$N_{k}\to\infty$ along which the difference of the averages in (34) is bounded
away from zero. By Proposition 5.5 there exists a subsequence
$(N_{k}^{\prime})$ of $(N_{k})$, such that for every $m\in{\mathbb{N}}$ we
have

(35)
$\limsup_{k\to\infty}\big{(}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}|f(n)-g_{m}(n)|^{2}+\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}]}|f(n)-g_{m}(n)|^{2}\big{)}\leq
1/m$

for some periodic sequence $g_{m}\colon{\mathbb{N}}\to{\mathbb{U}}$. Since
(34) clearly holds when we replace $f$ with the periodic sequence $g_{m}$,
using (35), we deduce that (34) also holds for $f$ as long as we average over
the sequence of intervals $[N_{k}^{\prime}]$. This is a contradiction, since
we have assumed that along the sequence $(N_{k})$ the difference of the
averages in (34) is bounded away from zero, and $(N_{k}^{\prime})$ is a
subsequence of $(N_{k})$. ∎

We immediately deduce the following.

######  Proposition 6.2.

Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be a multiplicative function such
that $f\sim\chi$ for some primitive Dirichlet character $\chi$. Let also
$N_{k}\to\infty$. Then the Furstenberg system of $f$ for Cesàro averages along
$(N_{k})$ is well defined if and only if it is well defined for logarithmic
averages along $(N_{k})$, and the two Furstenberg systems are equal (meaning
the corresponding $T$-invariant measures agree).

###### Proof.

Suppose that the Furstenberg system of $f$ for Cesàro averages is defined
along the sequence $(N_{k})$ (the argument is similar if we assume it is
defined for logarithmic averages). Then Proposition 6.1 implies that the
Furstenberg system for logarithmic averages is also well defined along
$(N_{k})$ and the measures that define the two Furstenberg systems coincide,
since they agree on all cylinder functions defined in the remark after
Definition 3.5. ∎

###  6.2. Correlation limits and isomorphism when $f\sim\chi$

The goal of this subsection is to show in Proposition 6.4 that different
Furstenberg systems of a fixed multiplicative function that pretends to be a
Dirichlet character are isomorphic. This is a consequence of the correlation
identities stated in the following lemma.

######  Lemma 6.3.

Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be a multiplicative function such
that $f\sim\chi$ for some primitive Dirichlet character $\chi$. If
$N_{k,1}\to\infty$ and $N_{k,2}\to\infty$ are sequences, then there exist
subsequences $(N_{k,1}^{\prime})$ and $(N_{k,2}^{\prime})$ and
$\alpha\in[0,1)$, such that the following holds: For all $\ell\in{\mathbb{N}}$
and $n_{1},\ldots,n_{\ell},k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, both
correlation limits below exist and we have

(36)
$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,1}^{\prime}]}\,\prod_{j=1}^{\ell}(e(\alpha)\cdot
f)^{k_{j}}(n+n_{j})=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,2}^{\prime}]}\,\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j}),$

where we let $f^{k}:=\overline{f}^{|k|}$ for $k<0$. A similar identity also
holds for logarithmic averages.

######  Remark.

More generally, if $f\sim n^{it}\cdot\chi$ for some $t\in{\mathbb{R}}$ and
primitive Dirichlet character $\chi$, then we get a similar result with the
identity (36) replaced by the identity

(37)
$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,1}^{\prime}]}\,n^{it^{\prime}}\cdot\prod_{j=1}^{\ell}(e(\alpha)\cdot
f)^{k_{j}}(n+n_{j})=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,2}^{\prime}]}\,n^{it^{\prime}}\cdot\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j}),$

where $t^{\prime}:=-t\cdot\sum_{j=1}^{\ell}k_{j}$. To see this, use (36) for
the multiplicative function $f\cdot n^{-it}\sim\chi$ together with the fact
that $\lim_{n\to\infty}((n+h)^{it}-n^{it})=0$ for every $h\in{\mathbb{N}}$.

###### Proof.

We give the argument for Cesàro averages, a similar argument applies to
logarithmic averages.

By Proposition 5.5 (and the remark following it), for every $m\in{\mathbb{N}}$
there exist periodic sequences $f_{m}$, subsequences $(N^{\prime}_{k,1})$ of
$(N_{k,1})$ and $(N_{k,2}^{\prime})$ of $(N_{k,2})$, and
$\alpha_{1,m},\alpha_{2,m}\in[0,1)$, such that

(38)
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,1}^{\prime}]}|f(n)-e(\alpha_{1,m})\cdot
f_{m}(n)|^{2}\leq 1/m$

and

(39)
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k,2}^{\prime}]}|f(n)-e(\alpha_{2,m})\cdot
f_{m}(n)|^{2}\leq 1/m.$

(Note that the function $f_{m}$ is the same in both cases.) By working with
further subsequences of $(N^{\prime}_{k,1})$ and $(N_{k,2}^{\prime})$ along
which the sequences $e(\alpha_{1,m})$ and $e(\alpha_{2,m})$ converge, we can
assume that (38) and (39) hold with $e(\alpha_{1})$ in place of
$e(\alpha_{1,m})$ and $e(\alpha_{2})$ in place of $e(\alpha_{2,m})$ for some
$\alpha_{1},\alpha_{2}\in[0,1)$. We let

(40) $\alpha:=\alpha_{2}-\alpha_{1}.$

Using a diagonal argument, we can find further subsequences that we denote
again by $(N_{k,1}^{\prime})$ and $(N_{k,2}^{\prime})$ such that all the
correlation limits in (36) exist. Using (38) and (39), we see that it suffices
to verify (36) with $f(n)$ replaced by $e(\alpha_{1})\cdot f_{m}(n)$ on the
first average and by $e(\alpha_{2})\cdot f_{m}(n)$ on the second average.
Taking into account (40), it suffices to verify that

$\lim_{k\to\infty}\Big{|}{\mathbb{E}}_{n\in[N_{k,1}^{\prime}]}\,\prod_{j=1}^{\ell}f_{m}^{k_{j}}(n+n_{j})-{\mathbb{E}}_{n\in[N_{k,2}^{\prime}]}\,\prod_{j=1}^{\ell}f_{m}^{k_{j}}(n+n_{j})\Big{|}=0$

for all $\ell\in{\mathbb{N}}$ and
$n_{1},\ldots,n_{\ell},k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$. Since the
sequence $f_{m}$ is periodic, both averages have limits and these limits
coincide, therefore, the asserted identity follows. ∎

######  Proposition 6.4.

Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be a multiplicative function such
that $f\sim\chi$ for some primitive Dirichlet character $\chi$. Then any two
Furstenberg systems of $f$ for Cesàro or logarithmic averages are isomorphic.

######  Remark.

With substantial additional effort, the result will be extended to arbitrary
pretentious multiplicative functions in the next subsection, where we prove
part (iii) of Theorem 2.8.

###### Proof.

By Proposition 6.2, we have that Furstenberg systems for Cesàro and
logarithmic averages coincide, hence it suffices to treat the case of Cesàro
averages. Let $(X,\mu,T)$ and $(X,\mu^{\prime},T)$ be two Furstenberg systems
for Cesàro averages taken along the sequences $N_{k,1}\to\infty$ and
$N_{k,2}\to\infty$, respectively. Using Lemma 6.3 and (7) to translate (36)
into identities for the two Furstenberg systems, we get that there exists
$\alpha\in[0,1)$ such that

(41) $\int\prod_{j=1}^{\ell}T^{n_{j}}(e(\alpha)\cdot
F_{0})^{k_{j}}\,d\mu=\int\prod_{j=1}^{\ell}T^{n_{j}}F_{0}^{k_{j}}\,d\mu^{\prime}$

holds for all $\ell\in{\mathbb{N}}$ and
$n_{1},\ldots,n_{\ell},k_{1},\ldots,k_{\ell}\in{\mathbb{Z}}$, where as usual
$F_{0}$ is the $0^{\text{th}}$-coordinate projection. This easily implies that
the two systems are isomorphic with the isomorphism given by the map
$M_{e(\alpha)}\colon X\to X$ defined by (11). Indeed, $M_{e(\alpha)}$ is
bijective, satisfies the commutation relation (12), and since $F_{0}\circ
M_{e(\alpha)}=e(\alpha)\cdot F_{0}$, equation (41) implies that
$\mu^{\prime}=(M_{e(\alpha)})_{\ast}\mu$, (the two measures agree on the
linearly dense subset of $C(X)$ given by the cylinder functions, see remark
after Definition 3.5). ∎

###  6.3. Proof of Theorems 2.7-2.8

Theorem 2.7 is an immediate consequence of Theorem 2.8, so we focus on the
proof of the latter.

####  6.3.1. Proof of part (i)

Let $f\sim\chi$ for some primitive Dirichlet character $\chi$. Suppose that
the Furstenberg system is defined using Cesàro averages along the sequence
$(N_{k})$. By Proposition 5.5, there exists a subsequence $(N_{k}^{\prime})$
of $(N_{k})$ such that $f$ is Besicovitch rationally almost periodic for
Cesàro averages along $(N_{k}^{\prime})$. By Theorem 3.2, the system
$(X,\mu,T)$ is an ergodic procyclic system.

It remains to prove that if $f\neq 1$, then the system is non-trivial. Suppose
first that $f\sim 1$. Since $f\neq 1$, there exist $p\in{\mathbb{P}}$ and
$s\in{\mathbb{N}}$ such that $f(p^{s})\neq 1$. Then Theorem 2.11 applies and
gives that $1/p\in\operatorname{Spec}(X,\mu,T)$. On the other hand, if
$f\not\sim 1$, then the conductor $q$ of the primitive Dirichlet character
$\chi$ satisfies $q>1$. Then for every $p\in{\mathbb{P}}$ with $p\mid q$ we
have by Theorem 2.11 that $1/p\in\operatorname{Spec}(X,\mu,T)$. In either case
we get that the system is non-trivial.

A similar argument applies to logarithmic averages since Proposition 5.5,
Theorem 3.2, and Theorem 2.11, also apply to logarithmic averages.
Alternatively, we can use Proposition 6.4, which implies that in this case,
Furstenberg systems taken with respect to Cesàro and logarithmic averages
coincide.

####  6.3.2. Proof of part (ii)

We give all the arguments for Cesàro averages, the proof carries over to
logarithmic averages, since we have proved variants for all the results needed
for logarithmic averages.

Before embarking into the proof of the various claims, we do some preparatory
work. Let $f\sim n^{it}\cdot\chi$ for some $t\neq 0$ and some primitive
Dirichlet character $\chi$, and suppose that $(X,\mu,T)$ is a Furstenberg
system of $f$ for Cesàro averages, defined along the sequence $(N_{k})$. Then
$f=\tilde{f}\cdot n^{it}$, where $\tilde{f}:=f\cdot n^{-it}\sim\chi$. Upon
passing to a subsequence $(N_{k}^{\prime})$ of $(N_{k})$ we get that the
Furstenberg systems of the sequences $\tilde{f}$ and $n^{it}$, taken along
$(N_{k}^{\prime})$, are both well defined. We denote the Furstenberg system of
$\tilde{f}$ along $(N_{k}^{\prime})$ by $(Y,\nu,S)$ ($Y:={\mathbb{U}}^{Z}$ and
$S$ is again the shift map) and recall that, by part (i), it is isomorphic to
an ergodic procyclic system. Furthermore, by [24, Corollary 5.5], for $t\neq
0$, the Furstenberg system of $n^{it}$ along $(N_{k}^{\prime})$ defines a
continuous measure on $X$ that is supported on the diagonal set
$\\{(z)_{k\in{\mathbb{Z}}}\colon z\in\mathbb{S}^{1}\\}$, on which $T$ acts as
an identity. Hence, the Furstenberg system of $n^{it}$ can be identified with
the system $(Z:=\mathbb{S}^{1},\lambda,\operatorname{id})$, where $\lambda$ is
a continuous measure on $\mathbb{S}^{1}$ (in fact, $\lambda$ is equivalent and
not equal to the Lebesgue measure, but we shall not use this). Since
$f=\tilde{f}\cdot n^{it}$, it follows that the system $(X,\mu,T)$ is a factor
of a joining of the systems $(Y,\nu,S)$ and $(Z,\lambda,\operatorname{id})$,
with factor map

(42) $\pi(y,z):=M_{z}(y),\quad y\in Y,\,z\in\mathbb{S}^{1},$

where $M_{z}(y)(k):=z\cdot y(k),\ k\in{\mathbb{Z}}$. (We will make use of the
form of $\pi$ only at the end of our argument to get the product structure of
the measure $\mu$). Since the two systems are disjoint, we obtain that

(43) $(X,\mu,T)\text{ is a factor of the direct product of }(Y,\nu,S)\text{
and }(Z,\lambda,\operatorname{id}).$

After this preparation, we are now ready to prove the claimed properties.

Rational discrete spectrum. Since both systems $(Y,\nu,S)$ and
$(Z,\lambda,\operatorname{id})$ have rational discrete spectrum, so does their
direct product. By (43) the same holds for $(X,\mu,T)$.

Non-trivial rational spectrum. We show that if $\tilde{f}\neq 1$, then the
spectrum of the Furstenberg system $(X,\mu,T)$ of $f$ is non-trivial (hence,
if the spectrum is trivial, then $f(n)=n^{it}$, $n\in{\mathbb{N}}$, for some
$t\in{\mathbb{R}}$). Arguing by contradiction, suppose that it is trivial.
Then since it has rational discrete spectrum, it is an identity system.
Moreover, as remarked above, the Furstenberg system of $n^{-it}$ is an
identity system. Since $\tilde{f}=f\cdot n^{-it},$ we deduce that the
Furstenberg system $(Y,\nu,S)$ of $\tilde{f}$ is a factor of a joining of two
identity systems, thus, it is an identity system. This contradicts part (i),
which implies that since $\tilde{f}\sim\chi$ and $\tilde{f}\neq 1$, the system
$(Y,\nu,S)$ has non-trivial spectrum (since it is a non-trivial ergodic
procyclic system).

Non-ergodicity.171717Although a different proof for this fact can be given
using the “product structure” result proved later in this subsection, we
choose to also give this shorter and more direct argument. We show that the
system $(X,\mu,T)$ is non-ergodic. We first reduce to the case, where
$\chi=1$. Since the non-zero values of $\chi$ are roots of unity of fixed
order, there exists $q\in{\mathbb{N}}$ such that $\chi^{q}\sim 1$. Then the
Furstenberg system of $f^{q}$ along $(N_{k}^{\prime})$ is also well defined
and ergodic if $(X,\mu,T)$ was ergodic (since it defines a factor of
$(X,\mu,T)$, see the second remark after Definition 3.7). Moreover, $f^{q}\sim
n^{iqt}\cdot\chi^{q}\sim n^{iqt}$. So, it suffices to show that the
Furstenberg system of $f^{q}$ along $(N_{k}^{\prime})$ is non-ergodic.
Furthermore, by considering $f^{2q}$ in place of $f^{q}$ if needed, we can
assume that $f(2)\neq-2^{it}$. Thus, we have reduced matters to the case
$\chi=1$ and $f(2)\neq-2^{it}$.

Recall that we have $f=\tilde{f}\cdot n^{it}$ where $\tilde{f}=f\cdot
n^{-it}$, hence $\tilde{f}\sim 1$ and $\tilde{f}(2)\neq-1$. Moreover, (43)
holds. We want to show that the system $(X,\mu,T)$ is non-ergodic, so arguing
by contradiction, suppose that it is ergodic. Then

(44)
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,f(m+n)\cdot\overline{f(m)}=|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,f(m)|^{2}.$

(By (7), this is equivalent to the identity
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\int
T^{n}F_{0}\cdot\overline{F_{0}}\,d\mu=|\int F_{0}\,d\mu|^{2}$, which holds by
the von Neumann ergodic theorem.) Using the disjointness of the Furstenberg
system of $(n^{it})$ and $(\tilde{f}(n))$, we get that the right hand side
equals

$|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,m^{it}|^{2}\cdot|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m)|^{2}.$

On the other hand, for every $n\in{\mathbb{N}}$ we have
$\lim_{m\to\infty}(m+n)^{it}\cdot m^{-it}=1$, hence the left hand side in (44)
equals

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m+n)\cdot\overline{\tilde{f}(m)}=|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m)|^{2},$

where the last identity follows as before from the ergodicity of the
Furstenberg system of $\tilde{f}$ along $(N_{k}^{\prime})$ that was proved in
part (i) of Theorem 2.8. Combining the above identities, we deduce that

$|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,m^{it}|^{2}\cdot|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m)|^{2}=|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m)|^{2}.$

Since $\tilde{f}\sim 1$ and $\tilde{f}(2)\neq-1$, by Theorem 3.1, we have
$\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,\tilde{f}(m)\neq 0$,
hence the previous identity implies that

$|\lim_{k\to\infty}{\mathbb{E}}_{m\in[N_{k}^{\prime}]}\,m^{it}|=1.$

But one easily verifies (see the remark after Proposition 6.1) that this can
only happen if $t=0$: a contradiction.

Product structure. Recall that (43) holds and our plan is to use it in order
to apply Proposition B.1 in the appendix. To this end, we need to verify the
following.

Claim: There exists at least one $z\in\mathbb{S}^{1}$ such that the measure
$\nu$ is not invariant under $M_{z}$. Arguing by contradiction, suppose that
the claim fails. Then, for every $F\in C(Y)$, we have

$\int F(z\cdot y)\,d\nu=\int F(y)\,d\nu,\quad\text{for every
}z\in\mathbb{S}^{1}.$

Let $F_{0}$ be the $0^{\text{th}}$-coordinate projection on
$Y={\mathbb{U}}^{\mathbb{Z}}$. Applying the previous identity for
$F:=F_{0}^{m}$, $m\in{\mathbb{N}}$, and using that
$F_{0}(zy)=z\,y(0)=zF_{0}(y)$, we deduce

$(z^{m}-1)\int F_{0}^{m}\,d\nu=0,\quad\text{for every }z\in\mathbb{S}^{1}.$

Hence, $\int F_{0}^{m}\,d\nu=0$ for every $m\in{\mathbb{N}}$. Recall that
$(Y,\nu,S)$ is the Furstenberg system for Cesàro averages of $\tilde{f}$ along
the sequence $(N_{k}^{\prime})$. We deduce using (7) that

(45)
$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,(\tilde{f}(n))^{m}=0,\quad\text{
for every }m\in{\mathbb{N}}.$

Since $\tilde{f}\sim\chi$ for some Dirichlet character $\chi$, there exists
$q\in{\mathbb{N}}$ such that $\tilde{f}^{q}\sim 1$. Furthermore, by
considering $2q$ in place of $q$, if needed, we can assume that
$\tilde{f}(2)^{q}\neq-1$. Keeping these two facts in mind, we get that (45),
for $m:=q$, contradicts Theorem 3.1, which claims that we cannot have a
vanishing subsequential limit in this case. This completes the proof of the
claim.

Hence, Proposition B.1 in the appendix gives that there exists
$r\in{\mathbb{N}}$ such that the system $(X,\mu,T)$ is isomorphic to
$(Y,\nu,S)\times(\mathbb{S}^{1},\lambda_{r},\operatorname{id})$, where
$\lambda_{r}$ is the pushforward of the continuous measure $\lambda$ by
$z\mapsto z^{r}$.181818If $f(n)=n^{i}\cdot\chi$ where $\chi:={\bf
1}_{4{\mathbb{Z}}+1}-{\bf 1}_{4{\mathbb{Z}}+3}$, then $r=2$ and the group $G$
that appears in the proof of Proposition B.1 is equal to $\\{-1,1\\}$. Lastly,
note that since $\lambda_{r}$ is a continuous measure, and any two identity
transformations on Lebesgue spaces with continuous probability measures are
isomorphic (see for example [31, Theorem 17.41]), the system
$(\mathbb{S}^{1},\lambda_{r},\operatorname{id})$ is isomorphic to the system
$({\mathbb{T}},m_{\mathbb{T}},\operatorname{id})$. This completes the proof.

####  6.3.3. Proof of part (iii)

Suppose that $f\sim n^{it}\cdot\chi$ for some non-zero $t\in{\mathbb{R}}$ and
primitive Dirichlet character $\chi$. Let $(X,\mu_{1},T)$ and $(X,\mu_{2},T)$
be two Furstenberg systems of $f$ for Cesàro or logarithmic averages (not
necessarily both taken using Cesàro or logarithmic averages). Using part (ii),
we get that for $i=1,2$ the system $(X,\mu_{i},T)$ is isomorphic to the direct
product of the system $({\mathbb{T}},m_{\mathbb{T}},\operatorname{id})$ and
the system $(X,\mu_{i}^{\prime},T)$, which is some Furstenberg system of
$\tilde{f}:=f\cdot n^{-it}$ (with respect to Cesàro or logarithmic averages).
Since $\tilde{f}\sim\chi$, by Proposition 6.4, we have that the systems
$(X,\mu_{1}^{\prime},T)$ and $(X,\mu_{2}^{\prime},T)$ are isomorphic.
Combining the above, we get that the systems $(X,\mu_{1},T)$ and
$(X,\mu_{2},T)$ are isomorphic. This completes the proof.

####  6.3.4. Proof of part (iv)

Follows immediately from Proposition 6.2.

####  6.3.5. Proof of part (v)

In case (i), the asserted property follows from Proposition 5.5. In case (ii),
suppose that there exists a sequence $N_{k}\to\infty$ along which $f$ is
Besicovitch rationally almost periodic for Cesàro or logarithmic averages.
Upon passing to a subsequence, we can assume that the Furstenberg system of
$f$ for Cesàro or logarithmic averages along $(N_{k})$ is well defined. Then
by Theorem 3.2 this system is ergodic. This contradicts the non-ergodicity
established in part (ii).

###  6.4. Proof of Corollary 2.9

To establish the equivalence of (i), (ii), (iii) we argue as follows. The
implication $\eqref{I:chi1}\implies\eqref{I:chi3}$ follows from part (i) of
Theorem 2.8. The implication $\eqref{I:chi3}\implies\eqref{I:chi2}$ is
obvious. The implication $\eqref{I:chi2}\implies\eqref{I:chi1}$ follows from
part (ii) of Theorem 2.8.

To establish the equivalence of (i), (iv), (v) we argue as follows. The
implication $\eqref{I:chi1}\implies\eqref{I:chi5}$ follows from the first
assertion in part (v) of Theorem 2.8. The implication
$\eqref{I:chi5}\implies\eqref{I:chi4}$ is obvious. The implication
$\eqref{I:chi4}\implies\eqref{I:chi1}$ follows from the second assertion in
part (v) of Theorem 2.8.

###  6.5. Proof of Corollary 2.10

The implication $\eqref{I:nit1}\implies\eqref{I:nit5}$ is obvious. The
implication $\eqref{I:nit5}\implies\eqref{I:nit4}$ is simple and follows from
[24, Corollary 5.5] and [18, Section 1.3]. The implications
$\eqref{I:nit4}\implies\eqref{I:nit3}\implies\eqref{I:nit2}$ are trivial. It
remains to establish the implication $\eqref{I:nit2}\implies\eqref{I:nit1}$
Namely, we want to show that if some Furstenberg system of $f$ for Cesàro or
logarithmic averages has trivial rational spectrum, then $f(n)=n^{it}$,
$n\in{\mathbb{N}}$, for some $t\in{\mathbb{R}}$. This follows from part (ii)
of Theorem 2.8.

##  7\. Spectral results for pretentious multiplicative functions and
applications

In this section we prove Theorems 2.11-2.16.

###  7.1. Necessary conditions for spectrum

The next result gives a necessary condition that often helps us decide when a
certain number belongs to the spectrum of some Furstenberg system of a bounded
sequence. Furthermore, it gives an identity that we will use to prove Theorem
2.13 when we happen to know that this number is not in the spectrum.

######  Proposition 7.1.

Let $(X,\mu,T)$ be the Furstenberg system for Cesàro averages of a sequence
$a\colon{\mathbb{N}}\to{\mathbb{U}}$ taken along a sequence $(N_{k})$. If
$\alpha\in(0,1)$ is such that $\alpha\not\in\operatorname{Spec}(X,\mu,T)$,
then

(46)
$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,e(-n\alpha)\prod_{j=1}^{\ell}a_{j}(n+n_{j})=0$

for all $\ell\in{\mathbb{N}}$, $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, and
$a_{1},\ldots,a_{\ell}\in\\{a,\overline{a}\\}$. A similar statement also holds
when we consider logarithmic averages.

###### Proof.

Arguing by contradiction, suppose that the conclusion fails for some
$\ell\in{\mathbb{N}}$, $n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, and
$a_{1},\ldots,a_{\ell}\in\\{a,\overline{a}\\}$. Using van der Corput’s lemma
(the variant needed follows from [35, Lemma 3.1]), we get

$\lim_{H\to\infty}{\mathbb{E}}_{h\in[H]}\,e(-h\alpha)\Big{(}\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,\prod_{j=1}^{\ell}a_{j}(n+n_{j})\prod_{j=1}^{\ell}\overline{a_{j}}(n+h+n_{j})\Big{)}\neq
0.$

(Note that all the limits as $k\to\infty$ exist because $a$ is assumed to
admit correlations along $(N_{k})$.) Using (7) to translate this equation to
the Furstenberg system, we get that

$\lim_{H\to\infty}{\mathbb{E}}_{h\in[H]}\,e(-h\alpha)\int F\cdot
T^{h}\overline{F}\,d\mu\neq 0,$

where $F:=\prod_{j=1}^{\ell}T^{n_{j}}F_{j}$ for some
$F_{j}\in\\{F_{0},\overline{F_{0}}\\}$ and $F_{0}$ is the
$0^{\text{th}}$-coordinate projection. This implies that $\alpha$ is on the
spectrum of $(X,\mu,T)$, in fact,

$G:=\lim_{H\to\infty}{\mathbb{E}}_{h\in[H]}\,e(-h\alpha)\,T^{h}\overline{F},$

where the limit is taken in $L^{2}(\mu)$, is a non-zero
$e(\alpha)$-eigenfunction. This contradicts our assumption
$\alpha\not\in\operatorname{Spec}(X,\mu,T)$ and completes the
proof.191919Alternatively, one can argue that if the conclusion fails, then we
get a nontrivial joining of the system with the rotation by $\alpha$ on the
circle. Now, if the limit in (46) is non-zero, then the conditional
expectation (with respect to the Furstenberg system in the joining) of the
eigenfunction (on the circle) corresponding to $\alpha$ is non-zero. Hence,
$\alpha$ belongs to the spectrum of the system $(X,\mu,T)$. ∎

The following simpler condition often suffices to deduce that a certain
rational number belongs to the spectrum of a Furstenberg system of a bounded
sequence (we caution the reader though, that the condition is far from
necessary).

######  Corollary 7.2.

Let $(X,\mu,T)$ be the Furstenberg system for Cesàro averages of a sequence
$a\colon{\mathbb{N}}\to{\mathbb{U}}$ taken along $N_{k}\to\infty$. Suppose
that for some $q\geq 2$ and $r\in\\{0,\ldots,q-1\\}$ we have

$\lim_{k\to\infty}\big{(}{\mathbb{E}}_{n\in[N_{k}/q]}\,a(qn+r)-{\mathbb{E}}_{n\in[N_{k}]}\,a(n)\big{)}\neq
0.$

Then $p/q$ belongs to the spectrum of $(X,\mu,T)$ for some
$p\in\\{1,\ldots,q-1\\}$. A similar statement holds for logarithmic averages,
and in this context we can replace $\mathbb{E}^{\log}_{n\in[N_{k}/q]}$ with
$\mathbb{E}^{\log}_{n\in[N_{k}]}$.

###### Proof.

Our assumption is equivalent to

(47) $\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,(1-q\,{\bf
1}_{q{\mathbb{N}}+r}(n))\,a(n)\neq 0.$

Upon substituting the identity

$q\,{\bf
1}_{q{\mathbb{N}}+r}(n)=\sum_{p=0}^{q-1}e\Big{(}p\cdot\frac{n-r}{q}\Big{)}$

in (47), we deduce that

$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,e\Big{(}-n\cdot\frac{p}{q}\Big{)}\,a(n)\neq
0$

for some $p\in\\{1,\ldots,q-1\\}$. It follows from Proposition 7.1 (we only
need to appeal to the $\ell=1$ case) that $p/q$ is in the spectrum of the
system. ∎

###  7.2. Preparation for the proof of Theorem 2.11

If $\chi$ is a primitive Dirichlet character with conductor $q$, then it has a
unique Furstenberg system, which is periodic with minimal period $q$, and its
spectrum is spanned by multiples of $1/q$. The next result shows that if
$f\sim\chi$ for some primitive Dirichlet character $\chi$, then the spectrum
of every Furstenberg system of $f$ is at least as large as the spectrum of the
Furstenberg system of $\chi$.

######  Proposition 7.3.

Let $(X,\mu,T)$ be the Furstenberg system for Cesàro averages of a
multiplicative function $f\colon{\mathbb{N}}\to{\mathbb{U}}$ that satisfies
$f\sim\chi$ for some primitive Dirichlet character $\chi$ with conductor
$q\geq 2$. Then for every $p\in{\mathbb{P}}$ with $p\mid q$ we have
$1/p\in\operatorname{Spec}(X,\mu,T)$.

###### Proof.

We assume that the Furstenberg system is defined along a sequence $(N_{k})$
and by passing to a further subsequence we can assume that the Furstenberg
system of all other multiplicative functions defined subsequently are also
taken along $(N_{k})$.

Suppose first that $q=p^{k}$ for some $p\in{\mathbb{P}}$. Since $f\sim\chi$ we
have $f\cdot\overline{\chi}\sim 1$. If $f(2^{s})\overline{\chi(2^{s})}\neq-1$
for some $s\in{\mathbb{N}}$, then by Theorem 3.1 we have

$\limsup_{k\to\infty}|{\mathbb{E}}_{n\in[N_{k}]}\,f(n)\cdot\overline{\chi(n)}|>0.$

Since $\chi$ is a primitive Dirichlet character with conductor $q>1$, it is
non-principal, and as a consequence has mean value zero. Since $\chi$ is
periodic with period $q$, using its Fourier expansion in ${\mathbb{Z}}_{q}$,
we deduce that

$\limsup_{k\to\infty}|{\mathbb{E}}_{n\in[N_{k}]}\,f(n)\cdot e(n\cdot r/q)|>0$

for some $r\in\\{1,\ldots,q-1\\}$. It follows from Proposition 7.1 that
$r/q\in\operatorname{Spec}(X,\mu,T)$ and since $q=p^{k}$ and the spectrum is
closed under integer multiplication, we deduce that
$1/p\in\operatorname{Spec}(X,\mu,T)$. Suppose now that
$f(2^{s})\cdot\overline{\chi(2^{s})}=-1$ for every $s\in{\mathbb{N}}$. Then
$\chi(2)\neq 0$, hence $p\neq 2$. We let $\tilde{f}:={\bf
1}_{2{\mathbb{Z}}+1}\cdot f\sim\chi$, then $\tilde{f}\cdot\overline{\chi}\sim
1$ and $(\tilde{f}\cdot\overline{\chi})(2)=0\neq-1$, and the previous case
gives that $1/p$ belongs to the spectrum of the Furstenberg system of
$\tilde{f}$ along $(N_{k})$. Since $\tilde{f}:={\bf 1}_{2{\mathbb{Z}}+1}\cdot
f$, the Furstenberg system of $\tilde{f}$ along $(N_{k})$ is a factor of a
joining of the Furstenberg system of $f$ along $(N_{k})$, which by Theorem 2.8
is an ergodic procyclic system, and the Furstenberg system of ${\bf
1}_{2{\mathbb{Z}}+1}$, which is periodic with period two. It follows that
$1/p$ is an integer combination of $1/2$ and elements in
$\operatorname{Spec}(X,\mu,T)$. Since $p\neq 2$, we deduce that
$1/p\in\operatorname{Spec}(X,\mu,T)$.

We consider now the general case where $q$ is not a prime power (our
assumption that $\chi$ is primitive is crucially used here). If
$q=p_{1}^{k_{1}}\cdots p_{\ell}^{k_{\ell}}$, $\ell\geq 2$, is the prime
factorisation of $q$, then it is a standard fact that $\chi$ can be decomposed
as $\chi=\chi_{1}\cdots\chi_{\ell}$ where $\chi_{1},\ldots,\chi_{\ell}$ are
primitive Dirichlet characters with conductors
$p_{1}^{k_{1}},\ldots,p_{\ell}^{k_{\ell}}$ respectively. Note that then
$f\cdot\overline{\chi_{1}}\cdots\overline{\chi_{\ell-1}}\sim\chi_{\ell}$, and
by the first case we get that $1/p_{\ell}$ belongs to the spectrum of the
Furstenberg system of
$f\cdot\overline{\chi_{1}}\cdots\overline{\chi_{\ell-1}}$ along $(N_{k})$,
which is a factor of a joining of the system $(X,\mu,T)$, which by Theorem 2.8
is an ergodic procyclic system, and the Furstenberg system of
$\overline{\chi_{1}}\cdots\overline{\chi_{\ell-1}}$. We deduce that
$1/p_{\ell}$ is an integer combination of elements of the form
$1/p_{i}^{k_{i}}$, $i=1,\ldots,\ell-1$ (these elements span the spectrum of
$\overline{\chi_{1}}\cdots\overline{\chi_{\ell-1}}$), and elements in
$\operatorname{Spec}(X,\mu,T)$. Since $p_{j}\neq p_{\ell}$ for
$j=1,\ldots,\ell-1$, it follows that
$1/p_{\ell}\in\operatorname{Spec}(X,\mu,T)$. Similarly we get that
$1/p_{i}\in\operatorname{Spec}(X,\mu,T)$ for $i=1,\ldots,\ell-1$. This
completes the proof. ∎

We will also need the following simple fact.

######  Lemma 7.4.

If $\chi$ is a primitive Dirichlet character with conductor $q\geq 2$, we
define the completely multiplicative function $\tilde{\chi}$ on prime numbers
$p$ by

$\tilde{\chi}(p):=\begin{cases}\chi(p),\quad&p\nmid q\\\ 1,\quad&p\mid
q.\end{cases}$

Then $\tilde{\chi}$ has a unique Furstenberg system $(X,\mu,T)$ and for
$p_{0}\in{\mathbb{P}}$ we have $1/p_{0}\in\operatorname{Spec}(X,\mu,T)$ if and
only if $p_{0}\mid q$.

###### Proof.

The uniqueness of the Furstenberg system of $\tilde{\chi}$ will be established
in the second part of the proof. For the moment we assume that $(X,\mu,T)$ is
some Furstenberg system of $\tilde{\chi}$.

Since $\tilde{\chi}\sim\chi$ and $\chi$ is a primitive Dirichlet character
with conductor $q$ and $p_{0}\mid q$, we have by Proposition 7.3 that
$1/p_{0}\in(X,\mu,T)$.

Suppose that $p_{0}\in{\mathbb{P}}$ is such that $p_{0}\nmid q$, we shall show
that $1/p_{0}\not\in\operatorname{Spec}(X,\mu,T)$. We define the
multiplicative function

$\tilde{\chi}_{m}(n):=\begin{cases}\tilde{\chi}(n),\quad&\text{if
}\,q^{m}\nmid n\\\ 0,\quad&\text{if }\,q^{m}\mid n.\end{cases}$

One easily verifies that $\tilde{\chi}_{m}$ is periodic with period $q^{m}$,
hence it has a unique Furstenberg system with spectrum a subset of the
subgroup generated by $1/q^{m}$, and

$\lim_{m\to\infty}\limsup_{N\to\infty}{\mathbb{E}}_{n\in[N]}|\tilde{\chi}_{m}(n)-\tilde{\chi}(n)|=0.$

By [2, Lemma 3.17], $\tilde{\chi}$ has a unique Furstenberg system, and it is
a factor of a joining of the periodic systems generated by $\tilde{\chi}_{m}$,
$m\in{\mathbb{N}}$. Hence, $\operatorname{Spec}(X,\mu,T)$ is contained in the
subgroup $\\{j/q^{m}\colon j=0,1\ldots q^{m}-1,m\in{\mathbb{N}}\\}$ of
${\mathbb{T}}$. Since $p_{0}\nmid q$, we deduce that
$1/p_{0}\not\in\operatorname{Spec}(X,\mu,T)$, completing the proof. ∎

Lastly, we will use the following elementary fact.

######  Lemma 7.5.

Let $G_{1},G_{2}$ be rational subgroups of ${\mathbb{T}}$. If
$p\in{\mathbb{P}}$ satisfies $1/p\not\in G_{j}$ for $j=1,2$, then $1/p$ does
not belong to the subgroup generated by $G_{1}$ and $G_{2}$.

###### Proof.

We argue by contradiction. Suppose that $1/p$ belongs to the subgroup
generated by $G_{1}$ and $G_{2}$. Then, for $j=1,2$, there exist
$a_{j},b_{j}\in{\mathbb{N}}$ such that $a_{j}/b_{j}\in G_{j}$,
$(a_{j},b_{j})=1$, and such that $1/p=a_{1}/b_{1}+a_{2}/b_{2}.$ Since $1/p$
does not belong to the subgroup $G_{1}$ and we also have $a_{1}/b_{1}\in
G_{1}$ and $(a_{1},b_{1})=1$, we easily deduce that $p$ does not divide
$b_{1}$. Similarly, we get that $p$ does not divide $b_{2}$. On the other
hand, we have $b_{1}b_{2}=p(a_{1}b_{2}+a_{2}b_{1})$, hence $p$ divides either
$b_{1}$ or $b_{2}$, a contradiction. This completes the proof. ∎

###  7.3. Proof of the implication
$\eqref{I:spectrum1}\implies\eqref{I:spectrum2}$ of Theorem 2.11

By Proposition 6.2, Furstenberg systems for Cesàro and logarithmic averages
coincide, so we only explain the argument for Cesàro averages. Suppose that
the Furstenberg system $(X,\mu,T)$ is taken along the sequence
$N_{k}\to\infty$.

Before proceeding further, let us make the following simple observation:
Suppose that $p_{1},\ldots,p_{k}\in\mathbb{P}$,
$s_{1},\ldots,s_{k}\in\mathbb{N},$ and define the multiplicative function $f$
by setting $f(p^{s}_{i}):=\alpha_{i,s}\in\mathbb{U}$ for $i=1,\ldots,k$ and
$s=1,\ldots,s_{i}$, and $f(p^{s}):=1$ for all other prime powers. Then $f(n)$
is completely determined from the knowledge of all remainders of $n$ modulo
$p_{i}^{s_{i}+1}$, $i=1,\ldots,k$, hence

(48) $f\colon\mathbb{N}\to\mathbb{U}\ \text{ is periodic with period }\
\prod_{i=1}^{k}p_{i}^{s_{i}+1}.$

In our argument below we will approximate $f$ by periodic multiplicative
functions of the previous form and deduce properties for $f$ from those of its
periodic approximants.

Suppose first that $f\sim 1$ and $p_{0}\in{\mathbb{P}}$ is such that
$f(p_{0}^{s})=1$ for every $s\in{\mathbb{N}}$. Our goal is to show that
$1/p_{0}\not\in\operatorname{Spec}(X,\mu,T)$.

By Proposition 5.5 (see the remark following the result), there exist a
subsequence $(N_{k}^{\prime})$ of $(N_{k})$ and constants
$\alpha_{m}\in[0,1)$, $m\in{\mathbb{N}}$, such that

(49)
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}|f(n)-e(\alpha_{m})\cdot
f_{m}(n)|^{2}\leq 1/m,$

where $f_{m}\colon{\mathbb{N}}\to{\mathbb{U}}$, $m\in{\mathbb{N}}$, are
defined as in Lemma 5.4 (for $\varepsilon:=1/m$) and satisfy property (i) of
Lemma 5.4 (with $\chi:=1$). Furthermore, since $f(p_{0}^{s})=1$ for every
$s\in{\mathbb{N}}$, we can also assume that

(50) $f_{m}(p_{0}^{s})=1\text{ for every }m,s\in{\mathbb{N}}.$

Moreover, since for $m\in{\mathbb{N}}$ the sequence $f_{m}$ satisfies (26)
with $\chi=1$, as shown in the proof of [9, Lemma 4] (we crucially use here
that $f\sim 1$), we have (in fact, we could replace
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}$ with
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}$)

(51)
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}|f_{m}(n)-f^{\prime}_{m}(n)|^{2}\leq
1/m,$

where for every $m\in{\mathbb{N}}$ the multiplicative functions
$f_{m}^{\prime}\colon{\mathbb{N}}\to{\mathbb{U}}$ are defined by

$f^{\prime}_{m}(p^{s}):=\begin{cases}f_{m}(p^{s}),\quad&\text{for }p\leq
r_{m},s\in{\mathbb{N}}\\\ 1,\quad&\text{for all other prime
powers},\end{cases}$

for $r_{m}\in{\mathbb{N}}$ sufficiently large that we can arrange to form an
increasing sequence. Note that (50) implies

(52) $f^{\prime}_{m}(p_{0}^{s})=1\quad\text{ for every }m,s\in{\mathbb{N}}.$

Let $f^{\prime}_{m,l}\colon{\mathbb{N}}\to{\mathbb{U}}$ be multiplicative
functions defined by

$f^{\prime}_{m,l}(p^{s}):=\begin{cases}f^{\prime}_{m}(p^{s}),\quad&\text{for
}p\leq r_{m},s\leq l\\\ 1,\quad&\text{for all other prime powers}.\end{cases}$

Then (52) implies that

(53) $f^{\prime}_{m,l}(p_{0}^{s})=1\,\text{ for every }m,l,s\in{\mathbb{N}}.$

Combining (48) with (53), we get that for every $m,l\in{\mathbb{N}}$, the
sequence $n\mapsto f^{\prime}_{m,l}(n)$ is periodic with period
$P_{m,l}:=\prod_{p\leq r_{m},p\neq p_{0}}p^{l+1}$. Arguing as in the proof of
[9, Lemma 4], we get

(54)
$\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N^{\prime}_{k}]}|f^{\prime}_{m}(n)-f^{\prime}_{m,l_{m}}(n)|^{2}\leq
1/m$

for $l_{m}\in{\mathbb{N}}$ sufficiently large that we can arrange to form an
increasing sequence. Combining (49), (51), (54), we get

(55)
$\lim_{m\to\infty}\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}|f(n)-e(\alpha_{m})\cdot
f^{\prime}_{m,l_{m}}(n)|^{2}=0.$

Now, for every $m\in{\mathbb{N}}$, the sequence $(e(\alpha_{m})\cdot
f^{\prime}_{m,l_{m}}(n))_{n\in{\mathbb{N}}}$ is periodic with period

$P_{m}:=\prod_{p\leq r_{m},p\neq p_{0}}p^{l_{m}+1}.$

Hence, the spectrum of its Furstenberg system $(X,\mu_{m},T)$ is contained in
the subgroup $G_{m}$ of ${\mathbb{T}}$ generated by $1/P_{m}$. Since $p_{0}$
does not divide $P_{m}$, we have that $1/p_{0}\not\in G_{m}$. In summary,

(56) $\operatorname{Spec}(X,\mu_{m},T)\subset G_{m}\,\text{ and
}\,1/p_{0}\not\in G_{m}\,\text{ for every }\,m\in{\mathbb{N}}.$

Note also that because $(r_{m})$ and $(l_{m})$ are increasing sequences, we
have that $P_{m}$ divides $P_{m+1}$, hence $(G_{m})$ is an increasing sequence
of rational subgroups.

By (55) and [2, Lemma 3.17], the system $(X,\mu,T)$ is a factor of a joining
of the periodic systems $(X,\mu_{m},T)$, $m\in{\mathbb{N}}$. Hence,
$\operatorname{Spec}(X,\mu,T)$ is contained in the subgroup generated by the
sequence of subgroups $\operatorname{Spec}(X,\mu_{m},T)$, $m\in{\mathbb{N}}$
(see the remark (iv) at the end of Section 3.3), which in turn, by (56), is
contained in the subgroup generated by the increasing sequence of rational
subgroups $G_{m}$, $m\in{\mathbb{N}}$. By (56) we have that $1/p_{0}\not\in
G_{m}$ for every $m\in{\mathbb{N}}$, hence
$1/p_{0}\not\in\operatorname{Spec}(X,\mu,T)$.

Lastly, we deal with the case where $f\sim\chi$ for some primitive Dirichlet
character $\chi$ with conductor $q\geq 2$. Let $p_{0}\in{\mathbb{P}}$ be such
that $p_{0}\nmid q$ and $f(p_{0}^{s})=\chi(p_{0}^{s})$ for every
$s\in{\mathbb{N}}$. Our goal is to show that
$1/p_{0}\not\in\operatorname{Spec}(X,\mu,T)$. We take $\tilde{\chi}$ as in
Lemma 7.4 and let $\tilde{f}:=f\cdot\overline{\tilde{\chi}}$. Then
$\tilde{f}\sim 1$ and $\tilde{f}(p_{0}^{s})=1$ for every $s\in{\mathbb{N}}$
(we used that $p_{0}\nmid q$ here). Hence, by the previous part we have that
$1/p_{0}$ does not belong to the spectrum of the Furstenberg system of
$\tilde{f}$ taken along a subsequence $(N_{k}^{\prime})$ of $(N_{k})$.
Furthermore, by Lemma 7.4, it follows that $1/p_{0}$ does not belong to the
spectrum of the (unique) Furstenberg system of $\tilde{\chi}$. Since
$f=\tilde{f}\cdot\tilde{\chi}$ (we used that $|\tilde{\chi}|=1$ here), arguing
as before, and using Lemma 7.5, we get that $1/p_{0}\not\in(X,\mu,T)$. This
completes the proof.

###  7.4. Proof of the implication
$\eqref{I:spectrum2}\implies\eqref{I:spectrum1}$ of Theorem 2.11

We assume that $f\sim\chi$ for some primitive Dirichlet character $\chi$ and
we have $f(p^{s})\neq\chi(p^{s})$ for some $p\in{\mathbb{P}}$ and
$s\in{\mathbb{N}}$. Our goal is to show that
$1/p\in\operatorname{Spec}(X,\mu,T)$. We do this in three steps. The bulk of
the proof is contained on the first step, where we work under the additional
hypothesis $f\sim 1$ and $f(2^{s_{0}})\neq-1$ for some $s_{0}\in{\mathbb{N}}$.
On the second step we use the first one to cover the complementary case $f\sim
1$ and $f(2^{s})=-1$ for every $s\in{\mathbb{N}}$. Finally, on the third step
we use the first two steps to cover the general case $f\sim\chi$ for some
Dirichlet character $\chi$.

####  7.4.1. The case $f\sim 1$ and $f(2^{s_{0}})\neq-1$ for some
$s_{0}\in{\mathbb{N}}$

We assume that $f\sim 1$ and $f(2^{s_{0}})\neq-1$ for some
$s_{0}\in{\mathbb{N}}$, this is in addition to our hypothesis that
$f(p^{s})\neq 1$ for some $s\in{\mathbb{N}}$. Note first that since $f\sim 1$,
Proposition 6.2 implies that the Furstenberg systems of $f$ for Cesàro and
logarithmic averages coincide. So, we only have to treat the case of
logarithmic averages, and this turns out to offer a substantial advantage
since it enables us to replace
$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}/p]}$ with
$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}$ throughout.

So, let $(X,\mu,T)$ be a Furstenberg system of $f$ for logarithmic averages
taken along the sequence $N_{k}\to\infty$. Note that then the limit
$L:=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,f(n)$ exists and since
$f\sim 1$ and $f(2^{s_{0}})\neq-1$ for some $s_{0}\in{\mathbb{N}}$, we get by
Theorem 3.1 that $L\neq 0$ (see the remarks following the theorem).

Arguing by contradiction, suppose that
$1/p\notin\operatorname{Spec}(X,\mu,T)$. By Corollary 7.2, this implies

(57)
$L=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,f(pn+j)=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,f(n),\quad\text{for
every }\,j\in\\{0,\ldots,p-1\\}.$

Using these identities, our goal is to show that $f(p^{s})=1$ for every
$s\in{\mathbb{N}}$, which would contradict our hypothesis.

For convenience, we use the convention

$\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,a(n):=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,a(n)$

whenever the limit exists. In what follows we will use the following basic
property of logarithmic averages: If $a\colon{\mathbb{N}}\to{\mathbb{U}}$ is
such that the limit $\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,a(n)$
exists, then we also have that the limit
$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[pN_{k}]}\,a(n)$ also exists and the
two limits are equal. Using this fact to justify the first identity below and
(57) to justify the second, we get

$\displaystyle\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(n)$
$\displaystyle=\frac{1}{p}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(pn)+\frac{1}{p}\sum_{j=1}^{p-1}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(pn+j)$
$\displaystyle=\frac{1}{p}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(pn)+L\cdot\frac{p-1}{p}.$

Repeating this process one more time in order to evaluate
$\frac{1}{p}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(pn)$, we get

$\displaystyle\frac{1}{p}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(pn)$
$\displaystyle=\frac{1}{p^{2}}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(p^{2}n)+\frac{1}{p^{2}}\sum_{j=1}^{p-1}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(p(pn+j))$
$\displaystyle=\frac{1}{p^{2}}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(p^{2}n)+L\cdot\frac{p-1}{p}\cdot\frac{f(p)}{p},$

where to get the second identity we used the multiplicativity of $f$ and (57).
Combining these two identities gives

$\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(n)=\frac{1}{p^{2}}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(p^{2}n)+L\cdot\frac{p-1}{p}\Big{(}1+\frac{f(p)}{p}\Big{)}.$

Repeating this process $M-2$ more times, we deduce

${\mathbb{E}}_{n\in{\mathbf{N}}}\,f(n)=\frac{1}{p^{M}}\mathbb{E}^{\log}_{n\in{\mathbf{N}}}\,f(p^{M}n)+L\cdot\frac{p-1}{p}\,\sum_{s=0}^{M-1}\frac{f(p^{s})}{p^{s}}.$

Letting $M\to\infty$ gives

$L={\mathbb{E}}_{n\in{\mathbf{N}}}\,f(n)=L\cdot\frac{p-1}{p}\cdot\sum_{s=0}^{\infty}\frac{f(p^{s})}{p^{s}}.$

Since $L\neq 0$, we deduce that

(58) $\sum_{s=1}^{\infty}\frac{f(p^{s})}{p^{s}}=\frac{1}{p-1}.$

Since $\sum_{s=1}^{\infty}\frac{1}{p^{s}}=\frac{1}{p-1}$ and
$\Re(f(p^{s}))\in[-1,1]$, comparing real parts in (58) we get
$\Re(f(p^{s}))=1$ for every $s\in{\mathbb{N}}$. Since
$f(p^{s})\in{\mathbb{U}}$, we deduce that $f(p^{s})=1$ for every
$s\in{\mathbb{N}}$, completing the proof in this first case.

####  7.4.2. The case $f\sim 1$ and $f(2^{s})=-1$ for every
$s\in{\mathbb{N}}$.

We assume that $f\sim 1$ and $f(2^{s})=-1$ for every $s\in{\mathbb{N}}$. Let
$p\in{\mathbb{P}}$ be such that $f(p^{s_{0}})\neq 1$ for some
$s_{0}\in{\mathbb{N}}$ and our goal is to show that
$1/p\in\operatorname{Spec}(X,\mu,T)$. Suppose that the Furstenberg system of
$f$ for logarithmic averages is taken along $N_{k}\to\infty$.

We first show that $1/2\in\operatorname{Spec}(X,\mu,T)$. Since $f(2^{s})=-1$
for every $s\in{\mathbb{N}}$, by Theorem 3.1, we have
$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,f(n)=0$. On the other hand,
the multiplicative function

(59) $\tilde{f}:=f\cdot{\bf 1}_{2{\mathbb{Z}}+1}$

also satisfies $\tilde{f}\sim 1$, and in addition satisfies
$\tilde{f}(2)\neq-1$. Hence, Theorem 3.1 gives that for a subsequence
$(N_{k}^{\prime})$ of $(N_{k})$ along which the next limit exists, we have

$0\neq\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}]}\,\tilde{f}(n)=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}/2]}\,f(2n+1)=\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}]}\,f(2n+1),$

where the last identity holds because we use logarithmic averages. Hence,

$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}]}\,f(2n+1)\neq\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}^{\prime}]}\,f(n).$

Since $(N_{k}^{\prime})$ is a subsequence of $(N_{k})$, Corollary 7.2 implies
that $1/2\in\operatorname{Spec}(X,\mu,T)$.

Suppose now that $p\neq 2$, we will show that
$1/p\in\operatorname{Spec}(X,\mu,T)$. We define again the multiplicative
function $\tilde{f}$ as in (59). It satisfies $\tilde{f}\sim 1$ and
$\tilde{f}(p^{s_{0}})\neq 1$, together with $\tilde{f}(2)\neq-1$. So, by the
case treated in Section 7.4.1, we have that if $(X,\tilde{\mu},T)$ is the
Furstenberg system of $\tilde{f}$ along a subsequence $(N_{k}^{\prime})$ of
$(N_{k})$, then $1/p\in\operatorname{Spec}(X,\tilde{\mu},T)$. Now, notice that
since $\tilde{f}=f\cdot{\bf 1}_{2{\mathbb{Z}}+1}$, we have that
$(X,\tilde{\mu},T)$ is a factor of a joining of the system $(X,\mu,T)$, which
is an ergodic procyclic system by part (i) of Theorem 2.8, and the ergodic
rotation on two elements. Hence, $1/p$ belongs to the subgroup spanned by
$\operatorname{Spec}(X,\mu,T)$ and $1/2$. Since $(X,\mu,T)$ is ergodic,
$\operatorname{Spec}(X,\mu,T)$ is a group, and as we showed just above,
$1/2\in\operatorname{Spec}(X,\mu,T)$. Combining these facts, we deduce that
$1/p\in\operatorname{Spec}(X,\mu,T)$.

####  7.4.3. The case $f\sim\chi$.

We now treat the case where $f\sim\chi$ for some primitive Dirichlet character
$\chi$ with conductor $q$. If $p\in{\mathbb{P}}$ satisfies $p\mid q$, then by
Proposition 7.3 we have that $1/p\in\operatorname{Spec}(X,\mu,T)$. So it
remains to show that if $p\in{\mathbb{P}}$ is such that $p\nmid q$ and
$f(p^{s_{0}})\neq\chi(p^{s_{0}})$ for some $s_{0}\in{\mathbb{N}}$, then
$1/p\in\operatorname{Spec}(X,\mu,T)$. Suppose that the Furstenberg system
$(X,\mu,T)$ of $f$ for logarithmic averages is taken along $N_{k}\to\infty$.

We define the multiplicative function

$\tilde{f}:=f\cdot\overline{\chi}.$

Since $f\sim\chi$, we have $\tilde{f}\sim 1$. Furthermore, since
$f(p^{s_{0}})\neq\chi(p^{s_{0}})$ and $|\chi(p^{s_{0}})|=1$, we have
$\tilde{f}(p^{s_{0}})\neq 1$. Hence, if $(N_{k}^{\prime})$ is a subsequence of
$(N_{k})$ along which the Furstenberg system $(X,\tilde{\mu},T)$ of
$\tilde{f}$ for logarithmic averages is defined, the case treated in Section
7.4.2 implies that $1/p\in\operatorname{Spec}(X,\tilde{\mu},T)$.

Since $\tilde{f}=f\cdot\overline{\chi}$, the system $(X,\tilde{\mu},T)$ is a
factor of a joining of the system $(X,\mu,T)$ and the Furstenberg system of
$\chi$, the first is an ergodic procyclic system by part (i) of Theorem 2.8
and the second is a periodic system with period $q$. Hence, $1/p$ belongs to
the subgroup generated by $G_{1}:=\operatorname{Spec}(X,\mu,T)$ and the set
$G_{2}:=\\{j/q\colon j=0,\ldots,q-1\\}$. Note that $G_{1},G_{2}$ are rational
subgroups of ${\mathbb{T}}$ and since $p\nmid q$ we have $1/p\not\in G_{2}$.
To conclude the proof, we note that if $1/p\not\in G_{1}$, then since we also
have $1/p\not\in G_{2}$, we get by Lemma 7.5 that $1/p$ does not belong to the
subgroup generated by $G_{1}$ and $G_{2}$, a contradiction. We deduce that
$1/p\in G_{1}=\operatorname{Spec}(X,\mu,T)$, completing the proof.

####  7.4.4. The case $f\sim n^{it}\cdot\chi$.

We justify here the second remark after Theorem 2.11. Suppose that $f\sim
n^{it}\cdot\chi$ for some $t\neq 0$ and primitive Dirichlet character $\chi$
with conductor $q$. Let $p\in{\mathbb{P}}$ be such that either $p\mid q$ or
$f(p^{s})\neq p^{ist}\cdot\chi(p^{s})$ for some $s\in{\mathbb{N}}$ and our
goal is to show that $1/p\in\operatorname{Spec}(X,\mu,T)$. Suppose that the
Furstenberg system $(X,\mu,T)$ of $f$ for Cesàro averages is taken along
$N_{k}\to\infty$, the argument is similar for logarithmic averages.

Let $\tilde{f}:=f\cdot n^{-it}$. Then our assumptions imply that
$\tilde{f}\sim\chi$ and either $p\mid q$ or $\tilde{f}(p^{s})\neq\chi(p^{s})$
for some $s\in{\mathbb{N}}$. Hence, if $(N_{k}^{\prime})$ is a subsequence of
$(N_{k})$ along which the Furstenberg system $(X,\tilde{\mu},T)$ of
$\tilde{f}$ for Cesàro averages is defined, the case treated in Section 7.4.3
implies that $1/p\in\operatorname{Spec}(X,\tilde{\mu},T)$.

Since $\tilde{f}:=f\cdot n^{-it}$, the system $(X,\tilde{\mu},T)$ is a factor
of a joining of the system $(X,\mu,T)$ and the Furstenberg system of
$n^{-it}$. The first system is an ergodic procyclic system by part (i) of
Theorem 2.8 and the second is isomorphic to an identity transformation in
${\mathbb{T}}$ by [24, Corollary 5.5] . Hence, the two systems are disjoint,
and since the second system has trivial spectrum, it follows that
$\operatorname{Spec}(X,\tilde{\mu},T)$ is contained in
$\operatorname{Spec}(X,\mu,T)$. Since
$1/p\in\operatorname{Spec}(X,\tilde{\mu},T)$, we deduce that
$1/p\in\operatorname{Spec}(X,\mu,T)$, completing the proof.

###  7.5. Proof of Theorem 2.12

Part (i) follows from part (i) of Theorem 2.8.

We prove part (ii). Since $f\sim\chi$ for some primitive Dirichlet character
$\chi$ with conductor $q$, Proposition 6.2 implies that Furstenberg systems of
$f$ for Cesàro and logarithmic averages coincide. We are going to work with
logarithmic averages in order to have part (i) of Corollary 2.3 available to
us.

The inclusion $\operatorname{Spec}(X,\mu,T)\subset\Lambda$ follows from the
following stronger fact that applies to general multiplicative functions that
pretend to be Dirichlet characters.

Claim. Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be a multiplicative function
such that $f\sim\chi$ for some primitive Dirichlet character $\chi$ with
conductor $q$. If

$A:=\\{p\in{\mathbb{P}}\colon\text{either }p\mid q\text{ or
}f(p^{s})\neq\chi(p^{s})\text{ for some }s\in{\mathbb{N}}\\}$

and $\Lambda$ is the subgroup generated by $\\{1/p^{s}\colon p\in
A,s\in{\mathbb{N}}\\}$, then $\operatorname{Spec}(X,\mu,T)\subset\Lambda$.

Let us see how we prove the claim. Since by the first part $(X,\mu,T)$ is an
ergodic procyclic system, $\operatorname{Spec}(X,\mu,T)$ is a subgroup of the
rationals in ${\mathbb{T}}$. Any such subgroup is generated by elements of the
form $1/p^{s}$ for $p\in{\mathbb{P}}$ and $s\in{\mathbb{N}}$ (see, for
example, [43]). By Theorem 2.11, if $p\not\in A$, then
$1/p\not\in\operatorname{Spec}(X,\mu,T)$, hence
$1/p^{s}\not\in\operatorname{Spec}(X,\mu,T)$ for every $s\in{\mathbb{N}}$.
Combining the above facts, we deduce that $\operatorname{Spec}(X,\mu,T)$ is
contained in the subgroup generated by elements of the form $1/p^{s}$ for
$p\in A$, $s\in{\mathbb{N}}$, that is,
$\operatorname{Spec}(X,\mu,T)\subset\Lambda$. This proves the claim.

So, it remains to establish the inclusion
$\Lambda\subset\operatorname{Spec}(X,\mu,T)$. Since $f\sim\chi$ and for $p\in
A$ we either have $p\mid q$ or $f(p)\neq\chi(p)$, Theorem 2.11 implies that
$1/p\in\operatorname{Spec}(X,\mu,T)$ for every $p\in A$. Using this, our
assumption that $f(p)\neq 0$, and the fact that $f$ is completely
multiplicative, we deduce from part (i) of Corollary 2.3 that
$1/p^{s}\in\operatorname{Spec}(X,\mu,T)$ for every $p\in A$ and
$s\in{\mathbb{N}}$. By part (i) of Theorem 2.8, the system $(X,\mu,T)$ is
ergodic, hence its spectrum is a subgroup of ${\mathbb{T}}$. Combining the
above, we get $\Lambda\subset\operatorname{Spec}(X,\mu,T)$, completing the
proof.

###  7.6. Proof of Theorem 2.13

By Proposition 7.1, it suffices to show that under the stated assumptions, we
have $\alpha\not\in\operatorname{Spec}(X,\mu,T)$ for every Furstenberg system
$(X,\mu,T)$ of $f$ for Cesàro averages. In case (i), this follows from the
fact that $\alpha$ is irrational and $\operatorname{Spec}(X,\mu,T)$ contains
only rational values, which is an immediate consequence of Theorem 2.7. In
case (ii), this follows from Theorem 2.11.

###  7.7. Proof of Theorem 2.14

We use Lemma 6.3, in its more general version mentioned on the remark after
the lemma, and follow the notation there. Note that since
$\sum_{j=1}^{\ell}k_{j}=0$, in (37) we have $t^{\prime}=0$ and we can replace
the constant $e(\alpha)$ with $1$. The result then follows from the fact that
a sequence $a\colon{\mathbb{N}}\to{\mathbb{U}}$ is convergent if any two
subsequences of $a$ have further subsequences whose difference converges to
$0$.

###  7.8. Proof of Theorem 2.15

Suppose that $a\colon{\mathbb{N}}\to{\mathbb{U}}$ satisfies the Sarnak
conjecture for Cesàro averages along $(N_{k})$, and $b=a\cdot f$, where
$f\colon{\mathbb{N}}\to{\mathbb{U}}$ is a pretentious multiplicative function.
Let $w\colon{\mathbb{N}}\to{\mathbb{U}}$ be a completely deterministic
sequence along $(N_{k})$. By Theorem 2.7, the sequence $f$ is completely
deterministic. It follows from the remark made after Definition 3.9, that the
sequence $f\cdot w$ is also completely deterministic along $(N_{k})$. Hence,
our assumption on $a$ gives that

$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,a(n)\,f(n)\,w(n)=0.$

This shows that the sequence $a\cdot f$ satisfies the Sarnak conjecture for
Cesàro averages along $(N_{k})$ and completes the proof. A similar argument
works for logarithmic averages since Theorem 2.7 also applies to this setting.

###  7.9. Proof of Theorem 2.16

Suppose that $a\colon{\mathbb{N}}\to\\{-1,1\\}$ satisfies the Chowla-Elliott
conjecture for Cesàro averages along $(N_{k})$, and $b=a\cdot f$, where
$f\colon{\mathbb{N}}\to{\mathbb{U}}$ is a pretentious multiplicative function.
It suffices to show that

$\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,(a\cdot
f)^{\epsilon_{1}}(n+n_{1})\cdots(a\cdot f)^{\epsilon_{\ell}}(n+n_{\ell})=0$

for all $\ell\in{\mathbb{N}}$, distinct
$n_{1},\ldots,n_{\ell}\in{\mathbb{N}}$, and
$\epsilon_{1},\ldots,\epsilon_{\ell}\in\\{-1,1\\}$ (recall our notation
$f^{-1}:=\overline{f}$). If this fails, then for some $\ell\in{\mathbb{N}}$,
distinct $n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}_{+}$, and
$\epsilon_{1},\ldots,\epsilon_{\ell}\in\\{-1,1\\}$, we have convergence to a
nonzero constant, along a subsequence $(N_{k}^{\prime})$ of $(N_{k})$ on which
the Furstenberg systems of $a$ and $f$ are well defined. Our assumption
implies that the Furstenberg system of $a$ along $(N_{k}^{\prime})$ is
Bernoulli and by Theorem 2.7 the Furstenberg system of $f$ along
$(N_{k}^{\prime})$ has zero entropy. Hence, the two systems are disjoint. This
implies that the correlation along $(N_{k}^{\prime})$ of $a\cdot f$ that we
assumed to be non-zero, is equal to the product of the individual correlations
of $a$ and $f$, hence it is zero since $a$ satisfies the Chowla-Elliott
conjecture along $(N_{k}^{\prime})$. This is a contradiction. A similar
arguments works for logarithmic averages since Theorem 2.7 also applies to
this setting.

##  8\. Furstenberg systems of MRT functions - Cesàro averages

In this section we will prove structural results for Furstenberg systems of
MRT multiplicative functions (see Definition 2.3) when these systems are
defined using Cesàro averages. In particular we will prove Theorems 2.18 and
2.19.

Throughout this section and the next one we use the convention $e(t):=e^{it}$
for $t\in{\mathbb{R}}$.

###  8.1. Correlations of MRT functions for Cesàro averages

We will use the following result from [24, Lemmas 3.2-3.3] to rewrite
correlations of an MRT multiplicative function in a convenient form.

######  Lemma 8.1.

Let $f\colon{\mathbb{N}}\to\mathbb{S}^{1}$ be an MRT multiplicative function.
Following the terminology in Definition 2.3, let $N_{m}\to\infty$ with
$N_{m}\leq t_{m+1}$, $m\in{\mathbb{N}}$. Then

$\lim_{m\to\infty}\frac{|n\in[N_{m}]\colon|f(n)-n^{is_{m+1}}|>t_{m}^{-1}|}{N_{m}}=0.$

We also have the same conclusion when we consider the logarithmic density of
the set $\\{n\in[N_{m}]\colon|f(n)-n^{is_{m+1}}|>t_{m}^{-1}\\}$.

######  Remark.

The argument was given in [24] for the standard density, but it also applies
for the logarithmic density.

Let $(N_{m})$ be a strictly increasing sequence of integers with $t_{m}\leq
N_{m}\leq t_{m+1}$. In the computation of the correlations of the function $f$
on the interval $[N_{m}]$, the previous lemma allows us to replace $f(n)$ with
$n^{is_{m+1}}=e(s_{m+1}\log{n})$. We deduce that

(60)
$\lim_{m\to\infty}{\mathbb{E}}_{n\in[N_{m}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\lim_{m\to\infty}{\mathbb{E}}_{n\in[N_{m}]}\,e\big{(}s_{m+1}\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}$

holds for all $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$. Depending on how
we choose $N_{m}$ in relation with $s_{m+1}$ we get different Furstenberg
systems. In the following subsections we will use this formula to compute the
correlations of all MRT multiplicative functions for various choices of
$N_{m}$ and determine the structure of their corresponding Furstenberg systems
for Cesàro averages. This will lead to a proof of Theorems 2.18 and 2.19.

####  8.1.1. The case $N_{m}:=\lfloor\alpha s_{m+1}^{1/d}\rfloor$

The goal of this subsection is to compute the correlations of an MRT function
when we average over intervals $[N_{m}]$ that satisfy $N_{m}=\lfloor\alpha
s_{m+1}^{1/d}\rfloor$ for some $d\in{\mathbb{Z}}_{+}$ and $\alpha>0$. This is
the context of the next result.

######  Proposition 8.2.

Let $\alpha>0$, $d\in{\mathbb{N}}$. For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{m\to\infty}{\mathbb{E}}_{n\in[\alpha
s_{m+1}^{1/d}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}<d\\\
\int_{0}^{1}(G_{\alpha,d}(x))^{\sum_{j=1}^{\ell}k_{j}n_{j}^{d}}\,dx,&\quad\text{if
}\,i_{0}\geq d\end{cases}$

where

$G_{\alpha,d}(x):=e(1/(\alpha^{d}x^{d}))\quad\text{for }x\in(0,1).$

######  Remark.

Note that the function defined by $x\mapsto e(1/(\alpha
x^{d}))=e^{i/(\alpha^{d}x^{d})}$ for $x\neq 0$, and, say, $0\mapsto 0$, is
Riemann integrable in $[0,1]$, since it is bounded and Riemann integrable on
$[a,1]$ for every $a\in[0,1]$.

Proposition 8.2 is an immediate consequence of (60) and the next result.

######  Lemma 8.3.

Let $\alpha>0$, $d\in{\mathbb{N}}$. For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}<d\\\
\int_{0}^{1}(G_{\alpha,d}(x))^{\sum_{j=1}^{\ell}k_{j}n_{j}^{d}}\,dx,&\quad\text{if
}\,i_{0}\geq d\end{cases}$

where

$G_{\alpha,d}(x):=e(1/(\alpha^{d}x^{d}))\quad\text{for }x\in(0,1).$

###### Proof.

Let

$A:=\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}.$

For convenience, we approximate $\log(n+a)$ with $\log n$ plus a polynomial.
Since

$\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}\leq\frac{C_{a}}{M_{N}^{d+1}}\,\text{
for all }n\geq M_{N},$

we have

$\lim_{N\to\infty}\max_{n\in[N^{c},N^{1/d}]}N\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}=0$

for every $c\in{\mathbb{R}}_{+}$ such that $1/(d+1)<c<1/d$. So in order to
compute $A$ we can replace $\log(n+n_{j})$ with $\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{n_{j}^{i}}{n^{i}}$ throughout. Hence,

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log
n+\sum_{i=1}^{d}(-1)^{i-1}\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\,n^{-i}\big{)}\big{)}.$

Suppose first that $i_{0}<d$. Then $\sum_{j=1}^{\ell}k_{j}n_{j}^{i_{0}}\neq 0$
and by combining Corollary A.4 (it applies for this $i_{0}$ since
$N^{1/d}\prec N^{1/i_{0}}$) and Lemma 6.3, we get that $A=0$.

Suppose now that $i_{0}\geq d$. Then $\sum_{j=1}^{\ell}k_{j}n_{j}^{i}=0$ for
$i=0,\ldots,d-1$, and

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}KN/n^{d}\big{)},$

where

(61) $K:=\sum_{j=1}^{\ell}k_{j}n_{j}^{d}.$

Then

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}K\alpha^{-d}(\lfloor\alpha
N^{1/d}\rfloor+\epsilon(N))^{d}/n^{d}\big{)},$

where $\epsilon(N)\in\\{0,1\\}$. Note that the error we make by replacing
$(\lfloor\alpha N^{1/d}\rfloor+\epsilon(N))^{d}$ with $(\lfloor\alpha
N^{1/d}\rfloor)^{d}$ is bounded by $CN^{(d-1)/d}$ for some $C>0$, and this is
much smaller than $n^{d}$ for $n\in[N^{c},N^{1/d}]$ whenever
$1/d-1/d^{2}<c<1/d$. We deduce that $\epsilon(N)$ can be ignored in the
computation of the above limit without affecting the value of $A$. Hence,

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[\alpha
N^{1/d}]}\,e\big{(}K\alpha^{-d}\lfloor\alpha N^{1/d}\rfloor^{d}/n^{d}\big{)},$

where $K$ is as in (61). More conveniently,

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,e\big{(}K\alpha^{-d}N^{d}/n^{d}\big{)},$

assuming that the last limit exists, which is something we shall prove
shortly. The last limit can be rewritten as a limit of Riemann sums

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,(G_{\alpha,d}(n/N))^{K}$

where $G_{\alpha,d}\colon(0,1]\to\mathbb{S}^{1}$ is the Riemann integrable
function in the statement. Hence,

$A=\int_{0}^{1}(G_{\alpha,d}(x))^{K}\,dx.$

This completes the proof. ∎

####  8.1.2. The case $s_{m+1}^{1/(d+1)}\prec N_{m}\prec s_{m+1}^{1/d}$.

The goal of this subsection is to compute the correlations of an MRT function
when we average over intervals $[N_{m}]$ that satisfy $s_{m+1}^{1/(d+1)}\prec
N_{m}\prec s_{m+1}^{1/d}$ for some $d\in{\mathbb{Z}}_{+}$. This is the context
of the next result.

######  Proposition 8.4.

Let $d\in{\mathbb{Z}}_{+}$ be fixed. If $d=0$ suppose that $s_{m+1}\prec
N_{m}\leq t_{m+1}$ , and if $d\in{\mathbb{N}}$ suppose that
$s_{m+1}^{1/(d+1)}\prec N_{m}\prec s_{m+1}^{1/d}$. For every
$\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{m\to\infty}{\mathbb{E}}_{n\in[N_{m}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq d\\\ 1,&\quad\text{if }\,i_{0}>d.\end{cases}$

Proposition 8.4 is an immediate consequence of (60) and the next result.

######  Lemma 8.5.

Let $d\in{\mathbb{Z}}_{+}$ be fixed. Suppose that the sequence $(L_{N})$ of
positive integers satisfies $N\prec L_{N}$ if $d=0$, and $N^{1/(d+1)}\prec
L_{N}\prec N^{1/d}$ if $d\in{\mathbb{N}}$. For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[L_{N}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq d\\\ 1,&\quad\text{if }\,i_{0}>d.\end{cases}$

###### Proof.

Let $d\in{\mathbb{Z}}_{+}$ and $A$ be the limit we aim to compute. As in the
proof of Lemma 8.3 we start by approximating $\log(n+a)$ with $\log n$ plus a
polynomial. Since

$\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}\leq\frac{C_{a}}{M_{N}^{d+1}}\,\text{
for all }n\geq M_{N},$

and $N^{1/(d+1)}\prec L_{N}$, we have

$\lim_{N\to\infty}\max_{n\in[cL_{N},L_{N}]}N\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}=0$

for every positive $c\in(0,1)$. So using Lemma A.3, we get that in order to
compute $A$, we can replace $\log(n+n_{j})$ with $\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{n_{j}^{i}}{n^{i}}$ throughout.202020This is a
trivial but crucial reduction that does not work for logarithmic averages and
leads to different correlations and Furstenberg systems. Hence,

(62)
$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[L_{N}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log
n+\sum_{i=1}^{d}(-1)^{i-1}\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\,n^{-i}\big{)}\big{)}.$

We also get a similar identity with $d+1$ in place of $d$.

Suppose first that $i_{0}\leq d$. We have
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i_{0}}\neq 0$ and we get by combining identity
(62) with Corollary A.4 (it applies for this $i_{0}$ since $L_{N}\prec
N^{1/d}$, hence $L_{N}\prec N^{1/i_{0}}$) and Lemma 6.3, that $A=0$.

Suppose now that $i_{0}>d$. Then $\sum_{j=1}^{\ell}k_{j}n_{j}^{i}=0$ for
$i=0,\ldots,d$. In this case, using identity (62) with $d+1$ in place of $d$
we have

$A=\lim_{N\to\infty}{\mathbb{E}}_{n\in[L_{N}]}\,e\big{(}K\,N/n^{d+1}\big{)},$

where $K:=(-1)^{d}\,\sum_{j=1}^{\ell}k_{j}n_{j}^{d+1}$. Our growth assumption
$N^{1/(d+1)}\prec L_{N}$ implies that for all $c\in(0,1)$ we have

$\lim_{N\to\infty}\max_{n\in[cL_{N},L_{N}]}(N/n^{d+1})=0.$

It follows that

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[cL_{N},L_{N}]}\,e\big{(}K\,N/n^{d+1}\big{)}=1$

for all $c\in(0,1)$. Hence, $A=1$ by Lemma A.3.212121This last conclusion
fails badly for logarithmic averages, which is the reason why similar
computations for logarithmic averages lead to very different expressions. This
completes the proof. ∎

###  8.2. Ergodic models of MRT functions for Cesàro correlations

Having computed the correlations of MRT functions for certain ranges of the
parameter $N_{m}$ we would like to identify simple measure preserving systems
and functions that reproduce these correlations. When we manage to do this, it
will be an easy matter to show isomorphism between this system and the
Furstenberg system of the MRT function.

####  8.2.1. The case $N_{m}:=\lfloor\alpha s_{m+1}^{1/d}\rfloor$.

The next lemma identifies systems and functions that reproduce the
correlations of MRT functions when $N_{m}:=\lfloor\alpha
s_{m+1}^{1/d}\rfloor$.

######  Lemma 8.6.

For $\alpha>0$ and $d\in{\mathbb{Z}}_{+}$, let
$({\mathbb{T}}^{d+1},S_{\alpha,d},m_{{\mathbb{T}}^{d+1}})$ be the system given
in Definition 2.6. Let also $F\colon{\mathbb{T}}^{d+1}\to\mathbb{S}^{1}$ be
defined by

$F(x_{0},\ldots,x_{d}):=e(x_{d}).$

For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

(63)
$\int\prod_{j=1}^{\ell}S_{\alpha,d}^{n_{j}}F^{k_{j}}\,dm_{{\mathbb{T}}^{d+1}}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}<d\\\
\int_{0}^{1}(G_{\alpha,d}(x))^{\sum_{j=1}^{\ell}k_{j}n_{j}^{d}}\,dx,&\quad\text{if
}\,i_{0}\geq d,\end{cases}$

where

$G_{\alpha,d}(x):=e(1/(\alpha^{d}x^{d}))\quad\text{for }x\in(0,1).$

###### Proof.

For $x=(x_{0},\ldots,x_{d})\in{\mathbb{T}}^{d+1}$ direct computation gives
that

$F(S_{\alpha,d}^{n}x)=e\Big{(}\sum_{i=0}^{d-1}\binom{n}{i}x_{d-i}+\binom{n}{d}\,g_{\alpha,d}(x_{0})\Big{)},\qquad
n\in{\mathbb{N}}.$

Hence, for $x\in{\mathbb{T}}^{d+1}$ we have

$\prod_{j=1}^{\ell}F^{k_{j}}(S_{\alpha,d}^{n_{j}}x)=e\Big{(}\sum_{i=0}^{d-1}c_{i}\cdot
x_{d-i}\Big{)}\cdot(G_{\alpha,d}(x_{0}))^{c_{d}},$

where

$c_{i}:=\sum_{j=1}^{\ell}k_{j}\binom{n_{j}}{i},\quad i=0,\ldots,d.$

Note that $c_{i}=0$ for $i=0,\ldots,d-1$ if and only if
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}=0$ for $i=0,\ldots,d-1$, equivalently when
$i_{0}\geq d$, and in this case we have
$c_{d}=\sum_{j=1}^{\ell}k_{j}n_{j}^{d}$. It follows that the integral in (63)
is equal to
$\int_{0}^{1}(G_{\alpha,d}(x))^{\sum_{j=1}^{\ell}k_{j}\binom{n_{j}}{d}}\,dx$
if $i_{0}\geq d$, and is equal to $0$ otherwise, that is, when $i_{0}<d$. This
completes the proof. ∎

####  8.2.2. The case $s_{m+1}^{1/(d+1)}\prec N_{m}\prec s_{m+1}^{1/d}$.

Repeating the computation in the proof of Lemma 8.6 with the function $x_{0}$
in place of $g_{\alpha,d}(x_{0})$, gives the following result.

######  Lemma 8.7.

For $d\in{\mathbb{Z}}_{+}$ let
$({\mathbb{T}}^{d+1},S_{d},m_{{\mathbb{T}}^{d+1}})$ be the level $d$ unipotent
system given in Definition 2.4. Let also
$F\colon{\mathbb{T}}^{d+1}\to\mathbb{S}^{1}$ be defined by

$F(x_{0},x_{1},\ldots,x_{d}):=e(x_{d}).$

For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\int\prod_{j=1}^{\ell}S_{d}^{n_{j}}F^{k_{j}}\,dm_{{\mathbb{T}}^{d+1}}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq d\\\ 1,&\quad\text{if }\,i_{0}>d.\end{cases}$

###  8.3. Furstenberg systems of MRT functions for Cesàro averages

We are now ready to prove our main results regarding Furstenberg systems of
MRT functions for Cesàro averages.

###### Proof of Theorem 2.19.

We first show that the system $(X,\mu_{\alpha,d},T)$ is isomorphic to the
system $({\mathbb{T}}^{d+1},m_{{\mathbb{T}}^{d+1}},S_{\alpha,d})$.

Recall that $X={\mathbb{U}}^{\mathbb{Z}}$. Let $F_{0}\colon
X\to\mathbb{S}^{1}$ be the $0^{\text{th}}$-coordinate projection and
$F\colon{\mathbb{T}}^{d+1}\to\mathbb{S}^{1}$ be as in Lemma 8.6. Combining
Proposition 8.2 with Lemma 8.6, we deduce that

$\lim_{m\to\infty}{\mathbb{E}}_{n\in[\alpha
s_{m+1}^{1/d}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\int\prod_{j=1}^{\ell}S_{\alpha,d}^{n_{j}}F^{k_{j}}\,dm_{{\mathbb{T}}^{d+1}}.$

Hence,

(64)
$\int\prod_{j=1}^{\ell}T^{n_{j}}F_{0}^{k_{j}}\,d\mu_{\alpha,d}=\int\prod_{j=1}^{\ell}S_{\alpha,d}^{n_{j}}F^{k_{j}}\,dm_{{\mathbb{T}}^{d+1}}$

holds for every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$.

Next, we define the map $\Phi\colon{\mathbb{T}}^{d+1}\to X$ by

$\Phi(y):=(F(S_{\alpha,d}^{n}y))_{n\in{\mathbb{Z}}},\quad
y\in{\mathbb{T}}^{d+1}.$

We clearly have $\Phi\circ S_{\alpha,d}=T\circ\Phi$. Moreover, a direct
computation shows that the map $\Phi$ is injective. Lastly, since
$F=F_{0}\circ\Phi$, identity (64) implies that

$\int\prod_{j=1}^{\ell}T^{n_{j}}F_{0}^{k_{j}}\,d\mu_{\alpha,d}=\int\prod_{j=1}^{\ell}T^{n_{j}}F_{0}^{k_{j}}\,d(m_{{\mathbb{T}}^{d+1}}\circ\Phi^{-1}).$

Hence, $\mu_{\alpha,d}=m_{{\mathbb{T}}^{d+1}}\circ\Phi^{-1}$, since a linearly
dense subset of $C(X)$ has the same integral with respect to the two measures
(see second remark after Definition 3.5). This establishes the asserted
isomorphism.

Since the system $({\mathbb{T}}^{d+1},m_{{\mathbb{T}}^{d+1}},S_{\alpha,d})$
has trivial rational spectrum, so does the system $(X,\mu_{\alpha,d},T)$.

To prove that the system $(X,\mu_{\alpha,d},T)$ is not strongly stationary, it
suffices to show that

$\int\overline{F_{0}}\cdot TF_{0}\,d\mu\neq\int\overline{F_{0}}\cdot
T^{r}F_{0}\,d\mu$

for some $r\in{\mathbb{N}}$. Equivalently, by applying (64) (for $\ell=2$,
$n_{1}=0$, $n_{2}=r$, $k_{1}=-1$, $k_{2}=1$) and (63), it suffices to show
that for fixed $\alpha,d$ we have

$\int_{0}^{1}e(1/(\alpha^{d}x^{d}))\,dx\neq\int_{0}^{1}e(r/(\alpha^{d}x^{d}))\,dx$

for some non-zero $r\in{\mathbb{Z}}$. Direct computation shows that this is
the case, for example when $\alpha=d=1$ and $r=2$. ∎

###### Proof of Theorem 2.18.

We combine Proposition 8.4 with Lemma 8.7, and argue as in the proof of
Theorem 2.19. ∎

##  9\. Furstenberg systems of MRT functions - Logarithmic averages

In this section we will prove structural results for Furstenberg systems of
MRT multiplicative functions (see Definition 2.3) when these systems are
defined using logarithmic averages. In particular, we will prove Theorem 2.20.
In the case of logarithmic averages the correlations of MRT functions turn out
to be different than those for Cesàro averages, and this leads to
substantially different Furstenberg systems.

###  9.1. Correlations of MRT functions for logarithmic averages

The main goal of this subsection is to give in Proposition 9.1 an explicit
description of the correlations for logarithmic averages of MRT functions when
we average over sequences of intervals that grow as fractional powers of
$s_{m+1}$, a notion that we define next. Again, using Lemma 8.1 our starting
point is the following identity

(65)
$\lim_{m\to\infty}\mathbb{E}^{\log}_{n\in[N_{m}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\lim_{m\to\infty}\mathbb{E}^{\log}_{n\in[N_{m}]}\,e\big{(}s_{m+1}\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)},$

which holds for all $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$. As in the case
of Cesàro averages, depending on how we choose $N_{m}$ in relation with
$s_{m+1}$ we get different Furstenberg systems, and the next notion will help
us identify which sequences $N_{m}$ give rise to which systems.

######  Definition 9.1.

We define the fractional degree of a sequence of positive real numbers
$L_{N}\to+\infty$ to be

$\lim_{N\to\infty}\frac{\log{L_{N}}}{\log{N}}$

if the limit exists (could be $+\infty$).

######  Remarks.

$\bullet$ Note that

$\lim_{N\to\infty}\frac{\log{L_{N}}}{\log{N}}=\begin{cases}0,\quad&\text{if
}\,L_{N}\prec N^{\varepsilon}\text{ for every }\varepsilon>0\\\
c\in(0,+\infty),\quad&\text{if }\,N^{c-\varepsilon}\prec L_{N}\prec
N^{c+\varepsilon}\text{ for every }\varepsilon>0\\\ +\infty,\quad&\text{if
}\,N^{d}\prec L_{N}\text{ for every }d\in{\mathbb{N}}.\end{cases}$

$\bullet$ If $c\in(0,+\infty)$, then the sequence $(N^{c})$ has fractional
degree $c$. More generally, the same holds for the sequence
$(N^{c}(\log{N})^{b}+L^{\prime}_{N})$ for every $b\in{\mathbb{R}}$ and every
$(L^{\prime}_{N})$ with $L^{\prime}_{N}\prec N^{c}$.

######  Proposition 9.1.

Let $N_{m}:=s_{m+1}^{1/c}$ for some $c>0$. For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell}$, $n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$
be the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{m\to\infty}\mathbb{E}^{\log}_{n\in[N_{m}]}\prod_{j=1}^{\ell}f^{k_{j}}(n+n_{j})=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq c\\\ 1-c/i_{0},&\quad\text{if }\,c<i_{0}<+\infty\\\
1,&\quad\text{if }\,i_{0}=+\infty.\end{cases}$

Furthermore, the same identity holds if $(L_{N})$ is a sequence of positive
real numbers with fractional degree $1/c$ and $N_{m}:=L_{s_{m+1}}$.

######  Remark.

By taking $c:=1/2$ (or any other $c<1$) we get the following non-vanishing
property of the $2$-point correlations of $f$

$\lim_{m\to\infty}\mathbb{E}^{\log}_{n\in[s_{m+1}^{2}]}\ \overline{f(n)}\cdot
f(n+h)=\frac{1}{2}$

for every non-zero $h\in{\mathbb{Z}}$.

Proposition 9.1 is an immediate consequence of identity (65) and the next
result.

######  Lemma 9.2.

Let $(L_{N})$ be a sequence of positive real numbers with fractional degree
$1/c$ for some $c\in(0,+\infty)$. For every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq c\\\ 1-c/i_{0},&\quad\text{if }\,c<i_{0}<+\infty\\\
1,&\quad\text{if }\,i_{0}=+\infty.\end{cases}$

######  Remarks.

$\bullet$ For example, if $A$ is the value of the limit on the left hand side,
then for $c\in[1,2)$ (in which case $L_{N}$ has fractional degree in
$(1/2,1]$) we have $A=0$ if either $\sum_{j=1}^{\ell}k_{j}\neq 0$ or
$\sum_{j=1}^{\ell}k_{j}n_{j}\neq 0$ (since $i_{0}\leq 1\leq c$); we have
$A=1-c/2$ if $\sum_{j=1}^{\ell}k_{j}=\sum_{j=1}^{\ell}k_{j}n_{j}=0$ and
$\sum_{j=1}^{\ell}k_{j}n_{j}^{2}\ \neq 0$ (since $c<i_{0}=2$), and so on.

$\bullet$ Note that the limit $A$ remains the same if we replace
$n_{1},\ldots,n_{\ell}$ with $rn_{1},\ldots,rn_{\ell}$, for every
$r\in{\mathbb{N}}$ and $\ell\in{\mathbb{N}}$,
$n_{1}\ldots,n_{\ell}\in{\mathbb{Z}}$. Hence, the Furstenberg system of the
multiplicative function that has such correlations is strongly stationary.
This is unlike the case of Cesàro averages, where for $L_{N}:=N^{1/k}$,
$k\in{\mathbb{N}}$, we got that the corresponding correlations given in Lemma
8.6 were not dilation invariant.

###### Proof.

In order to have a specific example in mind, in the course of the proof the
reader may find it convenient to assume that $L_{N}=N^{1/c}$ where $c>0$. We
denote by $A$ the limit of the averages we want to compute.

We shall use the following basic estimate that follows by applying the Taylor-
Lagrange theorem for the function $\log{x}$: For every $a\in{\mathbb{R}}_{+}$,
$d\in{\mathbb{N}}$, and $M_{N}>0$, we have

(66) $\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{d}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}\leq\frac{C_{a}}{M_{N}^{d+1}}\,\text{
for all }n\geq M_{N}.$

The utility of this approximation is that it enables us to connect various
exponential sums that appear below to those treated in Corollary A.6.

Case 1 ($i_{0}\leq c$). If $i_{0}\leq c$, we claim that $A=0$. Let
$\varepsilon>0$. We also take $\varepsilon<1$. We apply (66) for
$d_{\varepsilon}:=[1/\varepsilon]$, $a:=n_{j}$, $j=1,\ldots,\ell$, and use
that $\lim_{N\to\infty}(N/M_{N}^{d_{\varepsilon}+1})=0$ for
$M_{N}:=N^{\varepsilon}$. We get that

$\lim_{N\to\infty}\max_{n\in[N^{\varepsilon},L_{N}]}N\Big{|}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})-\sum_{j=1}^{\ell}k_{j}\,\log{n}+\sum_{i=1}^{d_{\varepsilon}}(-1)^{i-1}\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\,n^{-i}\Big{|}=0.$

Using this and the estimate $\mathbb{E}^{\log}_{n\in[N]}\,{\bf
1}_{[1,N^{\varepsilon}]}(n)\leq\varepsilon$, we get that for

$A_{\varepsilon}:=\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\,\log{n}+\sum_{i=1}^{d_{\varepsilon}}(-1)^{i-1}\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\,n^{-i}\big{)}\big{)},$

we have

$|A-A_{\varepsilon}|\leq\varepsilon.$

So in order to show that $L=0$, it suffices to show that for every
$\varepsilon>0$ we have $A_{\varepsilon}=0$. Equivalently, it suffices to show
that (for every $\varepsilon>0$)

$\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\,\log{n}+\sum_{i=1}^{d_{\varepsilon}}(-1)^{i-1}\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\,n^{-i}\big{)}\big{)}=0.$

Recall that the defining property of $i_{0}$ implies that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}=0$ for $i=0,\ldots,i_{0}-1$. Since $c\geq
i_{0}$ and the fractional degree of $L_{N}$ is $1/c$ (so positive in
particular), we have $N^{\gamma}\prec L_{N}\prec
N^{\frac{1}{i_{0}}+\varepsilon}$ for some $\gamma>0$ and every
$\varepsilon>0$. Hence, Corollary A.6 applies and gives $A=0$.

Case 2 $(i_{0}>c)$. We claim that $A=1-c/i_{0}$ if $i_{0}$ is finite and $L=1$
if $i_{0}=+\infty$. Suppose first that $i_{0}$ is finite.

We decompose

$A=A_{1}+A_{2},$

where

$A_{1}:=\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,{\bf
1}_{[1,N^{1/i_{0}}]}(n)\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}$

and

$A_{2}:=\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[N^{1/c}]}\,{\bf
1}_{[N^{1/i_{0}},L_{N}]}(n)\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}.$

(Note that since $L_{N}$ has fractional degree $1/c>1/i_{0}$ we have
$N^{1/i_{0}}\prec L_{N}$ and the logarithmic averages over $[L_{N}]$ and
$[N^{1/c}]$ coincide.)

Case 2a (Computation of $A_{1}$). We claim that $A_{1}=0$. Indeed, we have

$A_{1}=c\cdot
i_{0}^{-1}\cdot\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[N^{1/i_{0}}]}\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}$

and by Case 1 the last limit is $0$.

Case 2b (Computation of $A_{2}$). It remains to show that $A_{2}=1-c/i_{0}$.
For $\varepsilon>0$ such that $1/i_{0}+\varepsilon<1/c$, let

(67)
$A_{2,\varepsilon}:=\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[N^{1/c}]}\,{\bf
1}_{[N^{1/i_{0}+\varepsilon},N^{1/c}]}(n)\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}.$

Since $A_{2}=\lim_{\varepsilon\to 0^{+}}A_{2,\varepsilon}$, it suffices to
compute $A_{2,\varepsilon}$ for these values of $\varepsilon>0$.

We apply (66) for $d:=i_{0}-1$ (which is $\geq 0$ since $i_{0}\geq 1$),
$a=n_{j}$, $j=1,\ldots,\ell$, and $M_{N}:=N^{1/i_{0}+\varepsilon}$ for
$\varepsilon$ small enough so that $M_{N}\prec L_{N}$. Note that since
$\lim_{N\to\infty}(N/M_{N}^{d+1})=0$, we have

$\lim_{N\to\infty}\max_{n\in[N^{1/i_{0}+\varepsilon},L_{N}]}N\Big{|}\log(n+a)-\log
n-\sum_{i=1}^{i_{0}-1}(-1)^{i-1}\frac{a^{i}}{n^{i}}\Big{|}=0.$

(If $i_{0}=1$ the sum over $i$ is empty.) For $j=1,\ldots,\ell$, we apply
these identities for $a:=n_{j}$, multiply them by $k_{j}$, and add them up. We
deduce using the defining property of $i_{0}$
($\sum_{j=1}^{\ell}k_{j}n_{j}^{i}=0$ for $i=0,\ldots,i_{0}-1$), that

$\lim_{N\to\infty}\max_{n\in[N^{1/i_{0}+\varepsilon},L_{N}]}N\Big{|}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\Big{|}=0.$

Hence,

$\lim_{N\to\infty}\max_{n\in[N^{1/i_{0}+\varepsilon},L_{N}]}\Big{|}e\big{(}N\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}-1\big{|}=0.$

If we combine this with (67) we get (we use here that the fractional degree of
$L_{N}$ is $1/c$ and the fact that we use logarithmic averages)

$A_{2,\varepsilon}=\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,{\bf
1}_{[N^{1/i_{0}+\varepsilon},L_{N}]}(n)=1-c/i_{0}-c\varepsilon.$

Hence, $A_{2}=\lim_{\varepsilon\to 0^{+}}A_{2,\varepsilon}=1-c/i_{0}$. This
completes the proof of the Case 2 when $i_{0}$ is finite.

If $i_{0}=+\infty$ (in this case $A=A_{2}$), then the previous argument gives
for every $i_{0}\in{\mathbb{N}}$ the following identity

$\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,{\bf
1}_{[N^{1/j_{0}},L_{N}]}(n)\,e\big{(}N\big{(}\sum_{j=1}^{\ell}k_{j}\log(n+n_{j})\big{)}\big{)}=1-c/j_{0}.$

Letting $j_{0}\to+\infty$ gives that $A=A_{2}=1$, as required. ∎

###  9.2. Ergodic models of MRT functions for logarithmic correlations

As was the case in Section 8.2, our goal is to identify simple measure
preserving systems and functions that reproduce the correlations in the number
theory setting obtained in Proposition 9.1. With a bit of guesswork we get
that the systems in Definition 2.7 help us do the job; in fact the next lemma
motivated their definition.

######  Lemma 9.3.

Let $c>0$. For $d\geq\lfloor c\rfloor$ let $(Y_{d},\nu_{d},S_{d})$ be the
level $d$ unipotent system given in Definition 2.4 and
$F_{d}\colon{\mathbb{T}}^{d+1}\to\mathbb{S}^{1}$ be defined by

$F_{d}(x_{0},\ldots,x_{d}):=e(x_{d}).$

Let also the system $(Z_{c},\nu^{\prime}_{c},R_{c})$ be as in Definition 2.7
and $G_{c}\in L^{\infty}(\nu^{\prime}_{c})$ be defined by

$G_{c}:=\sum_{d=\lfloor c\rfloor}^{\infty}{\bf 1}_{Y_{d}}\cdot F_{d}.$

Lastly, for every $\ell\in{\mathbb{N}}$,
$k_{1},\ldots,k_{\ell},n_{1},\ldots,n_{\ell}\in{\mathbb{Z}}$, let $i_{0}$ be
the minimum $i\in{\mathbb{Z}}_{+}$ such that
$\sum_{j=1}^{\ell}k_{j}n_{j}^{i}\neq 0$ and $i_{0}:=+\infty$ if no such $i$
exists. Then

$\int\prod_{j=1}^{\ell}R_{c}^{n_{j}}G_{c}^{k_{j}}\,d\nu^{\prime}_{c}=\begin{cases}0,&\quad\text{if
}\,0\leq i_{0}\leq c\\\ 1-c/i_{0},&\quad\text{if }\,c<i_{0}<+\infty\\\
1,&\quad\text{if }\,i_{0}=+\infty.\end{cases}$

###### Proof.

We have that

$\int\prod_{j=1}^{\ell}R_{c}^{n_{j}}G_{c}^{k_{j}}\,d\nu^{\prime}_{c}=\Big{(}1-\frac{c}{\lceil
c\rceil}\Big{)}C_{\lfloor c\rfloor}+c\sum_{d=\lceil
c\rceil}^{\infty}\Big{(}\frac{1}{d}-\frac{1}{d+1}\Big{)}\,C_{d},$

where for $d\in{\mathbb{Z}}_{+}$ we let

$C_{d}:=\int\prod_{j=1}^{\ell}S_{d}^{n_{j}}F_{d}^{k_{j}}\,d\nu_{d}.$

By Lemma 8.7 we have

$C_{d}=\begin{cases}0,&\quad\text{if }\,0\leq i_{0}\leq d\\\ 1,&\quad\text{if
}\,i_{0}>d.\end{cases}$

The asserted identity follows by combining the previous identities and simple
direct computation. ∎

###  9.3. Furstenberg systems of MRT functions for logarithmic averages

Combining the previous results it is now easy to prove our main result
regarding the structure of Furstenberg systems of MRT functions for
logarithmic averages.

###### Proof of Theorem 2.20.

We combine Proposition 9.1 with Lemma 9.3, and argue as in the proof of
Theorem 2.19. ∎

##  Appendix A Exponential sum estimates

In this appendix we gather some simple facts and exponential sum estimates
used in the proofs of the results regarding the structure of Furstenberg
systems of MRT multiplicative functions.

###  A.1. Cesàro averages

The next result is an immediate consequence of [25, Theorem 2.9].

######  Theorem A.1 (Kuzmin-van der Corput).

Let $q\geq 2$ be an integer and $c\in(0,1)$. Suppose that $C_{1},C_{2}$ are
constants (depending on $q$ and $c$ only) and $h\in C^{q}([1,+\infty))$ be a
function, such that for some $L,M>0$, we have

$C_{1}\,M\,L^{-r}\leq\max_{x\in[cL,L]}|h^{(r)}(x)|\leq C_{2}\,M\,L^{-r}$

for $r=1,\ldots,q$. Then there exists a positive constant $C_{3}$, depending
only on $c$, $C_{1}$, $C_{2}$ (and not on $L,M$), such that

$\big{|}{\mathbb{E}}_{n\in[cL,L]}\,e(h(n))\big{|}\leq
C_{3}((M/L^{q})^{1/Q}+1/M),$

where $Q:=4\cdot 2^{q-2}-2$.

We are only going to use the following consequence that applies to a special
class of functions $g$ appearing in our arguments.

######  Corollary A.2.

Let $g(x):=c_{0}\,\log x+\sum_{i=1}^{k}c_{i}x^{-i}$ for
$x\in{\mathbb{R}}_{+}$. Let also $i_{0}$ be the minimum $i\in\\{0,\ldots,k\\}$
such that $c_{i}\neq 0$. Then for every $q\geq 2$ and $c\in(0,1)$, there
exists a constant $C_{c,q,g}$ such that the following holds: If $L>0$ is large
enough (depending only on $c,q,g$), then for every $N>0$ we have

$\big{|}{\mathbb{E}}_{n\in[cL,L]}\,e(N\,g(n))\big{|}\leq
C_{c,q,g}\big{(}(N/L^{q+i_{0}})^{1/Q}+L^{i_{0}}/N\big{)},$

where $Q:=4\cdot 2^{q-2}-2$.

###### Proof.

Let $N>0$, $c>0$, and $h_{N}(x):=Ng(x)$. We have

$h_{N}^{(r)}(x)=N\big{(}\sum_{i=0}^{k}\,c_{i}\,c_{r,i}\,x^{-i-r}\big{)}$

for every $r\in{\mathbb{N}}$ for some non-zero constants
$c_{0,r},\ldots,c_{k,r}$. Bounding the derivatives of the function inside the
parenthesis (which does not depend on $N$), we get that there exist
$L_{0}\in{\mathbb{N}}$ and $C_{1},C_{2}>0$, depending only on $c,g,r$ but not
on $N$, such that for all $L\geq L_{0}$ we have (we also use the defining
property of $i_{0}$ here)

$C_{1}\,NL^{-i_{0}-r}\leq\max_{x\in[cL,L]}|h_{N}^{(r)}(x)|\leq
C_{2}\,NL^{-i_{0}-r}$

for $r=1,\ldots,q$. Using Theorem A.1 for $h:=h_{N}$ and $M:=NL^{-i_{0}}$, we
get that the asserted estimate holds for all $L\geq L_{0}$. ∎

######  Lemma A.3.

Let $L_{N}\to\infty$ be a sequence of positive integers. For
$N\in{\mathbb{N}}$ let $a_{N}\colon[L_{N}]\to{\mathbb{U}}$ be finite sequences
such that for every small enough $c>0$ we have

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[cL_{N},L_{N}]}\ a_{N}(n)=0.$

Then

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[L_{N}]}\,a_{N}(n)=0.$

######  Remark.

We crucially use here that $\frac{1}{L_{N}}\sum_{n\in[1,cL_{N}]}|a_{N}(n)|\leq
c$ and this converges to $0$ as $c\to 0^{+}$. An analogous property fails for
logarithmic averages, so this convergence criterion cannot be used for
logarithmic averages.

Combining the above, we get the following qualitative result that we will use
in subsequent sections.

######  Corollary A.4.

Let $g(x)=c_{0}\,\log x+\sum_{i=1}^{k}c_{i}x^{-i}$ be non-zero and $i_{0}$ be
the minimum $i\in\\{0,\ldots,k\\}$ such that $c_{i}\neq 0$. Let also $(L_{N})$
be a sequence of positive real numbers that satisfies $N^{\gamma}\prec
L_{N}\prec N^{1/i_{0}}$ for some $\gamma>0$. Then

$\lim_{N\to\infty}\,{\mathbb{E}}_{n\in[L_{N}]}\,e(N\,g(n))=0.$

######  Remark.

If $i_{0}=0$, then our only growth assumption is $N^{\gamma}\prec L_{N}$ for
some $\gamma>0$.

###### Proof.

Let $c\in(0,1)$. Choose $q\geq 2$ such that $(q+i_{0})\gamma>1$. Using
Corollary A.2 and our growth assumption on $L_{N}$, we get that

$\lim_{N\to\infty}\,{\mathbb{E}}_{n\in[cL_{N},L_{N}]}\,e(N\,g(n))=0.$

We deduce from this using Lemma A.3 that the asserted convergence to $0$
holds. ∎

###  A.2. Logarithmic averages

The next lemma will enable us to deduce convergence results for logarithmic
averages of variable sequences from corresponding results for Cesàro averages
once we have some additional uniformity.

######  Lemma A.5.

For $N\in{\mathbb{N}}$, let $L_{N}$ be a sequence of positive integers that
satisfies $N^{\gamma}\prec L_{N}$ for some $\gamma>0$, and let
$a_{N}\colon[L_{N}]\to{\mathbb{U}}$ be finite sequences. Suppose that for all
small enough $c,\delta>0$ we have

(68)
$\lim_{N\to\infty}\sup_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\big{|}{\mathbb{E}}_{k\in[cn,n]}\,a_{N}(k)\big{|}=0.$

Then

(69) $\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}]}\,a_{N}(n)=0.$

###### Proof.

We first note that our assumption (68) implies

(70)
$\lim_{N\to\infty}\sup_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\big{|}{\mathbb{E}}_{k\in[n]}\,a_{N}(k)\big{|}=0.$

Indeed, one immediately verifies that the difference between the two
expressions is bounded by a function of $c$ that converges to $0$ as $c\to
0^{+}$.

Note also that in order to show (69), it suffices to show that for all small
enough $\delta>0$ we have

(71)
$\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\,a_{N}(n)=0.$

Indeed, because of the logarithmic averaging, the difference between this
limit and the one in (69) is bounded by a function of $\delta$ that converges
to $0$ as $\delta\to 0^{+}$.

Now let $\varepsilon>0$ and $\delta>0$ be small enough. Equation (70) implies
that there exists $N_{0}$ such that for $N>N_{0}$ we have

(72)
$\sup_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\big{|}{\mathbb{E}}_{k\in[n]}\,a_{N}(k)\big{|}\leq\varepsilon.$

Using partial summation, we get for $N\geq N_{0}$ that

$\Big{|}\sum_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\frac{a_{N}(n)}{n}\Big{|}\leq
2+\sum_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\Big{|}\frac{1}{n^{2}}\sum_{k=1}^{n}\,a_{N}(k)\Big{|}\leq
2+\sum_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\frac{\varepsilon}{n},$

where the last estimate follows from (72). Note also that since
$N^{\gamma}\prec L_{N}$, we have
$\lim_{N\to\infty}\sum_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\,\frac{1}{n}=+\infty$.
We deduce that

$\limsup_{N\to\infty}\big{|}\mathbb{E}^{\log}_{n\in[L_{N}^{\delta},L_{N}^{1-\delta}]}\,a_{N}(n)\big{|}\leq\varepsilon.$

Since $\varepsilon$ is arbitrary, this implies (71) and completes the proof. ∎

######  Corollary A.6.

Let $g(x):=c_{0}\,\log x+\sum_{i=1}^{k}c_{i}x^{-i}$ be non-zero and $i_{0}$ be
the minimum $i\in\\{0,\ldots,k\\}$ such that $c_{i}\neq 0$. Let also $(L_{N})$
be a sequence of positive real numbers that satisfies $N^{\gamma}\prec
L_{N}\prec N^{\frac{1}{i_{0}}+\varepsilon}$ for some $\gamma>0$ and for every
$\varepsilon>0$. Then

$\lim_{N\to\infty}\,\mathbb{E}^{\log}_{n\in[L_{N}]}\,e(N\,g(n))=0.$

######  Remarks.

$\bullet$ If $i_{0}=0$, then our only growth assumption is $N^{\gamma}\prec
L_{N}$ for some $\gamma>0$.

$\bullet$ If $c\in(0,1/i_{0}]$, then the assumptions are satisfied when
$L_{N}:=N^{c}$ or $L_{N}:=N^{c}\log N$. If $i_{0}=0$, then the assumptions are
satisfied when $L_{N}=N^{c}$ for all $c\in(0,+\infty)$. Note that by Corollary
A.4, if we use Cesàro averages, then a similar result holds for
$c\in(0,1/i_{0})$ but not for $c=1/i_{0}$. For example, although
$\lim_{N\to\infty}\mathbb{E}^{\log}_{n\in[N]}\,e(N/n)=0$, we have
$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,e(N/n)\neq 0$.

###### Proof.

Let $c\in(0,1)$ and $\delta>0$ be arbitrary. Let $\gamma>0$ be such that
$N^{\gamma}\prec L_{N}$. Choose $q\geq 2$ such that $(q+i_{0})\gamma\delta>1$
and $\varepsilon>0$ such that $(1+\varepsilon)(1-\delta)<1$. Using Corollary
A.2 and our growth assumption on $L_{N}$, we get that there exists a constant
$C_{c,q}$ such that for all large enough $N$ we have

$\sup_{L_{N}^{\delta}\leq n\leq
L_{N}^{1-\delta}}\big{|}{\mathbb{E}}_{k\in[cn,n]}\,e(Ng(n))\big{|}\leq
C_{c,q}\big{(}N/N^{(q+i_{0})\gamma\delta})^{1/Q}+N^{(1+\varepsilon)(1-\delta)}/N\big{)},$

where $Q:=4\cdot 2^{q-2}-2$ if $i_{0}\geq 1$, and a similar estimate with $1$
in place of $N^{(1+\varepsilon)(1-\delta)}$ if $i_{0}=0$. Since
$(q+i_{0})\gamma\delta>1$ and $(1+\varepsilon)(1-\delta)<1$, we get

$\lim_{N\to\infty}\sup_{L_{N}^{\delta}\leq n\leq
L_{N}^{1-\delta}}\big{|}{\mathbb{E}}_{k\in[cn,n]}\,e(Ng(n))\big{|}=0.$

Applying Lemma A.5, we deduce

$\lim_{N\to\infty}\,\mathbb{E}^{\log}_{n\in[L_{N}]}\,e(N\,g(n))=0$

completing the proof. ∎

##  Appendix B Factor of a product Erg$\times$Id

In this appendix we prove an ergodic result that was used in the proof of part
(ii) of Theorem 2.8.

Our setting is the following: We consider two systems $(Y,\nu,S)$ and
$(Z,\lambda,\operatorname{id})$, where

  * •

$Y:={\mathbb{U}}^{\mathbb{Z}}$, $S\colon Y\to Y$ is the shift map, and $\nu$
is an ergodic $S$-invariant probability measure.

  * •

$Z:=\mathbb{S}^{1}$, and $\lambda$ is an arbitrary Borel probability measure
on $\mathbb{S}^{1}$.

For $z\in\mathbb{S}^{1}$, we recall that $M_{z}\colon Y\to Y$ denotes the
coordinatewise multiplication by $z$

(73) $(M_{z}y)(k):=(z\cdot y(k)),\quad k\in{\mathbb{Z}}.$

We define the factor $(X,\mu,T)$ of the direct product
$(Y,\nu,S)\times(Z,\lambda,\operatorname{id})$ as follows:

$X:={\mathbb{U}}^{\mathbb{Z}}\,\text{ and }\,T\colon X\to X\,\text{ is also
the shift map,}$

and the factor map $\pi\colon Y\times\mathbb{S}^{1}\to X$ is given by

(74) $\pi(y,z):=M_{z}(y),\quad y\in Y,\ z\in\mathbb{S}^{1},$

where $M_{z}$ is as in (73), and

(75) $\mu:=\pi_{*}(\nu\otimes\lambda)$

is the pushforward of the product measure $\nu\otimes\lambda$ by the factor
map $\pi$. Using the previous notation and assumptions, we have the following
result.

######  Proposition B.1.

Suppose that there exists $z\in\mathbb{S}^{1}$ such that the measure $\nu$ is
not invariant by $M_{z}$. Then there exists $r\in{\mathbb{N}}$, such that
$(X,\mu,T)$ is isomorphic to the direct product
$(Y,\nu,S)\times(Z,\lambda_{r},\operatorname{id})$, where $\lambda_{r}$ is the
pushforward of $\lambda$ by $z\mapsto z^{r}$.

###### Proof.

Our plan is as follows: We first define the positive integer
$r\in{\mathbb{N}}$ that appears in the conclusion of our statement and then to
each $x\in X$ we associate unique elements $\tilde{y}\in Y$ and $\tilde{z}\in
Z$ such that $M_{\tilde{z}}(\tilde{y})=x$. Once this is done, we prove that
the map $\Phi\colon X\to Y\times Z$, defined by
$\Phi(x):=(\tilde{y},\tilde{z}^{r})$, is the required isomorphism.

Definition of $r$. Let

(76) $G:=\\{z\in Z\colon\nu\text{ is invariant by }M_{z}\\}.$

Since $G$ is a closed subgroup of $\mathbb{S}^{1}$ and by assumption
$G\neq\mathbb{S}^{1}$, there exists $r\in{\mathbb{N}}$ such that

$G=\\{\zeta_{r}^{k},\,k=0,\ldots,r-1\\}$

where $\zeta_{r}$ is an order $r$ root of unity.

Definition of $\tilde{y}(x)$ and $\tilde{z}(x)$. Using the ergodicity of the
measure $\nu$ and the fact that the map $M_{z}$ preserves $C(Y)$, we deduce
that for $\nu$-almost all $y\in Y$ and for all $z\in Z$ we have

$\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,\delta_{M_{z}S^{n}y}=(M_{z})_{\ast}\nu,$

where $(M_{z})_{\ast}\nu$ denotes the pushforward of $\nu$ by $M_{z}$. It
follows from this and (74), (75), that for $\mu$ almost every $x\in X$ the
following limit exists

$\mu_{x}:=\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,\delta_{T^{n}x},$

and for $\nu\otimes\lambda$ almost every $(y,z)\in Y\times Z$ we have
$\mu_{M_{z}(y)}=(M_{z})_{\ast}\nu$. Next, for $\mu$ almost every $x\in X$, we
let

$H_{x}:=\\{w\in\mathbb{S}^{1}\colon(M_{w^{-1}})_{\ast}\mu_{x}=\nu\\}.$

Then for $\nu\otimes\lambda$ almost every $(y,z)\in Y\times Z$, using (74) and
(75), we get

$H_{M_{z}(y)}=\\{w\in\mathbb{S}^{1}\colon(M_{w^{-1}z})_{\ast}\nu=\nu\\}=zG.$

This establishes that for $\nu\otimes\lambda$ almost every $(y,z)\in Y\times
Z$ the set $zG$ is uniquely determined by the element $x:=M_{z}(y)\in X$.
Hence, for $\mu$ almost every $x\in X$ we can define

$\tilde{z}(x):=\text{ the unique element of }\ zG\ \text{ that has argument in
}\ [0,2\pi/r).$

Let us justify that the map $x\mapsto\tilde{z}(x)$ is measurable. There is a
Borel subset $X_{0}\subset X$ of full $\mu$ measure on which $x\mapsto\mu_{x}$
is Borel measurable, and such that for all $x\in X_{0}$, $\mu_{x}$ is of the
form $(M_{z})_{\ast}\nu$ for some $z\in\mathbb{S}^{1}$. Set

$\mathbb{S}^{1}_{r}:=\\{e(\theta):0\leq\theta<2\pi/r\\}\subset\mathbb{S}^{1}.$

Then the graph of $x\mapsto\tilde{z}(x)$, $x\in X_{0}$, is the preimage of
$\\{\nu\\}$ by the Borel map $(x,z)\mapsto(M_{z^{-1}})_{\ast}\mu_{x}$,
$(x,z)\in X_{0}\times\mathbb{S}^{1}_{r}$. So this graph is Borel-measurable in
$X_{0}\times\mathbb{S}^{1}_{r}$, and by [31, Theorem 14.12], the map
$x\mapsto\tilde{z}(x)$ is Borel measurable. We also set, for $x\in X_{0}$,

$\tilde{y}(x):=M_{\tilde{z}(x)^{-1}}(x)\in Y.$

Definition of $\Phi$. We define the map $\Phi\colon X_{0}\to Y\times Z$ as
follows

(77) $\Phi(x):=\bigl{(}\tilde{y}(x),\tilde{z}(x)^{r}\bigr{)},$

where $r\in{\mathbb{N}}$, $\tilde{y}(x)\in Y$, $\tilde{z}(x)\in Z$, are
defined as above. The map $\Phi$ is measurable and we are going to show next
that it establishes an isomorphism between the systems $(X,\mu,T)$ and
$(Y,\nu,S)\times(Z,\lambda_{r},\operatorname{id})$.

$\Phi$ is an isomorphism. We have to establish the following three claims.

Claim 1: $\Phi$ is injective. Note that for $\mu$ almost every $x\in X$ we
have

(78) $x=M_{\tilde{z}(x)}\bigl{(}\tilde{y}(x)\bigr{)}.$

Replacing if necessary $X_{0}$ by a smaller Borel set (but still of full
measure), we can assume that the above identity and (77) hold on $X_{0}$.
Suppose that $\Phi(x)=\Phi(x^{\prime})$ for some $x,x^{\prime}\in X_{0}$. Then
$\tilde{y}(x)=\tilde{y}(x^{\prime})$ and
$\tilde{z}^{r}(x)=\tilde{z}^{r}(x^{\prime})$. Since both $\tilde{z}(x)$,
$\tilde{z}(x^{\prime})$ have argument in $[0,2\pi/r)$, we deduce that
$\tilde{z}(x)=\tilde{z}(x^{\prime})$. Then (78) gives that $x=x^{\prime}$,
which establishes that the map $\Phi$ is injective on $X_{0}$.

Claim 2: We have $\Phi\circ T=(S\times\operatorname{id})\circ\Phi$.
Equivalently, using (77), it suffices to verify that for $\mu$ almost every
$x\in X$ we have

(79) $(\tilde{y}(Tx),\tilde{z}^{r}(Tx))=(S\tilde{y}(x),\tilde{z}^{r}(x)).$

Since $\mu_{Tx}=\mu_{x}$, we have

$\tilde{z}(Tx)=\tilde{z}(x)$

and

$\tilde{y}(Tx)=M_{\tilde{z}(x)^{-1}}(Tx)=SM_{\tilde{z}(x)^{-1}}(x)=S\tilde{y}(x).$

Combining these two identities we get (79).

Claim 3: We have $\Phi_{*}\mu=\nu\otimes\lambda_{r}$. Note first that for
$\nu\otimes\lambda$ almost every $(y,z)\in Y\times Z$ we have
$\tilde{z}\left(M_{z}(y)\right)^{-1}=z^{-1}w$ for some $w\in G$, a $r$-root of
unity depending only on $z$, and

$\tilde{y}\bigl{(}M_{z}(y)\bigr{)}=M_{\tilde{z}\left(M_{z}(y)\right)^{-1}z}(y)=M_{w}(y).$

Since $w\in G$, we get by (76), that for a fixed $z\in Z$, the pushforward of
$\nu$ by $y\mapsto\tilde{y}\bigl{(}M_{z}(y)\bigr{)}$ is still $\nu$. This
means that, for all $f\in C(Y)$, we have for all $z\in Z$

$\int_{Y}f\bigl{(}\tilde{y}\bigl{(}M_{z}(y)\bigr{)}\bigr{)}\,d\nu(y)=\int_{Y}f(y)\,d\nu(y).$

But then, integrating the above identity with respect to $z$ we get

$\int_{X}f\bigl{(}\tilde{y}(x)\bigl{)}\,d\mu(x)=\int_{Z}\left(\int_{Y}f\bigl{(}\tilde{y}\bigl{(}M_{z}(y)\bigr{)}\bigr{)}\,d\nu(y)\right)\,d\lambda(z)=\int_{Y}f(y)\,d\nu(y),$

which shows that the pushforward of $\mu$ by $x\mapsto\tilde{y}(x)$ is also
$\nu$.

Let us consider now the pushforward of $\mu$ by $x\mapsto\tilde{z}(x)^{r}$.
Observe that, for $\nu\otimes\lambda$ almost every $(y,z)\in Y\times Z$, we
have $\tilde{z}\bigl{(}M_{z}(y)\bigr{)}G=zG$, hence

$\tilde{z}\bigl{(}M_{z}(y)\bigr{)}^{r}=z^{r}.$

Therefore, for an arbitrary continuous $g\in C(\mathbb{S}^{1})$, we have

$\int_{X}g\big{(}\tilde{z}(x)^{r}\big{)}\,d\mu(x)=\int_{Y\times Z}g\
\bigl{(}\tilde{z}\bigl{(}M_{z}(y)\bigr{)}^{r}\bigr{)}\,d(\nu\otimes\lambda)=\int_{Z}g\bigl{(}z^{r}\bigr{)}\,d\lambda(z)=\int_{Z}g(z)\,d\lambda_{r}(z).$

We conclude that the pushforward of $\mu$ by the map
$x\mapsto\tilde{z}^{r}(x)$ is $\lambda_{r}$.

Combining the above, we have that the measure $\Phi_{*}\mu$ is a joining of
$(Y,\nu,S)$ and $(Z,\lambda_{r},\operatorname{id})$. Since, by assumption, the
system $(Y,\nu,S)$ is ergodic and the system
$(Z,\lambda_{r},\operatorname{id})$ is an identity system, the two systems are
disjoint. Hence, $\Phi_{*}\mu=\nu\otimes\lambda_{r}$, as required. ∎

Our argument gives that $r$ depends only on $\nu$ and the factor generated by
the map $x\mapsto\tilde{z}(x)$ from $(X,\mu,T)$ to
$(Z,\lambda_{r},\operatorname{id})$ is non-trivial (in which case $(X,\mu,T)$
is non-ergodic) if and only if $\lambda$ is not concentrated on the group of
$r$-roots of unity. This is always the case if $\lambda$ is a continuous
measure.

##  Appendix C Ergodic consequences of a result of Klurman

In this last appendix we justify some remarks made immediately after Theorem
2.7 and Conjecture 3.

The next result follows immediately from an argument given by Klurman [32] and
crucially uses results about 2-point correlations of Tao [45]. We sketch the
argument for completeness.

######  Proposition C.1.

[32, Proof of Lemma 4.3] Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be an
aperiodic multiplicative function. Then there exist $N_{k}\to\infty$ such that

(80) $\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,\overline{f(n)}\cdot
f(n+j)=0\quad\text{ for every }j\in{\mathbb{N}}.$

######  Remarks.

$\bullet$ As shown in [38, Theorem B.1], some aperiodic multiplicative
functions may have non-vanishing $2$-point correlations (see also the remark
following Proposition 9.1).

$\bullet$ More elaborate arguments of [34] give stronger results, see for
example Theorem 1.2 therein.

###### Proof.

We first claim that for every $m\in{\mathbb{N}}$ there exists a subsequence
$(N_{m,k})_{k\in{\mathbb{N}}}$ (which also depends on $f$) such that

$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{m,k}]}\,\overline{f(n)}\cdot
f(n+j)=0\quad\text{ for }j=1,\ldots,m.$

Indeed, suppose that the conclusion fails. Then there exist
$m\in{\mathbb{N}}$, $\varepsilon>0$, and $N_{0}=N_{0}(m,\varepsilon)$, such
that for all $N\geq N_{0}$ we have

$|\mathbb{E}^{\log}_{n\in[N]}\,\overline{f(n)}\cdot
f(n+j_{N})|\geq\varepsilon$

for some $j_{N}\in[m]$. This fact, combined with the argument used to prove
the second part of [32, Lemma 4.3] gives, without any change, that $f$ is
pretentious, a contradiction.

Using the previous claim, we get that for every $k\in{\mathbb{N}}$ there
exists $N_{k}\in{\mathbb{N}}$ such that

$|\mathbb{E}^{\log}_{n\in[N_{k}]}\,\overline{f(n)}\cdot f(n+j)|\leq
1/k\quad\text{ for }j=1,\ldots,k.$

Furthermore, we can assume that the sequence $(N_{k})_{k\in{\mathbb{N}}}$ is
strictly increasing. The result follows. ∎

######  Corollary C.2.

If $f\colon{\mathbb{N}}\to{\mathbb{U}}$ is a non-trivial aperiodic
multiplicative function, then at least one of its Furstenberg systems for
logarithmic averages has a Lebesgue component. In fact, if $F_{0}$ is the
$0^{\text{th}}$-coordinate projection, on some Furstenberg system $(X,\mu,T)$
of $f$ we have

(81) $\int\overline{F_{0}}\cdot T^{j}F_{0}\,d\mu=0\quad\text{ for every
}j\in{\mathbb{N}}.$

###### Proof.

Using (7) we see that $F_{0}\neq 0$ ($\mu$-a.e.) is equivalent to
${\mathbb{E}}_{n\in{\mathbb{N}}}\,|f(n)|^{2}\neq 0$, which in turn is
equivalent to the non-triviality assumption for $f$.

Let $(N_{k})$ be the subsequence given by Proposition C.1. Pick a subsequence
$(N_{k}^{\prime})$ along which $f$ admits correlations for logarithmic
averages and let $(X,\mu,T)$ be the corresponding Furstenberg system of $f$.
Then (7) and (80) imply that (81) holds. ∎

By combining Corollary C.2 with the main argument in [18] we deduce the
following subsequential variant of the logarithmically averaged Sarnak
conjecture for ergodic weights that applies to all aperiodic multiplicative
functions (for multiplicative functions that satisfy stronger aperiodicity
properties [18, Theorem 1.1] covers a more general result).

######  Theorem C.3.

Let $f\colon{\mathbb{N}}\to{\mathbb{U}}$ be an aperiodic multiplicative
function. Then there exists a subsequence $N_{k}\to\infty$ such that for every
deterministic, ergodic sequence $w\colon{\mathbb{N}}\to{\mathbb{U}}$, we have

$\lim_{k\to\infty}\mathbb{E}^{\log}_{n\in[N_{k}]}\,f(n)\,w(n)=0.$

######  Remark.

As remarked in [24, Section 5.2], there exists an aperiodic multiplicative
function $f\colon{\mathbb{N}}\to\mathbb{S}^{1}$ and a non-ergodic
deterministic sequence $w\colon{\mathbb{N}}\to{\mathbb{U}}$ such that the
averages $\mathbb{E}^{\log}_{n\in[N]}\,f(n)\,w(n)$ do not converge to $0$ as
$N\to\infty$. So passing to a subsequence is needed if one wants to allow $w$
to be an arbitrary deterministic weight sequence (not necessarily ergodic). On
the other hand, see Conjectures 2 and 3 for variants that use relaxed
assumptions.

To prove Theorem C.3, one works with the subsequence $N_{k}\to\infty$ provided
by Corollary C.2 and repeats the proof of [18, Theorem 1.1] given in Section
5.1 in [18]. The point is that Corollary C.2 serves as a substitute for the
2-point correlation result of Tao [45] (the latter is not applicable to all
aperiodic multiplicative functions).

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