j \right|^2~d\mu} + 4 \int_{X \setminus E_{j,N}}{\left| b_j \widetilde{\xi}_j \right|^2~d\mu} \right) \\ \le &~\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)^2 \left( \left( \frac{\varepsilon}{8N} \right)^2 + 4 \left( \frac{\varepsilon}{16N_1} \right)^2 \right) \le 2 \left( \frac{\varepsilon}{8N_1} \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \right)^2 .\end{align*} $$

Putting together our estimates, we have

$$ \begin{align*} \bigg\| \text{UC -}\lim_{g \in G} & {\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} f_{\psi}} - \widetilde{f}_{\psi} \bigg\|_2 \\ = &~\left\| \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right)T_{\psi(g)} \widetilde{f}_{\psi}} - \widetilde{f}_{\psi} \right\|_{2} \\ \le &~\left\| \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) T_{\psi(g)}\widetilde{f}_{\psi} - T_{\psi(g)} \sum_{i=1}^{N_1}{b_i\widetilde{\xi}_i}} \right\|_{2} \\ & + \left\| \text{UC -}\lim_{g \in G}{\sum_{i=1}^{N_1}{ \eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) \left(T_{\psi(g)}(b_i \widetilde{\xi}_i) - b_j \widetilde{\xi}_j \right)}} \right\|_{2} \\ & + \left\| \sum_{i=1}^{N_1}{b_i\widetilde{\xi}_i} - \widetilde{f}_{\psi} \right\|_{2} \\ < &~\frac{\varepsilon}{8} + N_1 \frac{\sqrt{2}\varepsilon}{8N_1} + \frac{\varepsilon}{8} \le \frac{(2\sqrt{2} + 5)\varepsilon}{16} < \frac{\varepsilon}{2}. \end{align*} $$

Substituting back into (12), we have

(13) $$ \begin{align} \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right)} &> \int_X{f_c \cdot f \cdot \widetilde{f}_{\psi}~d\mu} - \frac{13\varepsilon}{16} \nonumber \\ & \ge \mu(A)^3 - \frac{13\varepsilon}{16}. \end{align} $$

Since $\text {UC -}\lim _{g \in G}\eta \left( \alpha _{\varphi (g)}, \alpha _{\psi (g)} \right)=1$ , it follows that the set

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

is syndetic in G. If not, there exists a Følner sequence $(\Phi _N)_{N\in \mathbb N}$ , such that $\mu (A\cap T_{\varphi (g)}^{-1}A\cap T_{\psi (g)}^{-1}A)\leq \mu (A)^3-\varepsilon $ for every $g\in \bigcup _{N\in \mathbb {N}}\Phi _N$ . But then,

$$ \begin{align*} \text{UC -}\lim_{g \in G}{\eta \left( \alpha_{\varphi(g)}, \alpha_{\psi(g)} \right) ~\mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right)} \leq \mu(A)^3-\varepsilon, \end{align*} $$

which contradicts the inequality in (13).

4.1 Characteristic factors related to Theorem 1.13

The goal of this subsection is to prove a stronger version of Propositions 3.5 and 3.8 with smaller characteristic factors. We will use the above extension theorem in order to express these characteristic factors in terms of $\mathcal {Z}_{\varphi ,\psi }(X)$ and the invariant $\sigma $ -algebras, $\mathcal {I}_{\varphi }(X)$ and $\mathcal {I}_{\psi }(X)$ . Then, using a result of Tao and Ziegler [Reference Tao and Ziegler29] (see Theorem 4.8 below), we will reduce matters further to studying the Conze–Lesigne factor $\mathcal {Z}^2(X)$ with respect to the action of G, which is already well understood for arbitrary countable discrete abelian groups (see [Reference Ackelsberg, Bergelson and Best2], [Reference Shalom27]).

We start with a lemma.

Lemma 4.6. Let $\textbf {X}=(X,\mathcal {X},\mu , (T_g)_{g\in G})$ be an ergodic G-system. Let $\mathcal {I}_{\varphi \times \psi }(X\times X)$ denote the $\sigma $ -algebra of $(T_{\varphi (g)}\times T_{\psi (g)})_{g\in G}$ -invariant sets in $X\times X$ . Then,

$$ \begin{align*} \mathcal {I}_{\varphi\times\psi}(X\times X)\preceq \mathcal{Z}_{\varphi}(X)\times \mathcal{Z}_{\psi}(X). \end{align*} $$

Proof. Let $f_1,f_2\in L^{\infty }(X)$ be arbitrary functions and $f(x,y)=f_1(x)f_2(y)$ . Then, by the mean ergodic theorem, we have that

$$ \begin{align*} E(f|\mathcal {I}_{\varphi\times\psi}(X\times X))(x,y) = \text{UC -}\lim_{g\in G} T_{\varphi(g)}f_1(x)\cdot T_{\psi(g)}f_2(y) \end{align*} $$

in $L^2(\mu \times \mu )$ . By the van der Corput lemma, $E(f|\mathcal {I}_{\varphi \times \psi }(X\times X))=0$ if

Since $\varphi (G)\times \psi (G)$ is measure-preserving, the above is equal to

$$ \begin{align*} \text{UC -}\lim_{h\in G}\left(\left|\int_X \Delta_{\varphi(h)}f_1(x) d\mu(x)\right|\right) \left(\left|\int_X \Delta_{\psi(h)}f_2(y) d\mu(y) \right|\right) \end{align*} $$

which by the Cauchy–Schwarz inequality is bounded above by

$$ \begin{align*} \left(\|f_1\|_{U^2(\varphi(G))}\cdot \|f_2\|_{U^2(\psi(G))}\right)^{1/2}. \end{align*} $$

We deduce that if $E(f|\mathcal {Z}_{\varphi }(X)\times \mathcal {Z}_{\psi }(X))=0$ , then $E\left(f|\mathcal {I}_{\varphi \times \psi }(X\times X)\right)=0$ . Since linear combinations of functions of the form

with $f_1,f_2\in L^{\infty }(X)$ are dense in $L^{\infty }(X\times X)$ , we deduce that the same holds for every bounded function on $X\times X$ , and this completes the proof.

Using Theorem 4.3, we can now prove the following useful result.

Lemma 4.7. Let G be a countable discrete abelian group, and let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system. Suppose that $\varphi ,\psi :G\rightarrow G$ are arbitrary homomorphisms, such that $(\psi -\varphi )(G)$ has finite index in G. Then, there exists an ergodic extension $\pi \colon \widetilde {X} \to X$ , such that

Proof. Let $\{\zeta _i\}_{i\in \mathbb {N}}$ be a relative orthonormal basis of eigenfunctions for $\mathcal {Z}_{\varphi }(X)$ and $\{\xi _i\}_{i\in \mathbb {N}}$ be the same for $\mathcal {Z}_{\psi }(X)$ . For every $i,j \in \mathbb N$ , let $\lambda _i:\varphi (G)\times X\rightarrow \mathbb {C}$ and $\mu _j:\psi (G)\times X\rightarrow \mathbb {C}$ denote the eigenvalues of $\zeta _i$ and $\xi _j$ , respectively. Our goal is to study the functions $f\in L^{\infty }(X^2)$ which are $(T_{\varphi (g)}\times T_{\psi (g)})_{g\in G}$ -invariant. By Lemma 4.6, we can write any such function as

where $c_{i,j}$ is a $\varphi (G)\times \psi (G)$ -invariant function. Since f is $T_{\varphi (g)}\times T_{\psi (g)}$ -invariant, we deduce that

Hypothetically, if $c_{i,j}$ was a constant, then unless it is zero (and then can be removed from the summation), the equation above implies that $\lambda _i(\varphi (g),\cdot ) = \mu _j(\psi (g),\cdot )=\chi (g)$ for some character $\chi \in \widehat G$ . In this special case, we can apply Theorem 4.3 in order to find an extension where $\lambda _i$ and $\mu _j$ are eigenvalues. This means that we can express the lift of to $\widetilde {X}$ as a product of a tensor product of G-eigenfunctions (whose eigenvalues are $\lambda _i$ and $\mu _j$ ) and a $\varphi (G)\times \psi (G)$ -invariant function, which completes the proof in this special case. Below, we generalise the above to arbitrary $c_{i,j}$ .

Let $C_{i,j} = \{(x,y)\in X\times X : c_{i,j}(x,y)\not = 0\}$ . Then, for every $(x,y)\in C_{i,j}$ and all $g\in G$ . Hence, $g\mapsto \lambda _i(\varphi (g),x)$ and $g\mapsto \mu _j(\psi (g),y)$ are equal to the same character $\chi \in \widehat G$ which factors through $\varphi $ and $\psi $ simultaneously for all $(x,y)\in C_{i,j}$ . Now, for every $\chi \in \widehat G$ , we let

$$ \begin{align*} J_{\chi}=\{(i,j)\in\mathbb{N}^2 : (\mu\times\mu)(\{(x,y)\in X\times X:\forall g\text{ }\lambda_i(\varphi(g),x)=\mu_j(\psi(g),y)=\chi(g)\}>0\} \end{align*} $$

and set

$$ \begin{align*} C:=\{\chi\in\widehat G : J_{\chi}\not = \emptyset\} \text{ and } J:=\bigcup_{\chi\in C} J_{\chi}. \end{align*} $$

Our first observation is that

(15) $$ \begin{align} f(x,y) = \sum_{(i,j)\in J} c_{i,j}(x,y) \zeta_i(x) \xi_j(y). \end{align} $$

Indeed, if $(i,j)\not \in J$ , then for every $\chi $ , $(i,j)\not \in J_{\chi }$ , but then from the computation above $\mu (C_{i,j})=0$ and $c_{i,j}=0$ for $(\mu \times \mu )$ -a.e. $(x,y)\in X\times X$ .

Claim. The set C is at most countable.

Proof of the claim

We use the fact that in a probability space there can be at most countably many disjoint sets of positive measure. Assume by contradiction that C is uncountable. Since there are only countably many $(i,j)\in \mathbb {N}^2$ , we deduce that there exists some $(i_0,j_0)$ which belongs to $J_{\chi }$ for all $\chi $ in an uncountable subset of $\widehat G$ . But since the sets

$$ \begin{align*} \{(x,y)\in X\times X:\forall g \in G,~\lambda_i(\varphi(g),x)=\mu_j(\psi(g),y)=\chi(g)\} \end{align*} $$

are disjoint for different $\chi $ ’s and of positive measure, we obtain a contradiction. This proves the claim.

Now, we return to the proof of the lemma. Since C is at most countable, we can apply Theorem 4.3. We see that there exists an ergodic extension $\pi :\widetilde {X}\rightarrow X$ , such that for every $\chi \in C$ , there exist G-eigenvalues $\chi ^{\varphi },\chi ^{\psi }:G\rightarrow S^1$ with $\chi ^{\varphi }(\varphi (g))=\chi (g)$ and $\chi ^{\psi }(\psi (g))=\chi (g)$ . Let $m_{\chi }^{\varphi },m_{\chi }^{\psi } :\widetilde {X}\rightarrow S^1$ be the corresponding eigenfunctions. Now, fix some $(i,j)\in J$ , and let $\chi \in C$ be, such that $\lambda _i(\varphi (g),x)=\mu _j(\psi (g),y)=\chi (g)$ whenever $c_{i,j}(x,y)\not =0$ . We deduce that

is a $\varphi (G)\times \psi (G)$ -invariant function. Since $c_{i,j}$ is also $\varphi (G)\times \psi (G)$ -invariant, we deduce by equation (15) that $f\circ \pi $ is a linear combination of products of eigenfunctions

and some $\varphi (G)\times \psi (G)$ -invariant functions. Equivalently, the lift of f to $\widetilde {X}\times \widetilde {X}$ is measurable with respect to the $\sigma $ -algebra

as required.

The following result of Tao and Ziegler [Reference Tao and Ziegler29] plays in important role in our work.

Theorem 4.8 ([Reference Tao and Ziegler29], Theorem 1.19)

Let G be a countable discrete abelian group, and let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g \in G})$ be a G-system. Let $H_1,H_2$ be two subgroups of G, and denote by $H_1+H_2$ the subgroup of G generated by $H_1$ and $H_2$ . Then, for every $d_1,d_2\in \mathbb {N}$ , one has

$$ \begin{align*} \mathcal{Z}_{H_1}^{d_1}(X) \land \mathcal {Z}^{d_2}_{H_2}(X)\preceq \mathcal {Z}^{d_1+d_2}_{H_1+H_2}(X). \end{align*} $$

In particular, by setting $d_1=d_2=1$ and using Lemma 3.6, we deduce:

We combine this with the results in Section 3 to deduce the following version of Theorem 3.5.

Theorem 4.10. Let G be a countable discrete abelian group and $\textbf {X} = (X, \mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system. Suppose that $\varphi , \psi : G \rightarrow G$ are arbitrary homomorphisms, such that $(\psi -\varphi )(G)$ has finite index in G. There exists an ergodic extension $\pi :(\widetilde {X},\widetilde {\mu })\rightarrow (X,\mu )$ , such that for any $f_0,f_1,f_2\in L^{\infty }(\mu )$

$$ \begin{align*} &\text{UC -}\lim_{g\in G} \int_{X} f_0\cdot T_{\varphi(g)} f_1\cdot T_{\psi(g)} f_2~ d\mu =\\ &\text{UC -}\lim_{g\in G} \int_{\widetilde{X}} \widetilde{f}_0\cdot T_{\varphi(g)} E(\widetilde{f}_1|\mathcal {Z}_{G}^2(\widetilde{X})\lor \mathcal {I}_{\varphi}(\widetilde{X})) \cdot T_{\psi(g)} E(\widetilde{f}_2|\mathcal {Z}_{G}^2(\widetilde{X})\lor \mathcal {I}_{\psi}(\widetilde{X}))~ d\widetilde{\mu} \end{align*} $$

in $L^2(\widetilde {X})$ , where $\widetilde {f}_i:=f_i\circ \pi $ denotes the lift of $f_i$ to the extension $\widetilde {X}$ .

Recall that the factors $\mathcal {Z}_{\varphi }(X)$ and $\mathcal {Z}_{\psi }(X)$ are relatively independent over $\mathcal {Z}_{\varphi ,\psi }(X)$ . To put this fact to use, we need to introduce a construction known as a fibre product:

Definition 4.11 (The fibre product over a factor.)

For $i=1,2$ , let $\textbf {Y}_i=(Y_i, \mathcal {Y}_i,\mu _i,(S^{(i)}_g)_{g\in G})$ be G-systems. Suppose that $\textbf {Y}=(Y,\mathcal {Y},\nu ,(S_g)_{g\in G})$ is a common factor, and let $\pi _i:Y_i\rightarrow Y$ , $i=1,2$ denote the factor maps. The fibre product of $\mathit{\boldsymbol Y}_1$ and $\mathit{\boldsymbol Y}_2$ over $\mathit{\boldsymbol Y}$ is the system , where

$$ \begin{align*} Y_1\times_Y Y_2 = \{(y_1,y_2)\in Y_1\times Y_2 : \pi_1(y_1)=\pi_2(y_2)\} \end{align*} $$

and

$$ \begin{align*} \mu_1\times_Y \mu_2 = \int_Y \mu_{1,y}\times \mu_{2,y} d\nu(y), \end{align*} $$

where

$$ \begin{align*} \mu_i = \int_{Y} \mu_{i,y} d\nu(y) \end{align*} $$

is the disintegration of the measure $\mu _i$ over Y for $i=1,2$ .

We will use the following result from [Reference Zimmer31]:

We also need the following result:

Host and Kra [Reference Host and Kra23] proved Theorem 4.14 for $\mathbb {Z}$ -actions, but the argument extends easily to arbitrary countable discrete abelian groups.

We now have all the requisite tools to prove Theorem 4.10.

Proof of Theorem 4.10

By the previous result, we see that if $f_0,f_1$ or $f_2$ are orthogonal to functions measurable with respect to the $\sigma $ -algebra $\mathcal {Z}_{\varphi }(X) \lor \mathcal {Z}_{\psi }(X)$ , then the averages above are zero. Therefore, by Theorem 4.12, the factor $\mathbf {Z}_{\varphi }(X)\times _{\mathbf {Z}_{\varphi ,\psi }(X)}\mathbf {Z}_{\psi }(X)$ is a characteristic factor. We may therefore assume without loss of generality that $\textbf {X}=\mathbf {Z}_{\varphi }(X)\times _{\mathbf {Z}_{\varphi ,\psi }(X)}\mathbf {Z}_{\psi }(X)$ . For the sake of simplicity of notations, we write $\mu _{\varphi ,\psi }$ for the measure $\mu _{Z_{\varphi }(X)}\times _{Z_{\varphi ,\psi }(X)} \mu _{Z_{\psi }(X)}$ on $Z_{\varphi (X)}\times _{Z_{\varphi ,\psi }(X)}Z_{\psi }(X)$ . By linearity, it suffices to prove the theorem in the case where

and

for some $f_1^{\varphi },f_2^{\varphi }:Z_{\varphi }(X)\rightarrow \mathbb {C}$ and $f_1^{\psi },f_2^{\psi } : Z_{\psi }(X)\rightarrow \mathbb {C}$ . Then,

(16)

By Proposition 3.5, (16) is equal to

(17) $$ \begin{align} \begin{aligned} \text{UC -}\lim_{g\in G} \underset{Z_{\varphi}(X)\times Z_{\psi}(X)}{\int} f_0&(x,y) \cdot T_{\varphi(g)}\left(f_1^{\varphi}\cdot E(f_1^{\psi}|\mathcal {Z}_{\varphi,\psi}(X))\right)(x) \\ & T_{\psi(g)} \left(E(f_2^{\varphi}|\mathcal {Z}_{\varphi,\psi}(X))\cdot f_2^{\psi}\right)(y)~d\mu_{\varphi,\psi}(x,y). \end{aligned} \end{align} $$

Note that we used the fact that $E(h|\mathcal {Z}_{\varphi ,\psi }(X))(x)=E(h|\mathcal {Z}_{\varphi ,\psi }(X))(y)$ for $\mu _{\varphi ,\psi }$ a.e. $x,y$ . By the mean ergodic theorem, applied to the transformation $T_{\varphi }\times T_{\psi }$ , the limit in (17) converges to

By Lemma 4.7, we can find an ergodic extension $\pi :\widetilde {X}\rightarrow X$ (independent of $f_0,f_1,f_2$ ), such that $\pi ^{-1}\left(\mathcal {I}_{\varphi \times \psi }(X)\right)$ is a sub- $\sigma $ -algebra of

. Now, by applying the same argument as above with $\widetilde {f}_0,\widetilde {f}_1$ and $\widetilde {f}_2$ instead of $f_0,f_1$ and $f_2$ , and using Theorem 4.14 in order to replace $\pi ^{-1}(\mathcal {Z}_{\varphi ,\psi }(X))$ with $\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})$ , we deduce that:

(18)

where $\widetilde {\mu }_{\varphi ,\psi }$ is the lift of $\mu _{\varphi ,\psi }$ to $\widetilde {X}$ .

We return to the proof of the theorem. By linearity, it is enough to show that if $E(\widetilde {f}_1|\mathcal {Z}_{G}^2(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X}))=0$ or $E(\widetilde {f}_2|\mathcal {Z}_{G}^2(\widetilde {X})\lor \mathcal {I}_{\psi }(\widetilde {X}))=0$ , then (18) is zero. By symmetry and Lemma 4.9, we may assume without loss of generality that $E(\widetilde {f}_1|\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X}))=0$ . Since $\mathcal {Z}_{\varphi }(\widetilde {X}),\mathcal {Z}_{\psi }(\widetilde {X})$ are relatively independent over $\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})$ , they are also relatively independent over the larger $\sigma $ -algebra $\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X})$ . We deduce, by Proposition 2.7, that

(19) $$ \begin{align} E(\widetilde{f}_1^{\varphi}|\mathcal {Z}_{\varphi,\psi}(\widetilde{X})\lor \mathcal {I}_{\varphi}(\widetilde{X}))\cdot E(\widetilde{f}_1^{\psi}|\mathcal {Z}_{\varphi,\psi}(\widetilde{X})\lor \mathcal {I}_{\varphi}(\widetilde{X}))=0. \end{align} $$

Claim. $E(\widetilde {f}_1^{\psi }|\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X})) =E(\widetilde {f}_1^{\psi }|\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})) $ .

Proof of the claim

$\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X})$ is a factor of $\mathcal {Z}_{\varphi }(\widetilde {X})$ . By Theorem 4.14, $\widetilde {f}_1^{\psi }$ is measurable with respect to $\mathcal {Z}_{\psi }(\widetilde {X})$ , and this and $\mathcal {Z}_{\varphi }(\widetilde {X})$ are relatively independent over $\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})$ , so the claim follows.

Equation (19) and the claim imply that

$$ \begin{align*} \widetilde{f}_1^{\varphi}\cdot E(\widetilde{f}_1^{\psi}|\mathcal {Z}_{\varphi,\psi}(\widetilde{X}))= \left(\widetilde{f}_1^{\varphi} - E(\widetilde{f_1}^{\varphi} |\mathcal {Z}_{\varphi,\psi}(\widetilde{X})\lor \mathcal {I}_{\varphi}(\widetilde{X}))\right)E(\widetilde{f}_1^{\psi}|Z_{\varphi,\psi}(\widetilde{X})) \end{align*} $$

is orthogonal to all functions measurable with respect to $\mathcal {Z}_{\varphi ,\psi }(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X})$ , and so it is also orthogonal to those measurable with respect to $\mathcal {Z}(\widetilde {X})\lor \mathcal {I}_{\varphi }(\widetilde {X})$ . Since $\pi ^{-1}(\mathcal {I}_{\varphi \times \psi }(X))$ is a sub- $\sigma $ -algebra of , this implies that (18) is equal to zero as required.

As a corollary, we also have the following stronger counterpart of Proposition 3.8.

Corollary 4.15. In the settings of Theorem 4.10. Let $\eta :Z(\widetilde {X})\rightarrow \mathbb {C}$ be a continuous function and $f_0,f_1,f_2\in L^{\infty }(X)$ . Let $\alpha _g$ denote the rotation of $g\in G$ on $Z(\widetilde {X})$ . If $a, b \in {\mathbb Z}$ are coprime, then

$$ \begin{align*} \text{UC -}\lim_{g\in G} \eta(\alpha_{g})\int_{\widetilde{X}} \widetilde{f}_0\cdot T_{ag} \widetilde{f}_1\cdot T_{bg} \widetilde{f}_2 ~d\widetilde{\mu} = \end{align*} $$

$$ \begin{align*}\text{UC -}\lim_{g\in G} \eta(\alpha_g)\int_{\widetilde{X}} \widetilde{f}_0\cdot T_{ag} E(\widetilde{f}_1|\mathcal{Z}_G^2(\widetilde{X})\lor \mathcal {I}_a(\widetilde{X})) \cdot T_{bg} E(\widetilde{f}_2|\mathcal {Z}_G^2(\widetilde{X})\lor \mathcal {I}_b(\widetilde{X}))~ d\widetilde{\mu} ,\end{align*} $$

where $\widetilde {f}_i = f_i \circ \pi $ is the lift of $f_i$ to $\widetilde {X}$ for $i=0,1,2$ .

Proof. Since $\eta $ is measurable with respect to $\mathcal {Z}(\widetilde {X})$ , it is a linear combination of characters. Therefore, it is enough to prove the equality in the special case where $\eta $ itself is a character. Then, since a and b are coprime, we can find $t,s\in \mathbb {Z}$ , such that $ta+sb=1$ . Set $h_0 =\widetilde {f}_0\cdot \eta ^{-(t+s)}$ , $h_1 = \widetilde {f}_1\cdot \eta ^s$ and $h_2=\widetilde {f}_2\cdot \eta ^t$ . Arguing as in Theorem 4.10, we have

(20) $$ \begin{align} &\text{UC -}\lim_{g\in G} \int_{\widetilde{X}} h_0\cdot T_{ag} h_1\cdot T_{bg} h_2 ~d\widetilde{\mu} =\nonumber\\ &\text{UC -}\lim_{g\in G} \int_{\widetilde{X}} h_0\cdot T_{ag} E(h_1|\mathcal{Z}_G^2(\widetilde{X}) \lor \mathcal {I}_a(\widetilde{X})) \cdot T_{bg} E(h_2|\mathcal{Z}_G^2(\widetilde{X}) \lor \mathcal {I}_b(\widetilde{X}))~ d\widetilde{\mu}. \end{align} $$

Now, since $\eta $ is measurable with respect to $\mathcal {Z}(\widetilde {X})$ , it is also measurable with respect to $\mathcal {Z}_G^2(\widetilde {X}) \lor \mathcal {I}_a(\widetilde {X})$ and $\mathcal {Z}_G^2(\widetilde {X}) \lor \mathcal {I}_b(\widetilde {X})$ , so the claim follows by rewriting $h_i$ in terms of $\eta $ and $\widetilde {f}_i$ on both sides of equation (20).

5.1 Previous limit formulas

Assuming all of the subgroups $aG$ , $bG$ , $(a+b)G$ and $(b-a)G$ have finite index in G, a limit formula was obtained in [Reference Ackelsberg, Bergelson and Best2] for the multiple ergodic averages in (21) by analysing a Mackey group associated to the abelian extension corresponding to the Conze–Lesigne factor (the relevant terminology is defined in the next subsection). For compact groups Z and H, let $\mathcal {M}(Z,H)$ denote the space of measurable functions $f : Z \to H$ equipped with the topology of convergence in measure (with respect to the Haar probability measure).

Theorem 5.1 ([Reference Ackelsberg, Bergelson and Best2], Theorem 7.1)

Let G be a countable discrete abelian group. Let $a,b\in \mathbb {Z}$ , such that $aG$ , $bG$ , $(a+b)G$ and $(b-a)G$ have finite index in G. Let $k_1' = -ab(a+b)$ , $k_2'=ab(a+b)$ and $k_3' = -ab(b-a)$ . Set $D=\text {gcd}(k_1',k_2',k_3')$ and $k_i=\frac {k_i'}{D}$ for $i=1,2,3$ . Let $c_1,c_2,c_3\in \mathbb {Z}$ so that $\sum _{i=1}^3 k_ic_i=1$ . Let $\textbf {X} = \textbf {Z}\times _{\sigma } H$ be as in Theorem 2.5(iii). There is a function $\psi :Z\times Z\rightarrow H$ , such that $\psi (0,z)=0$ for every $z\in Z$ and $t\mapsto \psi (t,\cdot )$ is a continuous map from Z to $\mathcal {M}(Z,H)$ , and for every $f_1,f_2,f_3\in L^{\infty }(\mu )$ ,

$$ \begin{align*} \text{UC -}\lim_{g\in G} f_1(T_{ag}x)f_2(T_{bg}x)f_3(T_{(a+b)g}x) = \int_{Z\times H^2} \prod_{i=1}^3 f_i(z+a_it,h+d_iu+a_i^2v+c_i\psi(t,z)~ du~dv~dt, \end{align*} $$

in $L^2(\mu )$ , where $x=(z,h)\in Z\times H$ , and $a_1=a,a_2=b,a_3=a+b$ .

Assuming that $(b-a)$ is even, the last author proved the following result.

Theorem 5.2 ([Reference Shalom27], Corollary 6.2)

Let G be a countable discrete abelian group. Let $a,b\in \mathbb {Z}$ be, such that $(b-a)$ is even and $(b-a)G$ has finite index in G. Let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system, such that $\textbf {X}=\mathbf {Z}^2(X)$ . Then, there exists an ergodic extension $\pi :Y\rightarrow X$ which is isomorphic to a $2$ -step nilpotent coset systemFootnote ³ and for every $f_1,f_2,f_3\in L^{\infty }(X)$ ,

$$ \begin{align*}&\text{UC -}\lim_{g\in G} \widetilde{f}_1(T_{ag}y\Gamma) f_2(T_{bg}y\Gamma) f_3(T_{(a+b)g}y\Gamma) =\\& \int_{\mathcal {G}/\Gamma} \int_{\mathcal {G}_2} \widetilde{f}_1(yy_1^ay_2^{\binom{a}{2}}) \widetilde{f}_2(yy_1^by_2^{\binom{b}{2}}\Gamma)\widetilde{f}_3(yy_1^{a+b}y_2^{\binom{a+b}{2}}\Gamma)~ d\mu_{\mathcal{G}_2}(y_2)~d\mu_{\mathcal{G}/\Gamma}(y\Gamma). \end{align*} $$

The above formula fails if $b-a$ is odd (see [Reference Shalom27, Example 6.3]).

Observe that in the formulas in Theorems 5.1 and 5.2, we can take $f_3\equiv 1$ and get a limit formula for the averages we are interested in. However, for the sake of our argument, we need a limit formula for every $a,b\in \mathbb {Z}$ regardless of the indices of the subgroups $aG$ , $bG$ and $(a\pm b)G$ and the parity of $b-a$ . Below, we remove the finite index assumptions in Theorem 5.1.

5.2 Mackey group

Let G be a countable discrete abelian group, and let $\textbf {X}= (X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system. Suppose that $X=Z^2(X)$ , then, by Theorem 2.5, we can write $\textbf {X} = \textbf {Z} \times _{\sigma } H$ , where $\textbf {Z} = (Z, \alpha )$ is the Kronecker factor, H is a compact abelian group and $\sigma :G\times Z\rightarrow H$ is a cocycle.