2.1.1 Divisibility properties of the spectrum

In this subsection, we discuss divisibility properties of the spectrum (see Definition 3.3) of Furstenberg systems of bounded completely multiplicative functions. Note that in all cases, we get stronger results when the Furstenberg systems are defined using logarithmic averages versus Cesàro averages, and it is not clear if equally strong results can be obtained for Cesàro averages (see a related question in §2.5).

Theorem 2.1 is proved in §4.1. The key idea is to study the action of the maps $\tau _r$ , defined in equation (10), on eigenfunctions of the system, and show that in the case of Furstenberg systems of completely multiplicative functions, non-zero functions are mapped to non-zero functions by $\tau _r$ . This is a consequence of Lemma 4.2, which we combine with Lemma 4.1 to prove Theorem 2.1.

For example, the sets $\{m/2^k\colon m=0,\ldots , 2^k-1, k\in {\mathbb N} \}$ and ${\mathbb Q}\cap [0,1)$ are divisible, and any non-trivial divisible subset of ${\mathbb T}$ has to be infinite.

The following is an immediate consequence of the previous result.

Note that if $\alpha =0$ or $\alpha =m/n$ and $(r,n)=1$ , then the content of Theorem 2.1 is empty. However, if $\alpha =1/p$ and $r=p^s$ for some $s\in {\mathbb N}$ , then we do get non-trivial consequences as the next result shows.

Further consequences of these results for the class of pretentious multiplicative functions will be given in Theorems 2.11 and 2.12 below.

2.1.2 Trivial rational spectrum implies strong stationarity

It turns out that for Furstenberg systems of bounded multiplicative functions defined using logarithmic averages, a trivial rational spectrum has very strong structural consequences. A notable one is strong stationarity, the dilation invariance property described in Definition 3.10, which also played a crucial role in the description of Furstenberg systems of general bounded multiplicative functions in [Reference Frantzikinakis and Host16].

Theorem 2.4 is proved in §4.2 and uses the ergodic limit formulae stated in Theorem 4.3, and a recent result about correlations of multiplicative functions of Tao and Teräväinen [Reference Tao and Teräväinen45] stated in Theorem 4.5.

Using the previous result and known results about the structure of strongly stationary systems (see [Reference Frantzikinakis14, Main Theorem] for part (i) and [Reference Frantzikinakis and Host16, Proposition 3.12(ii)] for part (ii)), we get the following structural result for Furstenberg systems of bounded multiplicative functions that have trivial rational spectrum.

The bulk of this result can also be deduced from [Reference Frantzikinakis and Host16, Proposition 3.12(ii)] and [Reference Frantzikinakis and Host17, Theorem 1.5].

The Liouville function is probably the most noteworthy example of a multiplicative function for which it is not known whether all its Furstenberg systems have trivial rational spectrum (the Chowla conjecture predicts that this is indeed the case), that is, it is not known whether any rational in $(0,1)$ is on the spectrum of any of its Furstenberg systems for Cesàro or logarithmic averages.

2.2.1 A known structural result

We start with a known result that gives substantial information for a rich class of pretentious multiplicative functions. It follows by combining [Reference Daboussi and Delange8, Theorem 6] and [Reference Bergelson, Kułaga-Przymus, Lemańczyk and Richter1, Theorem 1.7].

Theorem 2.6. [Reference Bergelson, Kułaga-Przymus, Lemańczyk and Richter1, Reference Daboussi and Delange8]

Let $f\colon {\mathbb N}\to {\mathbb U}$ be a multiplicative function and suppose that there exists a Dirichlet character $\chi $ such that the series

(1) $$ \begin{align} \sum_{p\in{\mathbb P}}\frac{1}{p}(1-f(p)\cdot \overline{\chi(p)}) \quad\text{converges}. \end{align} $$

Then, f is Besicovitch rationally almost periodic for Cesàro averages, it has a unique Furstenberg system with respect to Cesàro averages (hence also for logarithmic), and this unique system is isomorphic to an ergodic procyclic system.

When f pretends to be $1$ and condition (1) is not satisfied (this is the case in Example (viii) of §3.2.1), then it follows from [Reference Daboussi and Delange8, Corollary 2] that the mean value of f on some arithmetic progression does not exist and, as a consequence, f is not Besicovitch rationally almost periodic. Prior to our work, for such multiplicative functions, it was not clear what Furstenberg systems may arise, and it seemed plausible that they do not all have rational discrete spectrum. We show in Theorem 2.7 that this is not the case and in Theorem 2.8, we give more refined information about their structure.

2.2.2 New structural results

Our first main result applies to all pretentious multiplicative functions and shows that their Furstenberg systems have rational discrete spectrum. It extends Theorem 2.6, which covers multiplicative functions that satisfy condition (1).

Our main result for pretentious multiplicative functions is stated next and gives more refined structural information about their Furstenberg systems. In what follows, when we write $f\sim g$ , we mean ${\mathbb D}(f,g)<+\infty $ (see Definition 3.1).

Section 6 is devoted to the proof of Theorem 2.8 (Theorem 2.7 is an immediate consequence) and the proof is completed in §6.3. The argument uses several ingredients: we start in §5 with some preliminary work that leads to the decomposition result stated in Lemma 5.4, which in turn implies the subsequential Besicovitch rational almost periodicity property stated in Proposition 5.5. This basic tool is then exploited in §6 and together with several arguments of ergodic flavor, leads to the proof of Theorem 2.8.

We deduce from Theorem 2.8 some equivalent characterizations of various classes of pretentious multiplicative functions. The first one concerns multiplicative functions that pretend to be Dirichlet characters.

Corollary 2.9 is proved in §6.4.

Our second corollary concerns multiplicative functions that are equal to Archimedean characters.

Corollary 2.10 is proved in §6.5.

2.2.3 Spectral results

We give a result that, in conjunction with Theorem 2.1, helps us identify the spectrum of Furstenberg systems of pretentious multiplicative functions.

The implication $\mathrm {(i)}\implies \mathrm {(ii)}$ of Theorem 2.11 is proved in §7.3 and uses the periodic approximation property of Proposition 5.5 to deduce the result for f from its periodic approximants. The implication $\mathrm {(ii)}\implies \mathrm {(i)}$ is proved in §7.4. It is somewhat more involved and uses part (i) of Theorem 2.8, and a combination of elementary and ergodic considerations.

Our next result provides more refined spectral information for multiplicative functions that pretend to be Dirichlet characters and a complete characterization of the spectrum of their Furstenberg systems. We will use it to justify various claims we make in the examples given in §3.2.1. We assume complete multiplicativity to have Theorem 2.1 available for us, and explain in the remarks which consequences carry over to general multiplicative functions.

We prove Theorem 2.12 in §7.5. We essentially use the divisibility properties of the spectrum given in §2.1.1, Proposition 6.1, which asserts that in this case, Cesàro and logarithmic averages can be used interchangeably, and part (i) of Theorem 2.8, which asserts the ergodicity of the corresponding Furstenberg systems.

2.3.1 Correlations of pretentious multiplicative functions

We start with a vanishing property for weighted correlations of pretentious multiplicative functions.

Theorem 2.13. Let $f\colon {\mathbb N}\to {\mathbb U}$ be a multiplicative function with $f\sim \chi $ for some primitive Dirichlet character $\chi $ with conductor q. Suppose that either:

1. (i) $\alpha $ is irrational; or

2. (ii) $\alpha =k/p$ , $p\in {\mathbb P}$ , $(k,p)=1$ , $p\nmid q$ , and $f(p^s)=\chi (p^s)$ for every $s\in {\mathbb N}$ .

Then,

(2) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{n\in [N]}\, e(n\alpha)\, \prod_{j=1}^\ell f_j(n+n_j)=0 \end{align} $$

for all $n_1,\ldots , n_\ell \in {\mathbb Z}$ and $f_1,\ldots , f_\ell \in \{f,\overline {f}\}$ .

Theorem 2.13 is proved in §7.6.

Our next result establishes existence of certain correlations of pretentious multiplicative functions.

Theorem 2.14. Let $f\colon {\mathbb N}\to {\mathbb U}$ be a pretentious multiplicative function. If $k_1,\ldots , k_\ell $ are integers such that $\sum _{j=1}^\ell k_j=0$ , then the limit

(3) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{n\in [N]}\, \prod_{j=1}^\ell f^{k_j}(n+n_j) \end{align} $$

exists for all $n_1,\ldots , n_\ell \in {\mathbb Z}$ , where we use the notation $f^k:=\overline {f}^{|k|}$ for $k<0$ .

Theorem 2.14 is proved in §7.7.

2.3.2 Chowla and Sarnak conjecture

To facilitate discussion, we introduce the following notions and refer the reader to Definition 3.9 for the notion of completely deterministic sequences.

We give two results, which, roughly speaking, assert that if a bounded sequence satisfies the Sarnak or the Chowla–Elliot conjecture, then so does any multiple of the sequence by a pretentious multiplicative function. We caution the reader that although these claims are rather easy to prove when the multiplicative function satisfies Theorem 2.6, a priori, it is not even clear that such claims are expected to hold for general pretentious multiplicative functions. In fact, since all MRT functions satisfy the Sarnak and the Chowla–Elliott conjectures along some subsequence $(N_k)$ (this is a consequence of the main result in [Reference Gomilko, Lemańczyk and de la Rue23]), one has to first nullify the possibility that the MRT class contains a pretentious multiplicative function, which is a non-trivial task on its own.

We prove Theorem 2.15 in §7.8 by making essential use of Theorem 2.7.

We prove Theorem 2.16 in §7.9 by making essential use of Theorem 2.7.

2.4.1 The setting

We reproduce here the definition of the MRT class of completely multiplicative functions from [Reference Gomilko, Lemańczyk and de la Rue23, Definition 3.1].

It is shown in [Reference Matomäki, Radziwiłł and Tao37, Appendix B] that the MRT class is non-empty, and if $s_{m+1}>e^{t_m}$ , then f is an aperiodic multiplicative function. However, this is not a growth assumption that we impose in our defining axioms, nor is it known whether it follows from these axioms. However, it is shown in [Reference Gomilko, Lemańczyk and de la Rue23] that the axioms (i)–(iii) imply that for every $K\in {\mathbb N}$ , we have $s_{m+1}/t^K_m\to \infty $ .

It is a consequence of [Reference Gomilko, Lemańczyk and de la Rue23] that every element of the MRT class has uncountably many different Furstenberg systems, and depending on the choice of our averaging intervals $[N_m]$ , we may get substantially different structural properties, ranging from the identity system to the Bernoulli system. Our goal here is to give a complete classification of these Furstenberg systems for Cesàro and logarithmic averages, when, roughly speaking, $N_m$ grows as a fractional power of $s_{m+1}$ (that is, $\lim _{m\to \infty } N_m/s_{m+1}^\beta = c$ for some $\beta ,c>0$ ).

We stress that in the structural results that we give below, we fix an arbitrary MRT function f and describe the structure of different Furstenberg systems of this fixed f for Cesàro or logarithmic averages, depending on the choice of the sequence $N_m\to \infty $ that defines the Furstenberg system.

The following is a first non-trivial consequence of our results.

This follows at once from Theorem 2.18 (or the structural result obtained in [Reference Gomilko, Lemańczyk and de la Rue23]) and Theorem 2.7, since for every MRT function, not all of its Furstenberg systems for Cesàro averages have rational discrete spectrum.

2.4.2 Furstenberg systems of MRT functions for Cesàro averages

We first state structural results for Furstenberg systems of MRT functions defined using Cesàro averages. In the next subsection, we cover the case of logarithmic averages, which lead to systems with substantially different structural properties.

Definition 2.4. For $d\in {\mathbb Z}_+$ , we consider the measure-preserving system $ (Y_d,\nu _d,S_d), $ where $Y_d:={\mathbb T}^{d+1}$ , $\nu _d:=m_{{\mathbb T}^{d+1}}$ , and $S_d\colon {\mathbb T}^{d+1}\to {\mathbb T}^{d+1}$ is the transformation defined by

$$ \begin{align*} S_d(x_0,\ldots, x_d):=(x_0,x_1+x_0,\ldots, x_d+x_{d-1}), \quad x_0,\ldots, x_d\in {\mathbb T}. \end{align*} $$

We call it the level d unipotent system.

For $d=0$ , we have $S_0(x_0)=x_0$ on ${\mathbb T}$ with $m_{\mathbb T}$ . For $d=1$ , we have $S_1(x_0,x_1)= (x_0,x_1+x_0)$ on ${\mathbb T}^2$ with $m_{{\mathbb T}^2}$ , and so on. We introduce these systems to reproduce in an ergodic setting the correlations of MRT functions given in Proposition 8.4.

We start with a result previously obtained in [Reference Gomilko, Lemańczyk and de la Rue23] when $N_m=\lfloor s_{m+1}^{1/c}\rfloor $ and $c>0$ is not an integer (the case $c\in {\mathbb N}$ is covered by Theorem 2.19). We use somewhat different techniques to cover the more general case below.

Theorem 2.18 is proved in §8.3.

Our method of proof is somewhat different than that used in [Reference Gomilko, Lemańczyk and de la Rue23], the additional flexibility of our method enables us to give more refined results and also handle subsequent problems concerning logarithmic averages. We continue with a case that exhibits different structural behavior and gives examples of non-strongly stationary systems, a feature not present in the case of logarithmic averages.

Definition 2.6. For $\alpha>0$ and $d\in {\mathbb Z}_+$ , let $S_{\alpha ,d}\colon {\mathbb T}^{d+1}\to {\mathbb T}^{d+1}$ be the transformation defined by

$$ \begin{align*} S_{\alpha,d}(x_0,\ldots, x_d):=(x_0,x_1+g_{\alpha,d}(x_0),x_2+x_1,\ldots, x_d+x_{d-1}), \quad x_0,\ldots, x_d\in {\mathbb T}, \end{align*} $$

where $g_{\alpha ,d}\colon {\mathbb T}\to {\mathbb R}_+$ is defined by $g_{\alpha ,d}(x):=1 /(\alpha ^d\{x\}^d)$ for $x\neq 0$ .

This system helps us reproduce the correlations of MRT functions given in Proposition 8.2 and leads to the following result.

Theorem 2.19 is proved in §8.3.

2.4.3 Furstenberg systems of MRT functions for logarithmic averages

We describe structural properties of Furstenberg systems of MRT functions defined using logarithmic averages. The reader is advised to compare Theorems 2.18 and 2.19 with Theorem 2.20, and note that for the same choice of sequence of averaging intervals $([N_m])$ , the structure of the corresponding Furstenberg systems differs sharply when we use Cesàro versus logarithmic averages.

Definition 2.7. For $d\in {\mathbb Z}_+$ , let $(Y_d,\nu _d,S_d)$ be the level d unipotent system given in Definition 2.4. For $c>0$ , we define the system $(Z_c,\nu ^{\prime }_c, R_c)$ , where $Z_c$ consists of disjoint copies of $Y_d$ , $d\geq \lfloor c\rfloor $ , with the corresponding $\sigma $ -algebra, $R_c$ is defined to be $S_d$ on each piece $Y_d$ , and

$$ \begin{align*} \nu^{\prime}_c:=\bigg(1-\frac{c}{\lceil c\rceil}\bigg) \nu_{\lfloor c\rfloor } + c\sum_{d=\lceil c\rceil}^\infty \bigg(\frac{1}{d}-\frac{1}{d+1}\bigg)\, \nu_d. \end{align*} $$

The motivation for introducing this rather awkward system is that it helps us reproduce the correlations of MRT functions for logarithmic averages given in Proposition 9.1 and leads to the following result.

Theorem 2.20 is proved in §9.

3.2.1 Some examples

To get a sense of the variety of measure-preserving systems that arise as Furstenberg systems of pretentious multiplicative functions and their spectral properties, we provide a rather extensive assortment of examples. The terminology we use is explained in §§3.3 and 3.4. In all cases, the Furstenberg systems can be taken either for Cesàro or logarithmic averages. The properties recorded in Examples (i), (ii), (iv), (vii), (viii) follow from Theorem 2.12, the properties of Example (iii) are straightforward, the properties of Example (v) were established in [Reference Cellarosi and Sinai4] (see also [Reference Sarnak41, Theorem 9]), and those of Example (vi) can be found in [Reference Gomilko, Lemańczyk and de la Rue23, Corollary 5.5] and [Reference Frantzikinakis and Host17, §1.3]. Finally, we remark that Theorem 2.6 does not apply to Example (viii), so in this case, even showing that all Furstenberg systems have rational discrete spectrum is a non-trivial task. For the sake of brevity, in the following discussion, when we refer to the spectrum of a multiplicative function, we mean the spectrum of any of its Furstenberg systems.

1. (i) $f(2):=-1$ and $f(p):=1$ for $p\neq 2$ , and f is completely multiplicative. The spectrum of f is $m/2^s$ , $m\in {\mathbb Z}_+,s\in {\mathbb N}$ , and it has a unique Furstenberg system, which is an ergodic procyclic system.

2. (ii) More generally, let $f(p_1)=\cdots =f(p_\ell ):=-1$ and $f(p)=1$ for $p\notin \{p_1,\ldots , p_\ell \}$ , and f is completely multiplicative. Then, the spectrum of f consists of all integer combinations of the numbers $1/p_i^k$ , where $k\in {\mathbb N}$ , $ i\in \{1,\ldots , \ell \}$ , and it has a unique Furstenberg system, which is an ergodic procyclic system.

3. (iii) $f:=\chi $ is a Dirichlet character, or $f(n)=(-1)^{n+1}$ , or $f:=\mathbf {1}_{{\mathbb Z}\setminus 3{\mathbb Z}}- \mathbf {1}_{3{\mathbb Z}}$ . Then, f has finite rational spectrum and it has a unique Furstenberg system, which is an ergodic cyclic system.

4. (iv) $f:=\tilde {\chi }$ is a modified Dirichlet character, defined as in Lemma 7.4, where $\chi $ is a primitive Dirichlet character with conductor q. Then, f has a unique Furstenberg system, which is ergodic and procyclic, and its spectrum consists of all integer combinations of $1/p^k$ , $k\in {\mathbb N}$ , where $p\in {\mathbb P}$ divides q.

5. (v) $f:=\mu ^2$ is the indicator of the square free numbers. Then, f has a unique Furstenberg system, which is ergodic and procyclic, and its spectrum consists of all integer combinations of $1/p^2$ , where $p\in {\mathbb P}$ . In particular, for any prime p, we have that $1/p^2$ is on the spectrum of this system but $1/p^3$ is not. We also remark that this is an example of a multiplicative function that has positive topological entropy but its unique Furstenberg system has zero entropy.

6. (vi) $f(n):=n^{it}$ for some $t\neq 0$ . Then, f has uncountably many Furstenberg systems, all isomorphic to the identity transformation on ${\mathbb T}$ with the Lebesgue measure. In the case of logarithmic averages, it has a unique Furstenberg system, isomorphic to the identity transformation on ${\mathbb T}$ with the Lebesgue measure. In both cases, all Furstenberg systems have trivial rational spectrum.

7. (vii) $f(p):=e(1/p)$ , or $f(p):=1-1/p$ , $p\in {\mathbb P}$ , and f is completely multiplicative. Then, $f\sim 1$ and f has a unique ergodic procyclic Furstenberg system and its spectrum consists of all rational numbers in $[0,1)$ . If we change the value of f at a single prime to $1$ , say set $f(3):=1$ , then $1/3$ will no longer be in the spectrum, and the non-zero values in the spectrum will now consist of all rationals $p/q\in [0,1)$ with $(p,q)=1$ and $(q,3)=1$ .

8. (viii) $f(p):=e(1/ \log \log {p})$ , $p\in {\mathbb P}$ , and f is completely multiplicative. Then, $f\sim 1$ but f does not satisfy equation (1), so Theorem 2.6 does not apply. (The reason for this is that if $\theta _p:=1/\log \log {p}$ , $p\in {\mathbb P}$ , then $\sum _{p\in {\mathbb P}} ({\theta _p^2}/p) <+\infty $ , while $\sum _{p\in {\mathbb P}} ({\theta _p}/p) =+\infty $ .) We have that f does not have a mean value [Reference Daboussi and Delange8, Corollary 2] and hence has several Furstenberg systems, all of which are ergodic procyclic systems, isomorphic to each other, and their spectrum consists of all the rational numbers in $[0,1)$ .

It follows from [Reference Frantzikinakis and Host17, Theorem 1.5] that an irrational number cannot be on the spectrum of a Furstenberg system for logarithmic averages of any multiplicative function (pretentious or not) that takes values in ${\mathbb U}$ . For pretentious multiplicative functions, an alternative proof follows from Theorem 2.13, and it also applies to Furstenberg systems for Cesàro averages (a case not covered in [Reference Frantzikinakis and Host17]).

3.2.2 Mean values of multiplicative functions

We will need the following notion.

Definition 3.2. We say that a sequence $A\colon {\mathbb N}\to {\mathbb R}_+$ is slowly varying if for every ${c\in (0,1)}$ , we have

$$ \begin{align*} \lim_{N\to\infty} \sup_{n\in [N^c,N]}|A(n)-A(N)|=0. \end{align*} $$

For example, the sequence $A(n)=\log \log \log {n}$ , defined for $n\geq 3$ , is slowly varying, but the sequence $A(n)=\log \log {n}$ , defined for $n\geq 2$ , is not slowly varying.

We will use the following result that can be found in the form stated below in [Reference Elliott10, Theorems 6.2 and 6.3]. We will only apply this in the case of pretentious multiplicative functions.

3.3.1 Factors and isomorphisms

A homomorphism, also called a factor map, from a system $(X,{\mathcal X},\mu , T)$ onto a system $(Y, {\mathcal Y}, \nu , S)$ is a measurable map $\Phi \colon X\to Y$ , such that $\mu \circ \Phi ^{-1} = \nu $ and with $S\circ \Phi = \Phi \circ T$ valid for $\mu $ -almost every (a.e.) $x\in X$ . When we have such a homomorphism, we say that the system $(Y, {\mathcal Y}, \nu , S)$ is a factor of the system $(X,{\mathcal X},\mu , T)$ . If the factor map $\Phi \colon X\to Y$ is injective on a T-invariant set of full $\mu $ -measure, we say that $\Phi $ is an isomorphism and that the systems $(X,{\mathcal X}, \mu , T)$ and $(Y, {\mathcal Y}, \nu , S)$ are isomorphic.

The pushforward of $\mu $ by $\Phi $ is denoted by $\Phi _*\mu $ or $\mu \circ \Phi ^{-1}$ , and is defined by

$$ \begin{align*} \int F\, d(\Phi_*\mu)=\int F\circ \Phi\, d\mu \end{align*} $$

for every $F\in L^\infty (\nu )$ .

3.3.2 Spectrum, procyclic, and rational discrete spectrum systems

The notion of the spectrum and the class of systems with rational discrete spectrum play a crucial role in this article and we define these concepts next.

Hence, $\alpha \in \operatorname {\mathrm {Spec}}(X,\mu ,T)$ if it is the phase (taken in $[0,1)$ ) of an eigenvalue of the operator $f\mapsto Tf$ , acting on $L^2(\mu )$ . We will often identify $[0,1)$ with ${\mathbb T}$ , and using this identification, we have that the spectrum of a system is always a subset of ${\mathbb T}$ that is closed under multiplication by integers. When the system is ergodic, then the spectrum is a subgroup of ${\mathbb T}$ . We say that $e(\alpha )$ is a rational eigenvalue if $\alpha \in {\mathbb Q}$ .

It can be shown that a system has rational discrete spectrum if and only if its ergodic components are procyclic systems. Also, a procyclic system is an inverse limit of cyclic systems, and this happens if and only if it is isomorphic to an ergodic rotation on a procyclic group. Lastly, we remark that procyclic systems are isomorphic if and only if they have the same spectrum, but this is not the case for (non-ergodic) systems with rational discrete spectrum.

3.3.3 Joinings and disjoint systems

Following [Reference Furstenberg18], if $(X,{\mathcal X}, \mu ,T)$ and $(Y,{\mathcal Y}, \nu ,S)$ are two systems, we call a measure $\rho $ on $(X\times Y, {\mathcal X}\otimes {\mathcal Y})$ a joining of the two systems if $\rho $ is $T\times S$ -invariant and its projection onto the X and Y coordinates are the measures $\mu $ and $\nu $ , respectively. We say that the systems on X and on Y are disjoint if the only joining of the systems is the product measure $\mu \times \nu $ .

We will use the following well-known facts (the Furstenberg systems of the sequences $a,b\colon {\mathbb N}\to {\mathbb U}$ used below can be taken either for Cesàro or logarithmic averages).

1. (i) All identity systems are disjoint from all ergodic systems.

2. (ii) Bernoulli systems are disjoint from all zero entropy systems.

3. (iii) If $a,b\colon {\mathbb N}\to {\mathbb U}$ are sequences, then any Furstenberg system (see Definition 3.7) of the product sequence $a\cdot b$ is a factor of a joining of some Furstenberg system of a and another Furstenberg system of b.

4. (iv) The spectrum of an ergodic joining of two discrete spectrum systems is contained in the subgroup of ${\mathbb T}$ generated by the spectrum of the individual systems. (Similarly, we can define joinings of countably many systems and property (iv) extends to this more general setting.)

5. (v) If all Furstenberg systems of a sequence $a\colon {\mathbb N}\to {\mathbb U}$ are disjoint from all Furstenberg systems of a sequence $b\colon {\mathbb N}\to {\mathbb U}$ , then the spectrum of any Furstenberg system of the product sequence $a\cdot b$ is contained in the subgroup generated by the spectrum of a Furstenberg system of the sequence a and another Furstenberg system of the sequence b.

3.4.1 Sequence space systems and correlations

We start with two preparatory notions.

Definition 3.6. Let $ ([N_k])_{k\in {\mathbb N}}$ be a sequence of intervals with $N_k\to \infty $ . We say that a sequence $a\colon {\mathbb Z}\to {\mathbb U}$ admits correlations along $([N_k])$ (or simply along $(N_k)$ ) for Cesàro averages if the limits

(6) $$ \begin{align} \lim_{k\to\infty}{\mathbb E}_{n\in[N_k]} \prod_{j=1}^\ell a_j(n+n_j) \end{align} $$

exist for all $\ell \in {\mathbb N}$ , $n_1,\ldots , n_\ell \in {\mathbb Z}$ , and all $a_1,\ldots , a_\ell \in \{a,\overline {a}\}$ .

3.4.2 Furstenberg systems of sequences

If a sequence $a\colon {\mathbb Z}\to {\mathbb U}$ admits correlations for Cesàro averages along $(N_k)$ , then we use a variant of the correspondence principle of Furstenberg [Reference Furstenberg19, Reference Furstenberg20] to associate a sequence space system that captures the statistical properties of a along $(N_k)$ . We briefly describe this process next.

We consider the sequence $a=(a(n))$ as an element of X. Note that the conjugation closed algebra generated by functions of the form $x\mapsto x(k)$ , $x\in X$ , for $k\in {\mathbb Z}$ , separates points in X. We conclude that if the sequence $a\colon {\mathbb Z}\to {\mathbb U}$ admits correlations along $(N_k)$ , then, for all $F\in C(X)$ , the following limit exists:

$$ \begin{align*} \lim_{k\to\infty}{\mathbb E}_{n\in[N_k]}\, F(T^na). \end{align*} $$

Hence, the following weak-star limit exists:

(7) $$ \begin{align} \mu:=\lim_{k\to\infty}{\mathbb E}_{n\in[N_k]}\, \delta_{T^na} \end{align} $$

and we say that the point a is generic for $\mu $ along $([N_k])$ or $(N_k)$ .

Given a bounded sequence, our goal is to obtain structural properties of their Furstenberg systems. Ultimately, we would like to completely determine them up to isomorphism using as building blocks systems with algebraic structure, such as nilsystems, and systems that enjoy strong randomness properties, such as Bernoulli systems.

Similar notions as in the previous subsection can be defined for logarithmic averages.

Definition 3.8. By replacing Cesàro averages with logarithmic averages in Definitions 3.5–3.7, given a sequence $a\colon {\mathbb N}\to {\mathbb U}$ and a sequence of intervals $([N_k])$ along which a admits correlations for logarithmic averages, we can introduce Furstenberg systems of a along $(N_k)$ for logarithmic averages. If $(X,\mu ,T)$ is such a system, then an identity similar to equation (8) is satisfied with $\mathbb {E}^{\log }_{n\in [N_k]}$ in place of ${\mathbb E}_{n\in [N_k]}$ . Equivalently,

$$ \begin{align*} \mu=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in [N_k]}\, \delta_{T^na}, \end{align*} $$

where the limit is a weak-star limit.

3.4.3 Ergodic and completely deterministic sequences

Using the Furstenberg correspondence principle, we can naturally associate ergodic notions to bounded sequences—we record here two that are used in this article.

We will use the fact that if $a,b\colon {\mathbb N}\to {\mathbb U}$ are completely deterministic sequences for Cesàro or logarithmic averages along $(N_k)$ , then so is their product $a\cdot b$ . To see this, use property (iii) in §3.3.3 and the fact that zero entropy systems are closed under joinings and factors.

3.4.4 Strong stationarity

Lastly, we define the notion of strong stationarity that was introduced by Furstenberg and Katznelson in [Reference Furstenberg and Katznelson21], which turns out to be relevant for the structural analysis of measure-preserving systems associated with non-pretentious multiplicative functions.

Definition 3.10. A sequence space system $(X,\mu ,T)$ is strongly stationary if

(9) $$ \begin{align} \int \prod_{j=1}^\ell T^{n_j}F_j\, d\mu= \int \prod_{j=1}^\ell T^{rn_j}F_j\, d\mu \end{align} $$

for all $\ell ,r\in {\mathbb N}$ , $n_1,\ldots , n_\ell \in {\mathbb Z}$ , and $F_1,\ldots , F_\ell \in \{F_0,\overline {F_0}\}$ , where $F_0\in L^\infty (\mu )$ is the $0$ th-coordinate projection. (Equivalently, for every $r\in {\mathbb N}$ , the dilation map $\tau _r\colon X\to X$ in equation (10) is $\mu $ -preserving.) A sequence $a\colon {\mathbb N}\to {\mathbb U}$ is strongly stationary for Cesàro averages if all its Furstenberg systems for Cesàro averages are strongly stationary.

Similar definitions apply for logarithmic averages.