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Khintchine-type recurrence for 3-point configurations

Published online by Cambridge University Press:  05 December 2022

Ethan Ackelsberg
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA; E-mail: ackelsberg.1@osu.edu
Vitaly Bergelson
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA; E-mail: vitaly@math.ohio-state.edu
Or Shalom
Affiliation:
Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel; E-mail: or.shalom@mail.huji.ac.il

Abstract

The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi , \psi \}$ has the large intersections property. That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set

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

is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14).

One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ .

The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages

$$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$

In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10).

Type
Dynamics
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1 Introduction

Let $(G,+)$ be a countable discrete abelian group. A probability measure-preserving G-system, or simply G-system for short, is a quadruple $\textbf {X}=(X,\mathcal {X},\mu , (T_g)_{g\in G})$ , where $(X,\mathcal {X},\mu )$ is a standard Borel probability space (that is, up to isomorphism of measure spaces, X is a compact metric space, $\mathcal {X}$ is the Borel $\sigma $ -algebra and $\mu $ is a regular Borel probability measure) and $T_g :X \to X$ , $g \in G$ , are measure-preserving transformations, such that $T_{g+h}=T_g\circ T_h$ for every $g,h\in G$ and $T_0=Id$ . The transformation $T_g:X \to X$ gives rise to a unitary operator on $L^2(\mu )$ , which we also denote by $T_g$ , given by the formula $T_g f(x) = f(T_g x)$ . We say that a G-system is ergodic if the only measurable $(T_g)_{g\in G}$ -invariant functions are the constant functions.

1.1 Khintchine-type recurrence and the large intersections property

The starting point for the study of recurrence in ergodic theory is the Poincaré recurrence theorem, which states that, for any measure-preserving system $\left(X, \mathcal {X}, \mu , T \right)$ and any set $A \in \mathcal {X}$ with $\mu (A)> 0$ , there exists $n \in \mathbb N$ , such that $\mu (A \cap T^{-n}A)> 0$ .

Khintchine’s recurrence theorem strengthens and enhances Poincaré’s recurrence theorem by improving on the size of the intersections and the size of the set of return times.

Theorem 1.1 (Khintchine’s recurrence theorem [Reference Khintchine24])

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

$$ \begin{align*} \left\{ n \in \mathbb N : \mu \left( A \cap T^{-n}A \right)> \mu(A)^2 - \varepsilon \right\} \end{align*} $$

has bounded gaps.

Khintchine’s recurrence theorem easily extends to general semigroups, where the appropriate counterpart of ‘bounded gaps’ is the notion of syndeticity. In this paper, we deal with recurrence in countable discrete abelian groups. A subset A of a countable discrete abelian group G is said to be syndetic if there exists a finite set $F\subseteq G$ , such that $A+F = \{a+f : a\in A, f\in F\} = G$ .

It is natural to ask if recurrence theorems other than Poincaré’s recurrence theorem also have Khintchine-type enhancements. For instance, it follows from the IP Szemerédi theorem of Furstenberg and Katznelson [Reference Furstenberg and Katznelson20] and also from [Reference Austin3, Theorem B] that, for any abelian group G, any $k \in \mathbb N$ and any family of homomorphisms $\varphi _1, \dots , \varphi _k : G \to G$ , the following holds: if $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ is a G-system and $A \in \mathcal {X}$ has $\mu (A)> 0$ , then the set

$$ \begin{align*} \left\{ g \in G : \mu \left( A \cap T_{\varphi_1(g)}^{-1}A \cap \dots \cap T_{\varphi_k(g)}^{-1}A \right)> 0 \right\} \end{align*} $$

is syndetic.Footnote 1 With the goal of Khintchine-type enhancements in mind, this motivates the following definition:

Definition 1.2. A family of homomorphisms $\varphi _1, \dots , \varphi _k : G \to G$ has the large intersections property if the following holds: for any ergodic G-system $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ , any $A \in \mathcal {X}$ and any $\varepsilon> 0$ , the set

$$ \begin{align*} \left\{ g \in G : \mu \left( A \cap T_{\varphi_1(g)}^{-1}A \cap \dots \cap T_{\varphi_k(g)}^{-1}A \right)> \mu(A)^{k+1} - \varepsilon \right\} \end{align*} $$

is syndetic.

The large intersections property is closely related to the phenomenon of popular differences in combinatorics (see, e.g. [Reference Ackelsberg and Bergelson1, Reference Berger11, Reference Berger, Sah, Sawhney and Tidor12, Reference Mandache25, Reference Sah, Sawhney and Zhao26]).

Determining which families of homomorphisms have the large intersections property is a challenging problem with many surprising features. In the case $G = {\mathbb Z}$ and $\varphi _i(n) = in$ , the problem was resolved in [Reference Bergelson, Host and Kra8].

Theorem 1.3 ([Reference Bergelson, Host and Kra8], Theorems 1.2 and 1.3)

The family $\{n, 2n, \dots , kn\}$ has the large intersections property in ${\mathbb Z}$ if and only if $k \le 3$ .

Later work of Frantzikinakis [Reference Frantzikinakis18] and of Donoso et al. [Reference Donoso, Le, Moreira and Sun15] generalised this picture for arbitrary homomorphisms ${\mathbb Z} \to {\mathbb Z}$ , which take the form $n \mapsto an$ for some $a \in {\mathbb Z}$ .

Theorem 1.4 ([Reference Frantzikinakis18], special case of Theorem C; [Reference Donoso, Le, Moreira and Sun15], Theorem 1.5)

  1. 1. For any $a, b \in {{\mathbb Z}}$ , the families $\{an,bn\}$ and $\{an, bn, (a+b)n\}$ have the large intersections property (in ${\mathbb Z}$ ).

  2. 2. For any $k \ge 4$ and any distinct and nonzero integers $a_1, \dots , a_k \in {\mathbb Z}$ , the family $\{a_1n, \dots , a_kn\}$ does not have the large intersections property (in ${\mathbb Z}$ ).

Remark 1.5. Finitary combinatorial work of [Reference Sah, Sawhney and Zhao26, Theorem 1.6] suggests that the family $\{a_1n,a_2n,a_3n\}$ has the large intersections property if and only if $a_i+a_j=a_k$ for some permutation $\{i,j,k\}$ of $\{1,2,3\}$ .

In [Reference Bergelson, Tao and Ziegler10], Khintchine-type recurrence results are established in the infinitely generated torsion groups $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}/p{\mathbb Z}}$ .

Theorem 1.6 ([Reference Bergelson, Tao and Ziegler10], Theorems 1.12 and 1.13)

  1. 1. Fix a prime $p> 2$ . If $c_1, c_2 \in {\mathbb Z}/p{\mathbb Z}$ are distinct and nonzero, then $\{c_1g,c_2g\}$ has the large intersections property in $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}/p{\mathbb Z}}$ .

  2. 2. Fix a prime $p> 3$ . If $c_1, c_2 \in {\mathbb Z}/p{\mathbb Z}$ are distinct and nonzero and $c_1 + c_2 \ne 0$ , then $\{c_1g, c_2g, (c_1+c_2)g\}$ has the large intersections property in $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}/p{\mathbb Z}}$ .

Remark 1.7. It is conjectured in [Reference Bergelson, Tao and Ziegler10, Conjecture 1.14] that, if $c_1, c_2, c_3 \in {\mathbb Z}/p{\mathbb Z}$ are distinct and nonzero and $c_i+c_j \ne c_k$ for every permutation $\{i,j,k\}$ of $\{1,2,3\}$ , then $\{c_1g, c_2g, c_3g\}$ does not have the large intersections property in $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}/p{\mathbb Z}}$ .

Khintchine-type recurrence in general abelian groups was addressed in [Reference Ackelsberg, Bergelson and Best2] and [Reference Shalom27]. For 3-point linear configurations, the following was shown in [Reference Ackelsberg, Bergelson and Best2]:

Theorem 1.8 ([Reference Ackelsberg, Bergelson and Best2], Theorem 1.10)

Let G be a countable discrete abelian group. Let $\varphi ,\psi :G \to G$ be homomorphisms. If all three of the subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi ,\psi \}$ has the large intersections property.

Remark 1.9. Earlier work of Chu demonstrates that at least some finite index condition is necessary for large intersections. Namely, it follows from [Reference Chu13, Theorem 1.2] that the pair $\{(n,0), (0,n)\}$ , does not have the large intersections property in ${\mathbb Z}^2$ (see [Reference Ackelsberg, Bergelson and Best2, Example 10.2]). While we do not pursue optimal lower bounds for families lacking the large intersections property in this paper, Chu also showed that, for the pair $\{(n,0), (0,n)\}$ , the optimal lower bound is still polynomially large. In particular, $\mu \left( A \cap T_{(n,0)}^{-1}A \cap T_{(0,n)}^{-1}A \right)> \mu (A)^4 - \varepsilon $ for syndetically many n (see [Reference Chu13, Theorem 1.1]).

For more restricted 4-point configurations, the following result was shown in [Reference Ackelsberg, Bergelson and Best2] and independently in [Reference Shalom27]:

Theorem 1.10 ([Reference Ackelsberg, Bergelson and Best2], Theorem 1.11; [Reference Shalom27], Theorem 1.3)

Let G be a countable discrete abelian group. Let $a, b \in {\mathbb Z}$ be distinct, nonzero integers, such that all four of the subgroups $aG$ , $bG$ , $(a+b)G$ and $(b-a)G$ have finite index in G. Then $\{ag, bg, (a+b)g\}$ has the large intersections property.

1.2 Main results

In this paper, we refine the understanding of Khintchine-type recurrence for 3-point configurations in abelian groups and make substantial progress towards characterising the pairs of homomorphisms $\varphi , \psi : G \to G$ that have the large intersections property.

Our first result shows that the large intersections property holds for any pair of homomorphisms $\{\varphi ,\psi \}$ so long as at least two of the three subgroups in Theorem 1.8 have finite index in G. In particular, this shows that [Reference Ackelsberg, Bergelson and Best2, Conjecture 10.1] is false.

Theorem 1.11. Let G be a countable discrete abelian group. Let $\varphi , \psi : G \to G$ be homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G. Then for any ergodic G-system $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ , any $A \in \mathcal {X}$ and any $\varepsilon>0$ , 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.

As mentioned above (see Remark 1.9), the work of Chu [Reference Chu13] provides a counterexample to the large intersections property when all three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi - \varphi )(G)$ have infinite index in G. In this paper, we give additional counterexamples for the group $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}}$ with homomorphisms $g \mapsto ag$ and $g \mapsto bg$ for some $a, b \in {\mathbb Z}$ (see Theorem 1.14 below). A natural question to ask, then, is what happens when only one of the subgroups $\varphi (G)$ , $\psi (G)$ or $(\psi -\varphi )(G)$ has finite index. Namely:

Question 1.12. Let G be a countable discrete abelian group, and let $\varphi :G\rightarrow G$ , $\psi :G\rightarrow G$ be homomorphisms, such that at least one of the subgroups $\varphi (G)$ , $\psi (G)$ or $(\psi -\varphi )(G)$ has finite index in G. Is it true that, for any ergodic G-system $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ , any $A\in \mathcal {X}$ , and any $\varepsilon>0$ , 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?

Note that, by symmetry, it is enough to provide an answer to Question 1.12 under the assumption that $(\psi -\varphi )(G)$ has finite index. Indeed, suppose $\psi (G)$ has finite index in G. Then, since $(T_g)_{g \in G}$ is a measure-preserving action, we have the identity

$$ \begin{align*} \mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right) = \mu \left( A \cap T_{-\varphi(g)}^{-1}A \cap T_{(\psi-\varphi)(g)}^{-1}A \right) .\end{align*} $$

Hence, the pair $\{\varphi ,\psi \}$ has the large intersections property if and only if $\left\{\widetilde {\varphi }, \widetilde {\psi }\right\}$ has the large intersections property, where $\widetilde {\varphi } = - \varphi $ and $\widetilde {\psi } = \psi - \varphi $ . Moreover, we have $(\widetilde {\psi } - \widetilde {\varphi })(G) = \psi (G)$ , which is of finite index. A similar argument applies when $\varphi (G)$ has finite index.

When $G = {\mathbb Z}^2$ , we can use additional tools from linear algebra to classify all pairs of homomorphisms $\varphi $ and $\psi $ , which allows us to answer Question 1.12 affirmatively in this setting. In fact, we can give a precise description of the optimal size of intersections for all 3-point configurations in ${\mathbb Z}^2$ (see Subsection 1.4 below). However, our results rely heavily on properties of $2\times 2$ matrices, and it appears that the full generality of Question 1.12 for general abelian groups and general homomoprhisms is out of reach without developing new techniques.

On the other hand, in the special case $\varphi (g) = ag$ and $\psi (g) = bg$ for $a, b \in {\mathbb Z}$ , we answer Question 1.12 affirmatively:

Theorem 1.13. Let G be a countable discrete abelian group. Let $a,b\in \mathbb {Z}$ be integers, such that $(b-a)G$ has finite index in G. Then for any ergodic G-system $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ , any $A \in \mathcal {X}$ and any $\varepsilon>0$ , the set

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

is syndetic.

We also show that the assumption that $(b-a)G$ has finite index in G is necessary. To see this, we prove the following result:

Theorem 1.14. Let $G = \bigoplus _{n=1}^{\infty }{{\mathbb Z}}$ . Let $l \in \mathbb N$ . There exists a number $P = P(l)$ , such that, for any $a, b \in \mathbb N$ with $p \mid \gcd (a,b)$ for some prime $p \ge P$ , there is an ergodic G-system $\left( X, \mathcal {X}, \mu , (T_g)_{g \in G} \right)$ and a set $A \in \mathcal {X}$ with $\mu (A)> 0$ , such that

$$ \begin{align*} \mu(A\cap T_{ag}^{-1} A\cap T_{bg}^{-1} A)\leq \mu(A)^l \end{align*} $$

for every $g\ne 0$ .

Question 1.15. Can p in the statement of Theorem 1.14 be replaced by any natural number?

1.3 Applications to geometric progressions and other multiplicative patterns

One particularly interesting corollary of Theorem 1.13 is a multiplicative version of the following large intersection theorem in [Reference Bergelson, Host and Kra8]:

Theorem 1.16 ([Reference Bergelson, Host and Kra8], Corollary 1.5)

Let $E \subseteq {\mathbb Z}$ be a set of positive upper Banach density

$$ \begin{align*} d^*(E) = \limsup_{N - M \to \infty}{\frac{\left| E \cap \left\{M, M+1, \dots, N-1 \right\} \right|}{N-M}}> 0. \end{align*} $$

Then, for any $\varepsilon> 0$ , the set

$$ \begin{align*} \left\{ n \in {\mathbb Z} : d^* \left( E \cap (E - n) \cap (E - 2n) \right)> d^*(E) - \varepsilon \right\} \end{align*} $$

is syndetic.

Consider the group $G=(\mathbb {Q}_{>0},\cdot )$ . This is a multiplicative counterpart of $(\mathbb {Z},+)$ . In the group $(\mathbb Q_{>0}, \cdot )$ , the upper Banach density of a set $E \subseteq \mathbb Q_{>0}$ is given by

(1) $$ \begin{align} d^*_{\text{mult}}(E) = \sup_{\Phi}{\limsup_{N \to \infty}{\frac{|E \cap \Phi_N|}{|\Phi_N|}}}, \end{align} $$

where the supremum is taken over all Følner sequences $\Phi = (\Phi _N)_{N \in \mathbb N}$ in $(\mathbb Q_{>0}, \cdot )$ . An instructive class of examples of Følner sequences in $(\mathbb Q_{>0}, \cdot )$ is given by sequences of the form

$$ \begin{align*} \Phi_N = \left\{ b_N \prod_{i=1}^N{q_i^{r_i}} : -N \le r_i \le N \right\}, \end{align*} $$

where $(q_n)_{n \in \mathbb N}$ is a sequence of generators of $(\mathbb Q_{>0}, \cdot )$ and $(b_N)_{N \in \mathbb N}$ is any sequence in $\mathbb Q_{>0}$ . The subscript on $d^*_{\text {mult}}$ is to emphasise that this density is with respect to the multiplicative structure on $\mathbb Q_{>0}$ rather than its additive structure. Using an ergodic version of the Furstenberg correspondence principle (see [Reference Bergelson and Ferré Moragues6, Theorem 2.8]), we deduce the following result as an immediate consequence of Theorem 1.13:

Theorem 1.17. Let $E\subseteq \mathbb {Q}_{>0}$ be a set of positive multiplicative upper Banach density $d_{\text {mult}}^*(E)> 0$ , and let $k\in {\mathbb Z}$ . Then for any $\varepsilon>0$ , the sets

(2) $$ \begin{align} \left\{q\in\mathbb{Q}_{>0} : d^*_{\text{mult}}\left(E\cap q^{-k}E\cap q^{-(k+1)}E\right)>d^*_{\text{mult}}(E)^3-\varepsilon\right\} \end{align} $$

and

(3) $$ \begin{align} \left\{q\in\mathbb{Q}_{>0} : d^*_{\text{mult}}\left(E\cap q^{-1}E\cap q^{-k}E\right)>d^*_{\text{mult}}(E)^3-\varepsilon\right\} \end{align} $$

are syndetic.

Remark 1.18. The special case where $k=1$ in (2) or $k = 2$ in (3) is related to the existence of length 3 geometric progressions in sets of positive multiplicative density. Heuristically, if E were a random set, where each positive rational number $q\in \mathbb {Q}_{>0}$ is independently chosen to be inside E with probability $\alpha $ , then the expected number of geometric progressions of length 3 and quotient q would be $\alpha ^3$ . Now, fix any set E with $d^*_{mult}(E) = \alpha $ . Choosing $\varepsilon $ sufficiently small, our result implies that E contains almost as many geometric progressions with quotient q as a random set with the same density, $\alpha $ , for a syndetic set of quotients.

Theorem 1.14 shows that, if n and m share a large prime factor, then $\{q^n, q^m\}$ does not have the large intersections property in $(\mathbb Q_{>0}, \cdot )$ . What happens in the case that n and m are coprime is an interesting question that we are unable to answer with our current methods:

Question 1.19. Suppose $n, m \in \mathbb N$ are coprime. Does the pair $\{q^n, q^m\}$ have the large intersections property in $(\mathbb Q_{>0}, \cdot )$ ?

Since every $\mathbb {Z}$ -action can be lifted to a $(\mathbb {Q}_{>0}, \cdot )$ -action (indeed, $(\mathbb Q_{>0}, \cdot )$ is torsion-free, so ${\mathbb Z}$ embeds as a subgroup), we see from Theorem 1.3 above that $\{q,q^2,\dots , q^k\}$ does not have the large intersections property for $k \ge 4$ . However, we can still ask about geometric progressions of length $4$ .

Question 1.20. Does the triple $\{q, q^2, q^3\}$ have the large intersections property in $(\mathbb Q_{>0}, \cdot )$ ?

For a discussion of where our methods come up short for answering Questions 1.19 and 1.20, see Subsection 2.7 below.

1.3.1 Patterns in $(\mathbb N,\cdot )$

A notion of upper Banach density can be defined in the semigroup $(\mathbb N, \cdot )$ by the formula in (1), where the supremum is now taken over Følner sequences in $(\mathbb N, \cdot )$ . Examples of Følner sequences in $(\mathbb N, \cdot )$ include sequences of the form

$$ \begin{align*} \Phi_N = \left\{ b_N \prod_{i=1}^N{p_i^{r_i}} : 0 \le r_i \le N \right\}, \end{align*} $$

where $(p_n)_{n \in \mathbb N}$ is an enumeration of the prime numbers and $(b_N)_{N \in \mathbb N}$ is any sequence in $\mathbb N$ . In Section 8, we transfer Theorems 1.11 and 1.13 to the setting of cancellative abelian semigroups. As a consequence, we obtain the following result about geometric configurations in the multiplicative integers:

Theorem 1.21. Let $E \subseteq \mathbb N$ be a set of positive multiplicative upper Banach density, and let $k\in {\mathbb Z}$ . Then for any $\varepsilon>0$ , the sets

$$ \begin{align*} \left\{ m \in \mathbb N : d^*_{\text{mult}}\left(E\cap E/m^k \cap E/m^{k+1}\right)> d^*_{\text{mult}}(E)^3-\varepsilon \right\} \end{align*} $$

and

$$ \begin{align*} \left\{ m \in \mathbb N : d^*_{\text{mult}}\left(E\cap E/m \cap E/m^k \right)> d^*_{\text{mult}}(E)^3-\varepsilon \right\} \end{align*} $$

are (multiplicatively) syndetic in $(\mathbb N,\cdot )$ .

1.4 Applications to patterns in ${\mathbb Z}^2$

When $G = {\mathbb Z}^2$ , we are able to give a complete picture of the phenomenon of large intersections for 3-point matrix patterns, that is, patterns of the form $\{\vec {x}, \vec {x} + M_1\vec {n}, \vec {x} + M_2\vec {n}\}$ , where $\vec {x}, \vec {n} \in {\mathbb Z}^2$ and $M_1, M_2$ are $2\times 2$ matrices with integer entries (note that any homomorphism $\varphi : {\mathbb Z}^2 \to {\mathbb Z}^2$ can be expressed as a $2\times 2$ matrix with integer entries, so matrix patterns capture all possible configurations in ${\mathbb Z}^2$ that can be described within the framework of group homomorphisms).

Following [Reference Bergelson, Host and Kra8], we say that the syndetic supremum of a bounded real-valued ${\mathbb Z}^2$ -sequence $\left( a_{n,m} \right)_{(n, m) \in {\mathbb Z}^2}$ is the quantity

$$ \begin{align*} \text{synd-sup}_{(n,m) \in {\mathbb Z}^2}{a_{n,m}} := \sup{\left\{ a \in \mathbb R : \left\{ (n,m) \in {\mathbb Z}^2 : a_{n,m}> a \right\}~\text{is syndetic in} {{\mathbb Z}}^2 \right\}}. \end{align*} $$

For $2\times 2$ integer matrices $M_1$ and $M_2$ and $\alpha \in (0,1)$ , we define the ergodic popular difference density by

$$ \begin{align*} \text{epdd}_{M_1,M_2}(\alpha) := \inf{\text{synd-sup}_{\vec{n} \in {\mathbb Z}^2} \mu \left( A \cap T_{M_1\vec{n}}^{-1}A \cap T_{M_2\vec{n}}^{-1}A \right)}, \end{align*} $$

where the infimum is taken over all ergodic ${\mathbb Z}^2$ -systems $\left( X, \mathcal {X}, \mu , (T_{\vec {n}})_{\vec {n} \in {\mathbb Z}^2} \right)$ and sets $A \in \mathcal {X}$ with $\mu (A) = \alpha $ . This can be seen as an ergodic-theoretic analogue to the popular difference density defined in [Reference Sah, Sawhney and Zhao26]. It is natural to ask if $\text {epdd}_{M_1,M_2}(\alpha )$ coincides with the finitary combinatorial quantity $\text {pdd}_{M_1,M_2}(\alpha )$ . Standard tools for translating between ergodic theory and combinatorics, such as Furstenberg’s correspondence principle, are insufficient for resolving this question, and we do not know the answer in general. However, in special cases where $\text {pdd}_{M_1,M_2}(\alpha )$ is known, it is in agreement with the values of $\text {epdd}_{M_1,M_2}(\alpha )$ displayed in Table 1 below, and we suspect that $\text {pdd}_{M_1,M_2}(\alpha ) = \text {epdd}_{M_1,M_2}(\alpha )$ in the remaining cases (see Subsection 7.3 below for additional remarks on (combinatorial) popular difference densities for matrix patterns in ${\mathbb Z}^2$ ).

Table 1 Ergodic popular difference densities for 3-point matrix patterns in ${\mathbb Z}^2$

Theorem 1.11 provides a sufficient condition on the matrices $M_1$ and $M_2$ to guarantee that $\text {epdd}_{M_1, M_2}(\alpha ) \ge \alpha ^3$ for $\alpha \in (0,1)$ . We now seek to describe the quantity $\text {epdd}_{M_1, M_2}(\alpha )$ for any pair of $2\times 2$ integer matrices $M_1$ and $M_2$ . Table 1 summarises ergodic popular difference densities for all 3-point matrix configurations in ${\mathbb Z}^2$ (for matrices $M_1, M_2$ , we let $r(M_1,M_2)$ be a list of the ranks of $M_1$ , $M_2$ and $M_2 - M_1$ in decreasing order, and we denote by $[M_1,M_2]$ the commutator $M_1M_2 - M_2M_1$ ).

The cases $r(M_1, M_2) = (2,2,2)$ and $r(M_1, M_2) = (2,2,1)$ are covered directly by [Reference Ackelsberg, Bergelson and Best2, Theorem 1.10] and Theorem 1.11, respectively. Indeed, a matrix M has full rank if and only if the subgroup $M({\mathbb Z}^2) \subseteq {\mathbb Z}^2$ has finite index. More precisely,

$$ \begin{align*} [{\mathbb Z}^2 : M({\mathbb Z}^2)] = \begin{cases} \left| \det(M) \right|, & \text{if}~\det(M) \ne 0; \\ \infty, & \text{if}~\det(M) = 0. \end{cases} \end{align*} $$

The remaining cases are proved in Section 7.

1.5 Preliminary remarks on characteristic factors

In this paper, we approach multiple recurrence problems by determining and utilising the so-called characteristic factors, which are the factors that are responsible for the limiting behaviour of the quantity

$$ \begin{align*} \mu \left( A \cap T_{\varphi(g)}^{-1}A \cap T_{\psi(g)}^{-1}A \right) \end{align*} $$

in ergodic G-systems (see Subsection 2.2 for a discussion of factors in general and Definition 3.3 for a definition of characteristic factors). For ${\mathbb Z}$ -actions, there are two different approaches to characteristic factors for linear averages, developed independently by Host and Kra [Reference Host and Kra23] and by Ziegler [Reference Ziegler30], giving rise to factors that coincide (see [Reference Bergelson5, Appendix A]). However, in the context of G-actions, where G is an arbitrary (nonfinitely generated) countable discrete abelian group, the approaches of Host–Kra and of Ziegler may produce different factors (see Subsection 2.6 below for more details).

Our work, thus, leads to the general open question of how, in the setup of countable discrete abelian groups, the Host–Kra factors are related to the actual characteristic factors of the corresponding multiple ergodic averages (the factors obtained by Ziegler’s approach). Discerning the relationship between the Host–Kra factors and the characteristic factors may lead to a better understanding of the quantities

$$ \begin{align*} \mu(A\cap T_{\varphi_1(g)}^{-1} A\cap... \cap T_{\varphi_k(g)}^{-1} A), \end{align*} $$

where $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ is a G-system, $A \in \mathcal {X}$ and $\varphi _i:G\rightarrow G$ are homomorphisms or, more generally, polynomial maps.

1.6 Structure of the paper

The paper is organised as follows. In Section 2, we introduce notation and conventions that we use throughout the paper.

Proofs of the main results appear in Sections 36. First, in Section 3, we establish characteristic factors for the multiple ergodic averages

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

when $(\psi -\varphi )(G)$ has finite index in G and prove Theorem 1.11. Then, in Section 4, we use an extension trick to simplify the characteristic factors, and in Section 5, prove a new limit formula for the extension system, leading to a proof of Theorem 1.13. Finally, we prove Theorem 1.14 in Section 6.

The final two sections contain applications of the main results. Using Theorem 1.11 together with additional tools from [Reference Ackelsberg, Bergelson and Best2, Reference Bergelson, Host and Kra8, Reference Bergelson and Leibman9, Reference Chu13, Reference Donoso and Sun16], we compute ergodic popular difference densities for 3-point matrix patterns in ${\mathbb Z}^2$ . In Section 8, we extend the main results (Theorems 1.11 and 1.13) to the setting of cancellative abelian semigroups.

2 Preliminaries

The goal of this section is to introduce some notations and objects that will play an important role in this paper. Throughout this section, we let G denote an arbitrary countable discrete abelian group and $\textbf {X}=(X,\mathcal {X},\mu , (T_g)_{g\in G})$ a G-system.

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 $\textbf {X}$ .

Proposition 2.4 (cf. [Reference Bergelson, Tao and Ziegler10], Proposition 1.10)

Let G be a countable discrete abelian group, let $\textbf {X}$ be a G-system and let $k\geq 0$ . There exists a unique (up to isomorphism) factor $\mathbf {Z}^{k}(X)=\left( Z^{k}(X),\mathcal {Z}^{k}(X),\mu _k,(T^{(k)}_g)_{g \in G} \right)$ of $\textbf {X}$ with the property that for every $f\in L^{\infty } (X)$ , $\|f\|_{U^{k+1}(X)}=0$ if and only if $E(f|\mathcal {Z}^{k}(X))=0$ .

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.

Theorem 2.5. Let G be a countable discrete abelian group, and let $\textbf {X} = (X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be a ergodic G-system. Then,

  1. (i) For every $k\geq 1$ , $\mathcal {Z}^{k-1}(X)\preceq \mathcal {Z}^{k}(X)$ . In other words, $\mathbf {Z}^{k-1}(X)$ is a factor of $\mathbf {Z}^k(X)$ . In particular, $\mathcal {I}(X)\preceq \mathcal {Z}^k(X)$ for every $k\geq 0$ .

  2. (ii) The Kronecker factor of $\textbf {X}$ is isomorphic to a rotation on a compact abelian group. Namely, there exists a homomorphism $\alpha :G\rightarrow Z$ into a compact abelian group $(Z,+)$ , such that $\mathbf {Z}^1(X)$ is isomorphic to $(Z, (R_g)_{g \in G})$ , where $R_gz = z+\alpha (g)$ .

  3. (iii) For every $k\geq 1$ , if $\textbf {X}$ is ergodic, then $\mathbf {Z}^k(X)$ is an extension of $\mathbf {Z}^{k-1}(X)$ by a compact abelian group $(H,+)$ and a cocycle $\rho :G\times Z^{k-1}(X)\rightarrow H$ . Namely, $Z^k(X)=Z^{k-1}(X)\times H$ as measure spaces, and the action is given by $T^{(k)}_g (z,h) = (T^{(k-1)}_g z, h+\rho (g,z))$ .

Proof. The proof of $(i)$ is an immediate consequence of the monotonicity of the seminorms (see [Reference Host and Kra23, Corollary 4.4]). The proof of $(ii)$ in the generality of countable discrete abelian groups can be found in [Reference Ackelsberg, Bergelson and Best2, Lemma 2.4]. The proof of $(iii)$ can be found for $\mathbb {Z}$ -actions in [Reference Host and Kra23, Proposition 6.3], and the same proof works for arbitrary countable discrete abelian groups.

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.

Proposition 2.6 (Independent $\sigma $ -algebras)

Let $\textbf {X}=(X,\mathcal {X},\mu )$ be a probability space. Two $\sigma $ -algebras $\mathcal {A}_1$ and $\mathcal {A}_2$ on X are $\mu $ -independent if and only if the following equivalent conditions hold:

  1. (i) Any function $f\in L^{\infty }(X)$ measurable with respect to $\mathcal {A}_1$ and $\mathcal {A}_2$ simultaneously is a constant $\mu $ -almost everywhere.

  2. (ii) If $f\in L^{\infty }(X)$ is measurable with respect to $\mathcal {A}_1$ and $g\in L^{\infty }(X)$ is measurable with respect to $\mathcal {A}_2$ , then

    $$ \begin{align*} \int_X f\cdot g~d\mu = \int_X f~d\mu \cdot \int_X g~d\mu. \end{align*} $$

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 $ .

Proposition 2.7 (Relatively independent $\sigma $ -algebras)

Let $\textbf {X}=(X,\mathcal {X},\mu )$ be a probability space. Let $\mathcal {A}_1,\mathcal {A}_2$ be two $\sigma $ -algebras on X, and let $\mathcal {A}$ be a third $\sigma $ -algebra, such that $\mathcal {A}\preceq \mathcal {A}_1\land \mathcal {A}_2$ . Then, $\mathcal {A}_1$ and $\mathcal {A}_2$ are relatively independent with respect to $\mathcal {A}$ if the following equivalent conditions hold:

  1. (i) Any function $f\in L^{\infty }(X)$ measurable with respect to $\mathcal {A}_1$ and $\mathcal {A}_2$ simultaneously is measurable with respect to $\mathcal {A}$ .

  2. (ii) If f is measurable with respect to $\mathcal {A}_1$ and g is measurable with respect to $\mathcal {A}_2$ , then

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

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 Theorem 1.11

We first give a brief overview of the proof of Theorem 1.11. Let $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ be an ergodic G-system, and let $\varphi ,\psi :G\rightarrow G$ be arbitrary homomorphisms, such that $(\psi -\varphi )(G)$ has finite index in G. The key component in the proof of Theorem 1.11 is the analysis of the limit behaviour of the multiple ergodic averages

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

for $f_1,f_2\in L^{\infty }(X)$ . Standard arguments using the van der Corput lemma (Proposition 3.5) show that

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

where $\mathcal {Z}_{\varphi }(X)$ and $\mathcal {Z}_{\psi }(X)$ are the $\sigma $ -algebras of the Kronecker factors of X with respect to the actions of $\varphi (G)$ and $\psi (G)$ , respectively (see Subsection 2.5).

In Theorem 1.11, we assume, furthermore, that $\varphi (G)$ has finite index in G. In this case, the factor $\mathcal {Z}_{\varphi }(X)$ coincides with $\mathcal {Z}^1_G(X)$ , the Kronecker factor of X with respect to the action of G (see Lemma 3.6). Our main observation is that one can replace $\mathcal {Z}_{\psi }(X)$ in (9) with a smaller factor. As an illustration, we give the following example:

Example 3.1. Consider the additive group $G=\bigoplus _{j=1}^{\infty }\mathbb {Z}/4\mathbb {Z}$ . We use $i\in \mathbb {C}$ to denote the square root of $-1$ , and for every natural number $n\in \mathbb {N}$ , we let $C_n$ denote the group of roots of unity of degree n. We define an action of G on $X=\left(\prod _{j\in \mathbb {N}} C_4\right)\times C_2$ by

$$ \begin{align*} T_g (\textbf{x},y) = \left( (i^{g_j}x_j)_{j\in \mathbb{N}},y\cdot \prod_{j\in\mathbb{N}} (x_j^{2g_j}\cdot i^{g_j^2-g_j})\right), \end{align*} $$

where $\textbf {x}=(x_1,x_2,...)\in \prod _{j\in \mathbb {N}} C_4$ and $g=(g_1,g_2,...)$ is any representation of g in $\bigoplus _{j=1}^{\infty }\mathbb {Z}/4{\mathbb Z}$ . The system $(X, (T_g)_{g \in G})$ is a group extension of its Kronecker factor $Z_G(X) = \prod _{j\in \mathbb {N}} C_4$ by the cocycle

$$ \begin{align*} & \sigma : G \times \prod_{j\in\mathbb N}{C_4} \to C_2,\\ \sigma&(g,\textbf{x}) = \prod_{j\in\mathbb{N}} (x_j^{2g_j}\cdot i^{g_j^2-g_j}). \end{align*} $$

Let $\psi (g)=2g$ . We observe that the function $f(\textbf {x},y)=y$ is orthogonal to $L^2(Z^1(X))$ . On the other hand, we have

$$ \begin{align*} T_{2g}f(\textbf{x},y) = \sigma(2g,\textbf{x})\cdot y = \prod_{j\in\mathbb{N}} (x_j^{4g_j}\cdot i^{4g_j^2-2g_j}) \cdot y = \prod_{j\in\mathbb{N}} (-1)^{g_j} y = \prod_{j\in\mathbb{N}} (-1)^{g_j} f(\textbf{x},y). \end{align*} $$

In other words, f is an eigenfunction with respect to the action of $\psi (G)$ on X with eigenvalue $\lambda (2g) =\prod _{j\in \mathbb {N}} (-1)^{g_j}$ . Therefore, f is measurable with respect to $\mathcal {Z}_{\psi }(X)$ , and we see that $\mathcal {Z}^1(X)\not = \mathcal {Z}_{\psi }(X)$ . Now, let $\varphi (g)=g$ . We claim that f does not contribute to (8). Namely, we have that

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_g f_1 T_{2g} f = 0 \end{align*} $$

for every bounded function $f_1$ . Indeed, by (9), it is enough to check this equality in the case where $f_1$ is an eigenfunction with respect to the action of G. Let $\chi (g)$ be the eigenvalue of $f_1$ , we see that

$$ \begin{align*} \text{UC -}\lim_{g\in G} T_g f_1 T_{2g} f = f_1\cdot f \cdot \text{UC -}\lim_{g\in G} \chi(g)\cdot \lambda(2g). \end{align*} $$

The eigenfunctions of X take the form $h(\textbf {x},y) = \prod _{i=1}^n x_i^{l_i}$ for some $n\in \mathbb {N}$ and $l_1,...,l_n\in \{0,1,2,3\}$ . Therefore, $g\mapsto \chi (g)\lambda (2g)$ is a nontrivial character of G and so

$$ \begin{align*} \text{UC -}\lim_{g\in G} \chi(g)\lambda(2g)=0. \end{align*} $$

Remark 3.2. In the example above, the factor $\mathbf {Z}^1(X)$ is isomorphic to $\prod _{j\in \mathbb {N}} C_4$ equipped with the action $T^{(1)}_g x = (i^{g_j}\cdot x_j)_{j\in \mathbb {N}}$ , while $\mathbf {Z}^2(X)=\textbf {X}$ . On the other hand, for the $2G$ -system $\left( X, (T_g)_{g \in 2G} \right)$ , we have $\mathbf {Z}_{2G}^1(X) = \textbf {X}$ .

Example 3.1 suggests that a $\psi (G)$ -eigenfunction contributes to (8) if and only if its eigenvalue coincides with an eigenvalue of the G-action. In practice, we use a result of Frantzikinakis and Host [Reference Frantzikinakis and Host19] to decompose $f_2$ into a linear combination of eigenfunctions (see Proposition 3.12). However, since the action of $\psi (G)$ may not be ergodic, we have to include in our analysis the case where the $\psi (G)$ -eigenvalue, $\lambda (\psi (g))$ , is not a constant in X, but rather a $\psi (G)$ -invariant function. We let $\widetilde {\mathcal {Z}}_{\psi }(X)$ be the sub- $\sigma $ -algebra of $\mathcal {Z}_{\psi }(X)$ generated by all the $\psi (G)$ -eigenfunctions with eigenvalues $\lambda (\psi (\cdot ),x):X\rightarrow \widehat G$ that coincide with an eigenvalue with respect to the G-action for $\mu $ -a.e. $x\in X$ . We show that one can replace $\mathcal {Z}_{\psi }(X)$ with $\widetilde {\mathcal {Z}}_{\psi }(X)$ in (9). After replacing $\mathcal {Z}_{\psi }(X)$ by $\widetilde {\mathcal {Z}}_{\psi }(X)$ , the remainder of the proof of Theorem 1.11 follows by modifying previous arguments used for deducing Khintchine-type recurrence from knowledge of relevant characteristic factors (see, e.g. [Reference Ackelsberg, Bergelson and Best2, Section 8]).

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.

Lemma 3.6. Let $(X,\mathcal {X},\mu , (T_g)_{g\in G})$ be a G-system, and let $H\leq G$ be a subgroup of finite index. Then, for every $k\geq 1$ , one has $\mathcal {Z}^k_H(X) = \mathcal {Z}^k_G(X)$ .

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