Aleksander Gomilko

Mariusz Lemańczyk

Thierry De La Rue

On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao

It is shown that in a class of counterexamples to Elliott's conjecture by Matomäki, Radziwiłł and Tao [23], the Chowla conjecture holds along a subsequence.

Introduction

The celebrated Chowla conjecture [START_REF] Chowla | The Riemann Hypothesis and Hilbert's Tenth Problem[END_REF] from 1965 predicts that for the arithmetic Liouville function λ, we have [START_REF] Abdalaoui | A dynamical point of view on the set of B-free integers[END_REF] lim

N Ñ8 1 N ÿ 1ďnďN
λpn `a1 q ¨. . . ¨λpn `ak q " 0 for any choice of 0 ď a 1 ă . . . ă a k , k ě 1. As noticed by Sarnak [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF] this is equivalent to saying that the Liouville subshift X λ Ă t´1, 1u N is actually the full shift and λ is a generic point for the Bernoulli measure p1{2, 1{2q bN , which is the Haar measure on t´1, 1u N . When we consider a more general multiplicative function u : N Ñ S 1 which is aperiodic (i.e. its mean along any arithmetic progression exists and equals zero), and if all powers u k (k ě 1) are still aperiodic, then the analog of (1) for u becomes lim N Ñ8

1 N ÿ 1ďnďN u r1 pn `a1 q ¨. . . ¨ur k pn `ak qu s1 pn `b1 q ¨. . . ¨us ℓ pn `bℓ q " 0 for all powers r u , s t P N and ta 1 , . . . , a k u X tb 1 , . . . , b ℓ u " H, which in turn means that u is a generic point for pLeb S 1 q bN , the Haar measure on pS 1 q N . This more general form of Chowla conjecture is still a particular case of Elliott's conjecture [START_REF] Elliott | Multiplicative functions |g| ď 1 and their convolutions: An overview, Séminaire de Théorie des Nombres[END_REF], [START_REF] Elliott | On the correlation of multiplicative functions[END_REF], [START_REF] Elliott | On the correlation of multiplicative and the sum of additive arithmetic functions[END_REF] which deals with several (possibly different) multiplicative functions (one of which is aperiodic). Similar conjectures can be formulated for multiplicative functions taking values in the unit disk D, but in this case we have to consider properties of u relative to |u|, where the latter point is always generic for an ergodic measure (with respect to the left shift), often called the Mirsky measure, on r0, 1s N (see [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF] for the case of the Möbius function µ or e.g. [START_REF] Bergelson | Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics[END_REF] for a general case).

In [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF], Matomäki, Radziwiłł and Tao gave a class of examples of multiplicative and aperiodic1 u : N Ñ S 1 Y t0u for which the sequence

´1 N ÿ 1ďnďN upnqupn `1q ¯Ně1
does not converge to zero. This disproved the general form of Chowla conjecture for aperiodic u, and in particular, they disproved Elliott's conjecture. In their class |u| " µ2 but they also explained that their example could be modified to get a completely multiplicative S 1 -valued aperiodic function. Finally, this lead them to reformulate Elliott's conjecture in the sense that it is expected to be valid for a subclass of aperiodic functions, the so-called strongly aperiodic functions, and till today this stronger form of Elliott's conjecture is open and under intensive study [START_REF] Frantzikinakis | An averaged Chowla and Elliott conjecture along independent polynomials[END_REF], [START_REF] Frantzikinkis | Asymptotics for multilinear averages of multiplicative functions[END_REF], [START_REF] Matthiesen | Linear correlations of multiplicative functions[END_REF], [START_REF] Tao | The logarithmically averaged Chowla and Elliott conjectures for two-point correlations[END_REF]- [START_REF] Tao | The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures[END_REF].

In this note, we will have a closer look at the counterexample given by Matomäki, Radziwiłł and Tao. To avoid some technical, rather notational, issues, we will deal with the completely multiplicative, S 1 -valued version of their construction, which we call here the MRT class (see the complete description of such multiplicative functions in Section 3). If u P MRT then it cannot be generic for the Haar measure on pS 1 q N , but still we can ask for which measures on pS 1 q N it is quasi-generic. Each such measure yields a so-called measuretheoretic Furstenberg system (see Section 2). In particular, the arguments given in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] prove that there exists an increasing sequence of integers giving rise to a Furstenberg system which is measure-theoretically isomorphic to the action of the identity map on S 1 equipped with the Lebesgue measure. What kind of other dynamical systems can be obtained as Furstenberg systems for u in the MRT class is a natural question. Furthermore, in the topological setting, u determines a subshift X u Ă pS 1 q N and we can ask for its topological entropy. Our aim is to prove the following result.

Main Theorem. Let u be in the MRT class. Then, for each d ě 0, there is a Furstenberg system pX u , ν d , Sq of u which is measure-theoretically isomorphic to the unipotent system px d , x d´1 , . . . , x 0 q Þ Ñ px d , x d´1 `xd , . . . , x 0 `x1 q on T d`1 equipped with the pd `1q-dimensional Lebesgue measure.

Furthermore, the Bernoulli shift `pS 1 q N , pLeb S 1 q bN , S ˘is also a Furstenberg system of u, i.e.

(2)

the analog of the Chowla conjecture holds for u along a subsequence.

In particular,

(3) X u " pS 1 q N , 2 and (4) h top pX u , Sq " 8.

The proof of the above theorem is given in Section 4. It seems also that this result makes it legitimate to ask whether (some of) properties ( 2)-( 4) are valid for all totally aperiodic multiplicative functions u, |u| " 1. Note also that the theorem above sheds light on Frantzikinakis' question: see Problem 3.1 on the list of problems in [START_REF]Sarnak's Conjecture[END_REF] of whether the system px, yq Þ Ñ px, x `yq on T 2 can be a Furstenberg system of the Liouville function. The expected answer is of course negative (as the positive answer is in conflict with the Chowla conjecture) but, as our result shows, this unipotent system is a Furstenberg system for a class of multiplicative aperiodic functions. Moreover, our Main Theorem yields positive answers to questions raised in Problem 7.3 [START_REF]Sarnak's Conjecture[END_REF] in the class of MRT functions. More than that, since the ergodic components of ν d , d ě 2, are affine extensions of irrational rotations, the Main Theorem disproves Conjecture 2 from [START_REF] Frantzikinakis | Furstenberg systems of bounded multiplicative functions and applications[END_REF].

As each u P MRT is given "locally by Archimedean characters", in Section 5, we will deal with Furstenberg systems of Archimedean characters themselves and will describe their Furstenberg systems. Moreover, we will show that no u P MRT satisfies the analog of Sarnak's conjecture. Finally, in Section 6 we show that the analog of logarithmic Chowla conjecture for u (cf. ( 2)) holds along a subsequence and discuss further properties of Furstenberg systems of MRT arithmetic functions: strong stationarity and the absence of zero mean on typical short interval.

In Appendix we discuss Furstenberg systems given by ν d from the pure ergodic theory point of view. We explain their connections with the classical theory of transformations with quasi-discrete spectrum and give a new characterization of quasi-eigenfunctions which seems to be of independent interest.

Furstenberg systems of a bounded arithmetic function

Assume that u : N Ñ C is an arithmetic function, |u| ď 1. Denote by D the unit disk. On the space M `DN ˘of Borel probability measures on D N , we consider the weak ˚-topology, in which ν m Ñ ν if and only if

ş D N f dν m Ñ ş D N f dν for each f P CpD N q.
This topology turns M `DN ˘into a compact metrizable space. Let S denote the shift map on D N , and consider in M `DN ˘the sequence

pE N puqq N ě1 :" ˜1 N ÿ 0ďnăN δ S n u ¸Ně1
of empiric probability measures. By compactness, we can choose a converging subsequence E Nm puq Ñ ν, and we say that u is quasi-generic for ν along the sequence pN m q. It is not hard to see (cf. the Krylov-Bogolyubov theorem) that such a limit point ν is Sinvariant. Moreover, such a ν is always supported on the subshift X u generated by u, that is X u :" tS n u : n ě 0u.

The measure-theoretic dynamical system pX u , ν, Sq is called a Furstenberg system of u. Denote by V puq the set of all probability measures on X u for which u is quasi-generic:

V puq :" ! ν P M pX u , Sq : ν " lim mÑ8 E Nm puq for some N m Ñ 8
) .

Classically, we have the following:

Proposition 2.1. ([6]) (i) V puq Ă M pX u , Sq is closed (in the weak ˚-topology).
(ii) V puq is connected, whence either |V puq| " 1 or V puq is uncountable.

Let Z 0 : D N Ñ D be the 1-coordinate map: Z 0 pyq " yp1q for y " `ypnq ˘nPN P D N . In general, we define Z n :" Z 0 ˝Sn as the map y Þ Ñ ypn `1q. Then, given ν P V puq, we obtain a stationary process pZ 0 , Z 1 , . . .q with values in D N whose distribution is ν. Let κ " pZ 0 q ˚pνq be the distribution of the random variable Z 0 under ν. Then by the S-invariance of ν, κ is also the distribution of each coordinate Z n , n ě 0. An example of particular interest corresponds to the i.i.d. case, arising when ν is a product measure, i.e., when ν is of the form κ bN . We have the following observation. Proposition 2.2. Let κ be a probability measure on D. Assume that for each d ě 0 there exists ν d P V puq under which the distribution of pZ 0 , . . . , Z d q is κ bd`1 . Then κ bN P V puq.

Proof. By compactness, we can assume that ν d Ñ ρ, and by Proposition 2.1 (i), ρ P V puq. All we need to show is that

E ρ pZ q1 0 ˝Sj1 ¨. . . ¨Zq k 0 ˝Sj k q " k ź i"1 E κ Z qi 0
for each k ě 1, q i P Z and 0 ď j 1 ă . . . ă j k . But the equality above is true if ρ is replaced by ν d for d ě j k , and since Z q1 0 ˝Sj1 ¨. . . ¨Zq k 0 ˝Sj k P CpX u q, the result follows.

Remark 2.3. If κ bN P V puq then by the variational principle (see e.g. [START_REF] Walters | An introduction to ergodic theory[END_REF]Section 8.2]) h top pX u , Sq ě hpX u , κ bN , Sq " Hpκq. If the distribution of κ is continuous then immediately Hpκq " `8, whence h top pX u , Sq " `8 in this case. Proposition 2.2 can be useful if we want to show that the product measure yields a Furstenberg system of Bernoulli type ("Chowla holds along a subsequence"). Indeed, we only need to show the existence of Furstenberg systems which yield some finite degree of independence of the process pZ n q ně0 and such can be firstly of zero entropy and even very non-ergodic (i.e. belonging to Erg K ), cf. also [START_REF] Flaminio | Mixing k-fold independent processes of zero entropy[END_REF]. Together with Remark 2.3, it also gives a nice criterion to show that the topological entropy of u is infinite.

MRT multiplicative functions

In this section we describe more precisely the MRT class of completely multiplicative functions to which our Main Theorem applies. We start by giving a formal definition of this class, then we resume the construction given in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] by Matomäki, Radziwiłł and Tao of a completely multiplicative function meeting the required property. Finally, we give the key property of MRT arithmetic functions that will be used in the proof of our Main Theorem. Definition 3.1. A completely multiplicative function u : N Ñ S 1 belongs to the MRT class if there exist two increasing sequences of integers pt m q and ps m q such that, for each m ě 1, we have the following properties:

' t m ă s m`1 ă s 2 m`1 ď t m`1 , ' for each prime p P pt m , t m`1 s, uppq " p ism`1 , (5)
' for each prime p ď t m , ˇˇuppq ´pism`1 ˇˇă 1 t 2 m . (6)
Here is the method to get such a function. We just have to define uppq for each prime p and to construct the sequences pt m q and ps m q, which is done inductively as follows. Start by choosing an integer t 1 P N and set, for each prime p ď t 1 , uppq :" 1. Now, assume that for some m ě 1 we have already defined t m and uppq for each p ď t m . In the Cartesian product ś pďtm S 1 , we consider the sequence of points

´`p is ˘pďtm ¯sPN .

Since the numbers log p, p ď t m , are linearly independent over the integers, this sequence is dense in ś pďtm S 1 . Thus, we can choose s m`1 ą t m so that (6) is satisfied. We then choose t m`1 ě s 2 m`1 , and for t m ă p ď t m`1 we set uppq :" p ism`1 . Doing this inductively for each m ě 1 gives a completely multiplicative function u P MRT. Remark 3.1. It is also interesting to note that the growth of s m`1 {t m is necessarily superpolynomial: in fact, it follows from Propositions 4.3 and 4.7 that, for each β ą 0, t m ă s β m`1 for m large enough. It is also shown in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] that once s m`1 ą e tm for m ě 1, the resulting u is aperiodic.

We will use the following easy lemma. Lemma 3.2. Let u P MRT and pt m q, ps m q be as in Definition 3.1. Let m ě 1 and n ď t m`1 . If the number of prime factors of n less than or equal to t m (counting multiplicity) is bounded by t m , then [START_REF] Elliott | Multiplicative functions |g| ď 1 and their convolutions: An overview, Séminaire de Théorie des Nombres[END_REF] ˇˇupnq ´nism`1 ˇˇď 1 t m .

Proof. We write n as a product of primes

½ Bt pnq ď 1 N ÿ 1ďnďN ÿ pďt ½ p k t |n " ÿ pďt 1 N ÿ 1ďnďN ½ p k t |n ď ÿ pďt 1 p kt ă ÿ p 1 p kt Ý ÝÝ Ñ tÑ8 0 (since k t Ñ 8 as t Ñ 8).
Using Lemma 3.2 and Lemma 3.3, we obtain the following result.

Proposition 3.4. Let u P MRT and pt m q, ps m q be as in Definition 3.1. Let pN m q be an increasing sequence of integers with N m ď t m`1 for each m P N. Then

1 N m # " n P t1 . . . , N m u : ˇˇupnq ´nism`1 ˇˇą 1 t m * ď ε tm Ý ÝÝÝ Ñ mÑ8 0.

Proof of the Main Theorem

Here is an outline of the proof. First, we present in Section 4.1 a family of stationary processes, taking values in the unit circle, parametrized by an integer d ě 0. Such a process generates a measure-theoretic dynamical system isomorphic to the unipotent system appearing in the statement of the theorem. It is easily characterized by two conditions: each coordinate of the process is uniformly distributed on the circle, and some deterministic function φ d`1 of the coordinates of the process is constantly equal to 1. We completely describe in Proposition 4.1 and Remark 4.2 the distribution ν d of such a process, in particular we show that d `1 consecutive coordinates are independent. Then Proposition 4.3 provides a criterion for an arithmetic function u to be quasigeneric for this probability measure ν d . This criterion involves the functions φ d`1 , and we study in Section 4.2 two sequences of polynomials related with this family of functions. We show in Section 4.3 that for u P MRT, the criterion is fulfilled for each d ě 0. For this we use Proposition 3.4 which allows us to replace, in the appropriate range, upnq by n ism`1 . The criterion then becomes an evaluation of some exponential sums, that we can handle with the help of Kusmin-Landau Theorem (Theorem 4.8). Finally, using Proposition 2.2, we can conclude that u is also quasi-generic for the product measure `Leb S 1 ˘bN .

Processes in the unit circle

We recall the notation Z n " Z 0 ˝Sn (n ě 0) from Section 2, but we restrict ourselves to the set of sequences taking values in S 1 . We define inductively a new sequence of maps pY pdq q dě0 defined on pS 1 q N taking also their values in S 1 , by Y p0q :" Z 0 , Y p1q :" Z 1 {Z 0 , and in general for each d ě 0,

Y pd`1q :" Y pdq ˝S Y pdq .
We also define the auxilliary sequence pX d q dě0 taking values in the one-dimensional torus T :" R{Z, by e i2πX d :" Y pdq .

As e i2πX d`1 " Y pd`1q " e i2πpX d ˝S´X d q , we get

(9) @d ě 0, X d ˝S " X d`1 `Xd .
Moreover, by an easy induction on n using the above formula, we can recover the process each Z n from pX d q by the relation

(10) @n ě 0, Z n " e i2π ř n d"0 p n d qXd .
Proposition 4.1. Let ν be a shift-invariant probability measure on pS 1 q N . Assume that there exists d ě 0 such that, under ν,

• Y pdq is S-invariant (i.e. Y pd`1q " 1 ν-a.s.),
• the distribution of Y pdq is uniform on S 1 .

Then, for each n, Z n is uniformly distributed on S 1 , and Z 0 , Z 1 , . . . , Z d are independent.

Proof. As Y pdq is S-invariant, so is X d . Hence, in every ergodic component of ν, X d is a.s. constant. Moreover, since the distribution of X d under ν is the Lebesgue measure Leb T on T, in almost every ergodic component the value of X d is irrational. Let us fix such an ergodic component, say ν, and let α be the value taken by X d under ν. From ( 9), we get pX d´1 , . . . , X 0 q ˝S " pX d´1 `Xd , X d´2 `Xd´1 , . . . , X 0 `X1 q " pX d´1 `α, X d´2 `Xd´1 , . . . , X 0 `X1 q .

But when α is irrational, the map

px d´1 , . . . , x 0 q P T d Þ Ñ px d´1 `α, x d´2 `xd´1 , . . . , x 0 `x1 q
is uniquely ergodic [START_REF] Furstenberg | Strict ergodicity and transformation of the torus[END_REF], with the d-fold product pLeb T q bd as the only ergodic invariant measure. We deduce that, under ν, the distribution of pX d´1 , . . . , X 0 q must be pLeb T q bd . Integrating over the set of ergodic components, we get that under ν, the distribution of pX d , X d´1 , . . . , X 0 q is pLeb T q bpd`1q . Then, from Formula (10), we can write each 0 ď n ď d,

Z n " Zn´1 e i2πXn ,
where Zn´1 is measurable with respect to X 0 , . . . , X d´1 ( Zn´1 " 1 for n " 0).

From this it follows that Z n is uniformly distributed on S 1 conditionally to X 0 , . . . , X n´1 . This concludes the proof.

Remark 4.2. The proof shows in fact that, for each d ě 0, there is a unique shift-invariant measure ν d on pS 1 q N such that whenever the assumptions of Proposition 4.1 are satisfied, then the distribution of the stationary process pZ n q nPN is ν d . Under ν d , any d `1 consecutive coordinates of the process are independent and uniformly distributed on the circle, and for each n ě 0 the coordinate Z n`d`1 is a deterministic function of Z n , . . . , Z n`d given by the condition Y pd`1q " 1. Moreover, looking at the process `pX 0 , X 1 , . . . , X d q˝S n ˘ně0 , we see that the dynamical system `pS 1 q N , ν d , S ˘is measure-theoretically isomorphic to the unipotent system described in the statement of the Main Theorem.

It will be useful to introduce, for each d ě 0, the function φ d : pS 1 q d`1 Ñ S 1 such that Y pdq " φ d pZ 0 , . . . , Z d q. For d " 0, . . . , 4, these functions are given by

φ 0 pz 0 q " z 0 1 ; φ 1 pz 0 , z 1 q " z 1 z 0 ; φ 2 pz 0 , z 1 , z 2 q " z 0 z 2 z 2 1 ; φ 3 pz 0 , z 1 , z 2 , z 3 q " z 3 1 z 3 z 0 z 3 2 ; φ 4 pz 0 , z 1 , z 2 , z 3 , z 4 q " z 0 z 6 2 z 4 z 4 1 z 4 3 . (11)
In general, we can always write the function φ d as a quotient [START_REF] Frantzikinakis | An averaged Chowla and Elliott conjecture along independent polynomials[END_REF] φ d pz 0 , . . . , z d q " π d pz 0 , . . . , z d q πd pz 0 , . . . , z d q ,

where π d pz 0 , . . . , z d q and πd pz 0 , . . . , z d q are products of some z j 's (with possible multiplicity). These sequences of products are completely defined by the following recurrence property: π 0 pz 0 q :" z 0 , π0 pz 0 q :" 1, and for each d ě 1, (13) π d`1 pz 0 , . . . , z d`1 q :" π d pz 1 , . . . , z d`1 q πd pz 0 , . . . , z d q, and ( 14) πd`1 pz 0 , . . . , z d`1 q :" πd pz 1 , . . . , z d`1 q π d pz 0 , . . . , z d q.

Note that, as π d and πd only involve a product of variables, their definition can be extended to d`1 .

The next proposition is a key ingredient for the identification of some Furstenberg systems of multiplicative functions. Proposition 4.3. Let u " `upnq ˘nPN P pS 1 q N . Assume that, for some d ě 1 and some increasing sequence pN m q Ă N, we have

(15) 1 N m ÿ nďNm φ d`1 `upnq, upn `1q, . . . , upn `d `1q ˘Ý ÝÝÝ Ñ mÑ8 1,
and that

(16) @ℓ ě 1, 1 N m ÿ nďNm φ ℓ d `upnq, upn `1q, . . . , upn `dq ˘Ý ÝÝÝ Ñ mÑ8 0.
Then, along the subsequence pN m q, u is quasi-generic for the measure ν d described in Remark 4.2.

Proof. Let µ be a shift-invariant measure on pS 1 q N such that, along some subsequence of pN m q, u is quasi-generic for µ. From ( 15), we get that

1 " µ rφ d`1 pZ 0 , . . . , Z d`1 qs " µ " Y pd`1q
ı , hence Y pd`1q " 1 µ-a.s. And from ( 16), we get that for each ℓ ě 1,

0 " µ " φ ℓ d pZ 0 , . . . , Z d q ‰ " µ " pY pdq q ℓ ı ,
which shows that, under µ, Y pdq is uniformly distributed on S 1 . Then, with Proposition 4.1 and Remark 4.2, we conclude that µ " ν d .

Special sequences of polynomials

We now introduce two sequences pP d q dě0 and pQ d q dě0 of polynomials in the variable n, linked to the analysis of the preceding section by the following definition: for each d ě 0, The first polynomials of this family are given below (compare with the numerators and denominators in ( 11)).

P 0 pnq " n ; Q 0 pnq " 1 ;

P 1 pnq " n `1 ; Q 1 pnq " n ;
P 2 pnq " npn `2q ; Q 2 pnq " pn `1q 2 ; P 3 pnq " pn `1q 3 pn `3q ; Q 3 pnq " npn `2q 3 ; P 4 pnq " npn `2q 6 pn `4q ; Q 4 pnq " pn `1q 4 pn `3q 4 .

Note that, according to ( 13) and ( 14), these sequences of polynomials satisfy the following recurrence property:

P d`1 pnq " P d pn `1qQ d pnq, Q d`1 pnq " Q d pn `1qP d pnq. ( 18
)
Proposition 4.4. For each d ě 1, P d and Q d are both of degree 2 d´1 , and the degree of the difference R d :"

Q d ´Pd is 2 d´1 ´d.
The computation of the degree of P d and Q d is a straightforward induction using [START_REF] Hahn | Minimal dynamical systems with quasi-discrete spectrum[END_REF]. For the degree of R d , we will use the following lemma. Lemma 4.5. Let P and R be two polynomials, of degree p and r, respectively and with p ‰ r. Let R be the polynomial defined by Rpnq :" Rpn `1qP pnq ´RpnqP pn `1q.

Then the degree of R is r `p ´1.

Proof. Let us write the two terms of highest degree in P and R:

P pnq " a P n p `bP n p´1 `¨¨ä nd Rpnq " a R n r `bR n r´1 `¨¨ẅ
here a P a R ‰ 0. Then a direct computation shows that the two terms of highest degree in Rpn `1qP pnq are Rpn `1qP pnq " a R a P n r`p `pa R b P `aP b R `r a P a R q n r`p´1 `¨¨L ikewise, the two terms of highest degree in RpnqP pn `1q are RpnqP pn `1q " a R a P n r`p `pa R b P `aP b R `p a P a R q n r`p´1 `¨¨Ï t follows that the term of highest degree in Rpnq is pr ´pqa P a R n r`p´1 .

End of the proof of Proposition 4.4. We show by induction on d ě 1 that the degree of R d is 2 d´1 ´d. This is already true for d " 1 since R 1 " ´1 is of degree 0. Then, assume that the result holds for some d ě 1, and let us consider the polynomial R d`1 . We have But we know that the degree of P d is 2 d´1 , and by the induction assumption the degree of R d is 2 d´1 ´d. These degrees are different, therefore Lemma 4.5 applies, and shows that the degree of R d`1 is 2 d ´pd `1q.

R d`1 pnq " Q d`1 pnq ´Pd`1 pnq " Q d pn

Furstenberg systems of MRT multiplicative functions

Now, we consider a completely multiplicative arithmetic function u " `upnq ˘ně1 P MRT. Let pt m q and ps m q be the associated sequences, as in Definition 3.1. Let pN m q be an increasing sequence of integers with N m ď t m`1 , and let d ě 0 and ℓ ě 1 be fixed integers.

f d pxq " K d x d as x Ñ 8.
For each d ě 0, there exists L d ‰ 0 such that

(23) f 1 d pxq " L d x d`1 as x Ñ 8.
Moreover, there exists H d ą 0 such that f 1 d is monotone on rH d , `8q.

Proof. For d " 0, note that f 0 pxq " log x, so that f 1 0 pxq "

Q d P d ¨P 1 d pP d `Rd q ´Pd pP 1 d `R1 d q Q 2 d " P 1 d R d ´Pd R 1 d P d Q d .
Since [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF]. Finally, as a nonzero rational fraction, the second derivative f 2 d has finitely many zeros, from which we get the last claim of the lemma.

We can now state the precise asymptotics which, together with Proposition 4.3, will allow us to identify some Furstenberg systems of u. Proposition 4.7. Let d ě 0 be a fixed integer, and choose a real number β such that

1 d `1 ă β ă 1 d if d ě 1, 1 ă β ă 2 if d " 0.
Set, for each m ě 1, N m :" X ps m`1 q β \ . Then

(24) @ℓ ě 1, 1 N m ÿ 1ďnďNm e iℓsm`1f d pnq Ý ÝÝÝ Ñ mÑ8 0, and
(25) 1 N m ÿ 1ďnďNm e ism`1f d`1 pnq Ý ÝÝÝ Ñ mÑ8 1.
An important tool in the proof of the above proposition is the following theorem of Kusmin-Landau, which we state as in [START_REF] Rivat | Analytic Number Theory, in: Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics[END_REF]. Here, if x is a real number, }x} stands for the distance of x to the nearest integer.

Theorem 4.8 (Kusmin-Landau Theorem). If f : ra, bs Ñ R is C 1 , f 1 is mono- tone and }f 1 } ě λ 1 ą 0 then ˇˇˇˇˇÿ nPra,bs e i2πf pnq ˇˇˇˇˇď 2 πλ 1 .
Proof of Proposition 4.7. Choose α such that 1 d`1 ă α ă β. We can replace the average in [START_REF] Rivat | Analytic Number Theory, in: Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics[END_REF] and in [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF] by the average over s α m`1 ď n ď s β m`1 . For such an n, using [START_REF] Matomäki | Multiplicative functions in short intervals[END_REF], we get

|s m`1 f d`1 pnq| " O ˜1 s αpd`1q´1 m`1 ¸Ý ÝÝÝ Ñ mÑ8 0,
and this immediately gives [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF].

By Lemma 4.6, for m large enough f 1 d is monotone on

" s α m`1 , s β m`1 ı
, and there exists K ą 0 (depending only on ℓ and d) such that, for n in this interval,

}ℓs m`1 f 1 d pnq} ě K s m`1 n d`1 ě K 1 s βpd`1q´1 m`1
.

By Kusmin-Landau Theorem, we get ˇˇˇˇˇˇÿ

s α m`1 ďnďs β m`1 e iℓsm`1f d pnq ˇˇˇˇˇˇ" O ´sβpd`1q´1 m`1 ¯,
and finally ˇˇˇˇ1

N m ÿ 1ďnďNm e iℓsm`1f d pnq ˇˇˇˇ" O ´sβd´1 m`1 ¯Ý ÝÝÝ Ñ mÑ8 0.
Putting together (20), Proposition 4.7 and Proposition 4.3, we get the following result. Theorem 4.9. For any d ě 0, the measure-preserving system `pS 1 q N , ν d , S ȃppears as a Furstenberg system of u.

Using Proposition 2.2 and the fact that, under ν d , d `1 consecutive coordinates of the process pZ n q are independent and uniformly distributed on S 1 , we obtain the following.

Corollary 4.10. The system ´pS 1 q N , pLeb S 1 q bN , S ¯is a Furstenberg system of u.

This concludes the proof of the Main Theorem.

Archimedean characters and slowly varying arithmetic functions

We recall that an Archimedean character is a completely multiplicative function of the form n Þ Ñ n it with some fixed t P R. As ˇˇpn `1q it ´nit ˇˇ" ˇˇe it logp1`1{nq ´1ˇˇˇÝ ÝÝÑ A useful weakening of this property is the following: we say that the bounded arithmetic function u is mean slowly varying if

(26) 1 N ÿ nďN |upn `1q ´upnq| Ý ÝÝÝ Ñ N Ñ8 0.
Note that this is equivalent to the fact that upn `1q ´upnq Ñ 0 on a subset of n of density 1.

It is worth mentioning a result by Klurman [20, Theorem 1.8], who proved a conjecture by Kátai concerning mean slowly varying multiplicative functions (non necessarily bounded). From his theorem, it easily follows that the only mean slowly varying multiplicative functions taking values in the unit circle are the Archimedean characters.

Furstenberg systems of (mean) slowly varying functions

Proposition 5.1. The arithmetic function u : N Ñ D is mean slowly varying if and only if all Furstenberg systems of u are measure-theoretically isomorphic to the action of the identity on some probability space.

Proof. Assume first that u is mean slowly varying, and that u is quasi-generic for some S-invariant measure ν on D N along a subsequence pN m q. Then, by (26), we have

ν " |Z 1 ´Z0 | ‰ " lim mÑ8 1 N m ÿ nďNm |upn `1q ´upnq| " 0.
It follows that Z 1 " Z 0 ν-a.e., and more generally by S-invariance, for each k P N, we also have Z k`1 " Z k ν-a.e. Hence, ν is concentrated on the subset of sequences with identical coordinates, and S " Id ν-a.e. Conversely, assume that (26) fails. Then there exists a subsequence pN m q such that lim mÑ8 1 N m ÿ nďNm |upn `1q ´upnq| ą 0, and by compactness of M pD N q we can assume that u is quasi-generic for some S-invariant measure ν along pN m q. But then we have

ν " |Z 1 ´Z0 | ‰ " ν " |Z 0 ˝S ´Z0 | ‰ ą 0,
and the Furstenberg system of u defined by ν is not measure-theoretically isomorphic to the action of the identity.

Proposition 5.2. Let u : N Ñ D be a slowly varying arithmetic function, and let X u Ă D N be the subshift generated by u. Then the topological entropy of pX u , Sq is zero.

Proof. We start by observing that, as u is slowly varying, for each ε ą 0 there exists L ε such that whenever ℓ ě L ε ,

@n P N, 1 ℓ ÿ nďjăn`ℓ |upj `1q ´upjq| ă ε. (27)
Let w be a sequence in X u . As each subblock of w is a limit of subblocks of u, it follows that ( 27) is still valid if we replace u by w. But this in turn implies that any w P X u is mean slowly varying. Now, let ν be an ergodic shift-invariant measure on X u , and let w P X u be ν-generic (ν-almost all sequences in X u are ν-generic). Then w is mean slowly varying, and by Proposition 5.1, the measure-theoretic system pX u , ν, Sq is isomorphic to pX u , ν, Idq. In particular, its Kolmogorov entropy is zero.

By the variational principle (see e.g. [30, Section 8.2], the topological entropy of pX u , Sq must be also zero.

Remark 5.3. As a matter of fact, the above proof shows that whenever u is a slowly varying function, any w P X u is also a slowly varying function. More than that, we can make the following observation: by the definition of X u , each w P X u can be written as

w " lim kÑ8 S n k u,
where pn k q is a non-decreasing sequence of non-negative integers. Either pn k q is bounded, and then w " S n u for some n ě 0, or n k Ñ 8, and then as u is slowly varying, w must be of the form pw 1 , w 1 , w 1 , . . .q for some w 1 P D.

In the following remark we provide some more observations on mean slowly varying functions. ν. Indeed, note that for each k ě 1, ℓ r ě 1, j r ě 0 for r " 1, . . . , k, we have

ˇˇˇˇ1 N ÿ 1ďnďN u ℓ1 pn `j1 q ¨. . . ¨uℓ k pn `jk q ´1 N ÿ 1ďnďN u ℓ1`...`ℓ k pnq ˇˇˇˇÝ ÝÝÝ Ñ N Ñ8 0.
(c) We can repeat word for word all arguments for the logarithmic averages. In particular, if u is mean slowly varying and, for each ℓ ě 1, the limit [START_REF] Tao | The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures[END_REF] lim

N Ñ8 1 log N ÿ 1ďnďN 1 n u ℓ pnq
exists, then there is only one logarithmic Furstenberg system (see Section 6.1): there exists a shift-invariant measure ν such that

1 log N ÿ 1ďnďN 1 n δ S n u Ý ÝÝÝ Ñ N Ñ8
ν.

(d) For upnq " n it , we have the logarithmic assumption ( 28) satisfied (in fact, each such limit is zero whenever ℓ ‰ 0), so there is only one logarithmic Furstenberg system, as already noticed in [START_REF] Frantzikinakis | Furstenberg systems of bounded multiplicative functions and applications[END_REF] but we have uncountably many Furstenberg systems (cf. Corollary 5.5).

As it follows from Proposition 2.1, a slowly varying function u : N Ñ D either has one Furstenberg system, or it has uncountably many different Furstenberg systems. By different, we mean that we obtain different measures. The measure-theoretic dynamical systems given by these measures can however be all isomorphic which is the case in the example below.

We consider now a mean slowly varying function u such that |upnq| " 1 for all n ě 1. Assume, moreover, that

lim mÑ8 1 N m ÿ 0ďnăNm δ S n u " lim mÑ8 E Nm puq " ν
for an increasing sequence pN m q. It follows from Proposition 5.1 that ν is supported on the set tpz, z, . . .q : z P S 1 u. Moreover, Z k 0 pz, z, . . .q " z k . Hence, we can identify ν with a measure defined on S 1 such that

lim mÑ8 1 N m ÿ 1ďnďNm u k pnq " ż Z k 0 dν " ż S 1 z k dκ.
It follows that, varying k P Z, the LHS of the equation above provides the Fourier transform of the measure κ which we are seeking. In the particular case where upnq " n i , we have

1 N m ÿ 1ďnďNm n ik " N ik m 1 `ik `op1q,
thus the Fourier coefficients of κ are given by the limits of

N ki m 1`ik as m Ñ 8. The case k " 1 yields that N i m Ñ c
, where |c| " 1, and we obtain that ż

S 1 z k dκ " c k 1 `ik , k P Z.
Consider, κ 1 :" κ ˚δc . Then

ż S 1 z k dκ 1 " 1 1 `ik , k P Z.
Since p 1 1`ik q is an ℓ 2 -sequence, the measure κ 1 is absolutely continuous with respect to the (normalized) Lebesgue measure Leb S 1 , with density g equal to gpzq "

ř 8 k"´8 1
1`ik z k . In fact, noting that for all k P Z, ż 1 0 e 2πx e i2πkx dx " e 2π ´1 2πp1 `ikq , we see that, for x P r0, 1q, gpe i2πx q " 2πe 2πx e 2π ´1 . Finally, note that the sequence pn i q is dense in S 1 as the sequence plog nq is dense in r0, 2πq mod 2π.

Corollary 5.5. The family of Furstenberg systems of upnq " n i consists of uncountably many different systems given by all rotations of gpzqdz. All of them are isomorphic to the identity on the circle with Lebesgue measure (and ergodic components are Dirac measures on the circle). Moreover, X u " tpz, z, . . .q : z P S 1 u Y tS n u : n P Nu.

MRT arithmetic functions do not satisfy Sarnak's conjecture

As MRT arithmetic functions mostly behave like Archimedean characters on very large intervals, we will use some ideas presented in the preceding section to prove that MRT arithmetic functions do not satisfy Sarnak's conjecture: for any MRT function u there exists a zero entropy topological system which outputs a sequence having some correlation with u.

Let u be an MRT arithmetic function, and let pt m q and ps m q be as in Definition 3.1. We set, for each m P N, r m :" ts 3{2 m u. Then we define a new arithmetic function v " `vpnq ˘nPN P pS 1 q N by setting for each m: vpnq :"

# 1 for t m ă ď r m`1 , n ism`1 for r m`1 ă n ď t m`1 .
Then, for each m, we have

|vpn `1q ´vpnq| " # 0 if t m ă n ă r m`1 , O ´s´1{2 m`1 ¯if r m`1 ă n ă t m`1 .
Even though v may not be slowly varying because of the possible jumps in t m and in r m`1 , the property described in Equation ( 27) is still valid, as these jumps are bounded by 2 and are separated by gaps whose lengths tend to 8. Therefore, the proof of Proposition 5.2 also applies to v, and we have h top pX v , Sq " 0. But, since rm`1 tm`1 Ý ÝÝÝ Ñ mÑ8 0, in view of Proposition 3.4, we have

1 t m`1 ÿ 1ďnďtm`1 upnqvpnq Ý ÝÝÝ Ñ mÑ8 1.
We thus have found a topological dynamical system pX v , Sq of zero topological entropy, a point v P X v and a continuous map f : X v Ñ C (the conjugate of the zero-coordinate map), such that the sequence `f pS n vq ˘is not orthogonal to u, in the sense that

1 N ÿ 1ďnďN upnqf pS n vq ÝÑ { 0 as N Ñ 8.
6 Further properties of MRT functions

MRT functions satisfy logarithmic Chowla conjecture along a subsequence

The purpose of this section is to study what the Main Theorem becomes if we consider logarithmic averages instead of usual averages. A logarithmic Furstenberg system of an arithmetic function u : N Ñ D is defined as in Section 2 as a measure-theoretic dynamical system pX u , ν, Sq, where ν is now a weak ˚limit of a subsequence of the logarithmic empirical measures

E log N puq :" 1 L N ÿ 1ďnďN 1 n δ S n´1 u , with L N :" 1 `1 2 `¨¨¨`1 N (N ě 1).
We recall the classical relation between the logarithmic and usual averages, obtained by summation by parts:

(29) E log N puq " 1 L N ÿ 1ďnďN ´1 1 n `1 E n puq `1 L N E N puq.
We recall that the weak ˚-topology turns M ppS 1 q N q into a compact metrizable space, and that a possible metric is given by ∆pν, κq "

ÿ jě1 1 2 j ˇˇˇż f j dν ´ż f j dκ ˇˇˇ,
where pf j q jě1 is a countable family generating a dense subspace of CppS 1 q N q. We can for example take for pf j q jě1 the family of tZ r1 i1 ¨. . .¨Z rt it : t ě 1, i k ě 0, r k P Zu taken in any order. Then, the metric is globally bounded by 2. Note also that for convex combinations, we have [START_REF] Walters | An introduction to ergodic theory[END_REF] ∆ ˆż ν γ dP pγq, ż κ γ dP pγq ˙ď ż ∆pν γ , κ γ q dP pγq.

We need the following simple observation.

Lemma 6.1. Assume that µ k Ñ µ and let, for

d ě 1, κ d :" ř d`a d k"d α pdq k µ k βpdq ρ d , where α pdq k , β pdq ě 0, ř d`a d k"d α pdq k " 1 ´βpdq . If β pdq Ñ 0 then κ d Ñ µ.
Proof. This follows immediately from [START_REF] Walters | An introduction to ergodic theory[END_REF]. Now, we consider an MRT arithmetic function u, and we take pt m q and ps m q as in Definition 3.1. The proof of Main Theorem (see Section 4.3) yields the following: Lemma 6.2. Fix d ě 0 and choose

# 1 d`1 ă β d ă β 1 d ă 1 d if d ě 1, 1 ă β 0 ă β 1 0 ă 2 if d " 0.
Let ε ą 0. Then, for each m large enough,

∆ pE N puq, ν d q ă ε uniformly in s β d m`1 ď N ď s β 1 d m`1 .
Proof. Assume the result does not hold. Then for infinitely many integers m we can find s

β d m`1 ď N m ď s β 1 d m`1 such that ∆ pE Nm puq, ν d q ě ε.
On the other hand, the proof of Proposition 4.7 shows that, along such a sequence pN m q we must have

E Nm puq Ñ ν d ,
which yields a contradiction. Now, let us fix integers 1 ď D 1 ă D 2 . We consider the convergence of logarithmic empirical measures of u along the increasing sequence of integers pN m q :" ´Ys

1{D1 m`1 ]¯.
Given a small real number ε ą 0, set for each

D 1 ď d ď D 2 β d :" 1 d `1
`ε 2dpd `1q and β 1 d :"

1 d ´ε 2dpd `1q .
Using (29), we write E log Nm puq as a convex combination of the empirical measures E n puq, 1 ď n ď N . We partition t1, . . . , N m u as

Ů D1ďdďD2 I m d \ J m
, where

I m d :" ! n P N : s β d m`1 ď n ď s β 1 d m`1
) , and

J m :" t1, . . . , N m uz ğ D1ďdďD2 I m d .
By Lemma 6.2, for m large, E n puq is close to

ν d for n P I m d . The weight of I m d in the convex combination is 1 L Nm ÿ nPI m d 1 n `1 " mÑ8 D 1 pβ 1 d ´βd q " D 1 ˆ1 d ´1 d `1 ˙p1 ´εq.
It follows that the total weight of Ů D1ďdďD2 I m d is asymptotic, as m Ñ 8, to

D 1 ˆ1 D 1 ´1 D 2 ˙p1 ´εq ą 1 ´ε ´D1 D 2 ,
and then for m large enough the weight of J m is bounded by ε `D1 D2 . In view of Lemma 6.2 and (30), any weak ˚limit of E log Nm puq can be written as

D 1 p1 ´εq ÿ D1ďdďD2 ˆ1 d ´1 d `1 ˙νd `αρ,
where ρ is some shift-invariant probability measure on X u and 0 ď α ď ε `D1 D2 . Letting ε Ñ 0 and D 2 Ñ 8, we see (cf. Lemmma 6.1) that for any D 1 ě 1 there exists a logarithmic Furstenberg system of u whose invariant measure is

D 1 ÿ děD1 ˆ1 d ´1 d `1 ˙νd .
Note that, under this measure, the distribution of pZ 0 , . . . , Z D1 q is pLeb S 1 q bpD1`1q . By Proposition 2.2 (which is also valid in the logarithmic case), we can also find pLeb S 1 q bN as an invariant measure of a logarithmic Furstenberg system of u. Thus, we have proved the following result: Corollary 6.3. Each u P MRT satisfies the logarithmic Chowla conjecture along a subsequence.

Getting ν 0

Under the assumptions given in Definition 3.1, we can easily modify the above argument to incorporate ν 0 in the weak ˚limit of E log Nm puq: we just have to take D 1 " 0 and choose β 0 :" 1 `ε, β 1 0 :" 2 ´ε. The weak ˚limit we get has then the form

1 2 ˜ν0 `ÿ dě1 ˆ1 d ´1 d `1 ˙νd ¸.
If in the construction of the MRT sequence we add the extra assumption that s am m`1 ď t m`1 , with a m Ñ 8, which is compatible with the rest, then we can now take β 1 0 as large as we want, so that the weight of ν 0 in the weak ˚limit is as close to 1 as we want. Finally, we get a logarithmic Furstenberg system of u with ν 0 as an invariant measure.

It is not clear however if we can get ν d (d ě 1) for some logarithmic Furstenberg system of an MRT function.

Absence of zero mean on typical short interval

Motivated by Matomäki-Radziwiłł's theorem [START_REF] Matomäki | Multiplicative functions in short intervals[END_REF] Proof. Suppose that u has zero mean on short intervals. Assume that lim kÑ8 E M k puq " κ. Now, the system pS, X u , κq is the identity if and only if Z 0 " Z 0 ˝S κ-a.e.

Let ε ą 0 and choose H 0 so that for H ą H 0 , we have

lim sup kÑ8 1 M k ÿ 1ďmďM k ˇˇˇˇ1 H ÿ 0ďhăH upm `hq ˇˇˇˇă ε. Take H ą H 0 . Then lim sup kÑ8 1 M k ÿ 1ďmďM k ˇˇˇˇ1 H ÿ 0ďhăH upm `hq ˇˇˇˇ" lim sup kÑ8 1 M k ÿ 1ďmďM k ˇˇˇˇ1 H ÿ 0ďhăH Z 0 pS h`m uq ˇˇˇˇ" lim sup kÑ8 ż Xu ˇˇˇˇ1 H ÿ 0ďhăH Z 0 ˝Sh ˇˇˇˇd E M k puq " ż Xu ˇˇˇˇ1 H ÿ 0ďhăH Z 0 ˝Sh ˇˇˇˇd κ " ż Xu |Z 0 | dκ ě α ą 0, a contradiction.
Corollary 6.5. If u P MRT then u has no zero mean on short intervals.

Strong stationarity

We now show that, for each d ě 0, the stationary process pZ n q with distribution ν d is strongly stationary (see [START_REF] Jenvey | Strong stationarity and de Finetti's theorem[END_REF]), i.e. for each s 1 ă . . . ă s k and each r ě 1, the distributions of the vectors pZ s1 , . . . , Z s k q and pZ rs1 , . . . , Z rs k q are the same. Fix d ě 0 and we recall that the process pZ n q with the distribution ν d is the same as of the process pf ˝T n d q, see (34) in Appendix for the definition of T d and f px 1 , . . . , x d q " e i2πx d . Then f ˝T s d px 1 , . . . , x d q " e 2πipp s d´1 qx1`p s d´2 qx2`...`p s 0 qxdq .

It follows that for each s 1 ă . . . ă s k and each choice of integers j 1 , . . . , j k , we have

EZ j1 s1 ¨. . . ¨Zj k s k ‰ 0 ô k ÿ i"1 j i ˆsi ℓ ˙" 0 for ℓ " 0, . . . , d ´1 ô k ÿ i"1
s ℓ i " 0 for ℓ " 0, . . . , d ´1.

Therefore, EZ j1 s1 ¨. . . ¨Zj k s k " 0 ô EZ j1 rs1 ¨. . . ¨Zj k rs k " 0 for each r ě 1.

Moreover, EZ j1 s1 ¨. . . ¨Zj k s k ‰ 0 is equivalent to EZ j1 s1 ¨. . . ¨Zj k s k " 1. It follows that the process pZ n q with the distribution ν d is strongly stationary.

It follows that all Furstenberg systems which have been found in the paper are given by stationary processes which are strongly stationary. Note also that since every convex combination of strongly stationary processes remains strongly stationary, also the logarithmic Furstenberg systems determined in Section 6.1 are strongly stationary which fits perfectly to a general result of Frantzikinakis and Host [START_REF] Frantzikinakis | The logarithmic Sarnak conjecture for ergodic weights[END_REF], [START_REF] Frantzikinakis | Furstenberg systems of bounded multiplicative functions and applications[END_REF] about logarithmic Furstenberg systems of strongly aperiodic multiplicative functions.

We end by asking the following question, which has been suggested to us by both Nikos Frantzikinakis and Florian Richter: can we find a Furstenberg system of some u P MRT which is the direct product of a Bernoulli shift and a unipotent system?

Appendix

We aim at showing that from the dynamical point of view there is a close relation between the processes pY pdq q dě0 which appeared in Section 4 and the concept of quasi-eigenfunction in ergodic theory.

Algebraic constraints and quasi-eigenfunctions

Given a standard Borel probability space pX, B, µq, let MpXq denote the set of all measurable functions of modulus 1 defined on X. Endowed with the pointwise multiplication and the L 2 -topology, it becomes a Polish group. Given an automorphism T on pX, B, µq, set E 0 pT q :" tg P MpXq : g ˝T " gu and then inductively, define E d`1 pT q :" tf P MpXq : f ˝T {f P E d pT qu, d ě 0.

Note that each E d pT q is a group which is called the group of d-quasi-eigenfunctions. For w P MpXq and m ě 1, we set w pmq " w ˝T m´1 ¨. . . ¨w with w p0q " 1. Assume that g P E d pT q and let g ˝T " hg with h " g ˝T {g P E d´1 pT q. Then, for each ℓ 1 , . . . , ℓ k , we have

k ź i"1 g ˝T ℓi " k ź i"1 h pℓiq g " g k k ź i"1 h pℓiq " g k k ź i"1 ˜ℓi´1 ź j"0 h ˝T j ¸" g k k ź i"1 ℓi´1 ź j"0 ˆh ˝T h ˙pjq h " g k k ź i"1 h ℓi ℓi´1 ź j"0 ˆh ˝T h ˙pjq " g k h ř k i"1 ℓi k ź i"1 ℓi´1 ź j"0 ˆh ˝T h ˙˝T j´1 ¨. . . ¨ˆh ˝T h ˙" g k h ř k i"1 ℓi k ź i"1 ℓi´1 ź j"0 j´1 ź p"0 ˆph ˝T {hq ˝T h ˝T {h ˙ppq h ˝T h " g k h ř k i"1 ℓi k ź i"1 ℓi´1 ź j"0 ˆh ˝T h ˙j j´1 ź p"0 ˆph ˝T {hq ˝T h ˝T {h ˙ppq " g k h ř k i"1 ℓi ˆh ˝T h ˙řk i ℓ i pℓ i ´1q 2 k ź i"1 ℓi´1 ź j"0 j´1 ź p"0 ˆph ˝T {hq ˝T h ˝T {h ˙ppq " . . .
It follows that for the stationary process pg ˝T n q nPN the following holds: whenever ℓ i , ℓ 1 i satisfy

ř k i"1 ℓ j i " ř k i"1 ℓ 1j i for j " 0, 1, . . . , d, we have (31) k ź i"1 g ˝T ℓi " k ź i"1 g ˝T ℓ 1 i provided that g P E d pT q.
In fact, the processes given by quasi-eigenfunctions are the only satisfying the algebraic relation (31). Proposition 7.1. Assume that pZ n q is a stationary S 1 -valued process. Then for each d ě 0, Z 0 P E d pT q if and only if

(32) Z ℓ1 ¨. . . ¨Zℓ k " Z ℓ 1 1 ¨. . . ¨Zℓ 1 k
for each ℓ i , ℓ 1 i for which

ř k i"1 ℓ j i "
ř k i"1 ℓ 1j i for j " 0, 1, . . . , d assuming that (33) tℓ 1 , . . . , ℓ l u X tℓ 1 1 , . . . , ℓ 1 k u " H.

Proof. We can assume that Z n " Z 0 ˝T n for an automorphism T of pX, B, µq. Now, we proceed by induction. Since the first condition

(d " 0) is just Z 0 ˝T ℓ1 . . . ¨˝Z 0 ˝T ℓ k " Z 0 ˝T ℓ 1 1 ¨. . . ¨Z0 ˝T ℓ 1
k whichever numbers we take, we have (by taking k " 1, ℓ 1 " 1 and ℓ 1 1 " 0) Z 0 ˝T " Z 0 , so Z 0 P E 0 pT q. Notice that condition (33) is superfluous since if some of the numbers ℓ i are equal to ℓ 1 j , then we can just cancel out the relevant factors. So, assume now that Z 0 satisfies (32) and let f " Z 0 ˝T {Z 0 . We need to prove that f satisfies the relation (32) up to d (if so, then by the induction assumption f P E d pT q, whence Z 0 P E d`1 pT q). That is, we want to prove that

f ˝T ℓ1 ¨. . . ¨f ˝T ℓ k " f ˝T ℓ 1 1 ¨. . . ¨f ˝T ℓ 1 k provided that ř k i"1 ℓ j i " ř k i"1 ℓ 1j i for j " 0, 1, . . . , d. Equivalently, we want to show that Z 0 ˝T ℓ1`1 ¨. . . ¨Z0 ˝T ℓ k `1 ¨Z0 ˝T ℓ 1 1 ¨. . . ¨Z0 ˝T ℓ 1 k " Z 0 ˝T ℓ 1 1 `1 ¨. . . ¨Z0 ˝T ℓ 1 k `1 ¨Z0 ˝T ℓ1 ¨. . . ¨Z0 ˝T ℓ k .
But we clearly have

pℓ 1 `1q d`1 `. . . `pℓ k `1q d`1 `ℓ1d`1 1 `. . . `ℓ1d`1 k " pℓ 1 1 `1q d`1 `. . . `pℓ 1 k `1q d`1 `ℓd`1 1 `. . . `ℓd`1 k
(let alone if we replace d `1 by a smaller j), hence the result.

Remark 7.2. Assume additionally that T is totally ergodic (i.e. all nonzero powers are ergodic). Then, following [START_REF] Abramov | Metric automorphisms with quasi-discrete spectrum[END_REF], T has quasi-discrete spectrum if span `Ťdě0 E d pT q ˘" L 2 pX, B, µq (sometimes, T is called an Abramov automorphism). Quasi-discrete spectrum automorphisms are basically affine automorphisms of compact, Abelian, metric groups, see [START_REF] Hahn | Minimal dynamical systems with quasi-discrete spectrum[END_REF] for more details.

Remark 7.3. Knowing that Z 0 P E d pT q does not determine the whole process pZ m q mPN . Indeed, for example, if we take the ergodic decomposition of T , then the process pZ m q (considered with respect to an ergodic component) will still satisfy the same algebraic relation even though the distribution of the process may have changed.

Algebraic constraints and n-independence

We will be interested in the (very) non-ergodic case. Given d ě 1, consider the (unipotent) automorphism T d : T d Ñ T d , defined by: (34) T d px 1 , . . . , x d q :" px 1 , x 1 `x2 , . . . , x d´1 `xd q (hence T 1 is just the identity on T). Set f px 1 , . . . , x d q " e i2πx d . Then, `f T {f ˘px 1 , . . . , x d q " e i2πx d´1 , so by induction, it is easy to show that f P E d pT d q.

Denote by Erg K the class of automorphisms disjoint from all ergodic automorphisms. Proof. To prove that T d P Erg K , we first notice that its ergodic decomposition is given by the tori (on which we consider the relevant Lebesgue measure) T d´1

x " T d´1 with x P T irrational (and we consider Lebesgue measure3 on T) on which the action T d,x is given by px 2 , x 3 . . . , x d q Þ Ñ px 2 `x, x 2 `x3 , . . . , x d´1 `xd q.

If an ergodic automorphism S acting on pY, C, νq is non-disjoint with T d,x then since T d,x is an ergodic compact group extension of an irrational rotation, the associated Koopman operator U S must share a common eigenvalue with the Koopman operator U T d,x given by T d,x (cf. [START_REF] Glasner | Ergodic Theory via Joinings[END_REF], Chapter 6). As U S can have only countably many eigenvalues and the measure on the space of ergodic components is continuous, we can assume that S is disjoint with all ergodic components. Take any joining ρ of T d and S. Let Now, choose any integers r 1 ă r 2 ă . . . r d and q j P Z for j " 1, . . . , d. We want to show that the distribution of the vector pf q1 ˝T r1 d , . . . , f q d ˝T r d d q is pLeb T q bd and for that we need to check that (37) Epf q1 ˝T r1 d ¨. . . ¨f q d ˝T r d d q " 0 unless q 1 " . . . " q d " 0. Now, in view of (36), the negation of (37) is equivalent to d ÿ j"1 q j ˆrj k ˙" 0 for k " 0, 1, . . . , d ´1.

It is not hard to see that this system of linear equations is equivalent to d ÿ j"1 q j r k j " 0 for k " 0, 1, . . . , d ´1.

However, the determinant here is the Vandermonde determinant and since the numbers r j are pairwise different, our claim follows.

Remark 7.5. By the proof of Proposition 4.1 (see also Remark 4.2), under the assumptions of Proposition 4.1, we obtain that the dynamical system corresponding to the stationary process pZ n q is (up to isomorphism) just T d .

Remark 7.6. If T acting on pX, B, µq is additionally ergodic then E 0 pT q consists of the constants while E 1 pT q consists of the eigenfunctions of the Koopman operator U T acting on L 2 pX, B, µq. Hence, if pZ n q nPN is a stationary process satisfying (38) Z 0 ¨Z2 " Z 2 1 then, using also Proposition 7.1, the ergodic components of the corresponding dynamical system must have discrete spectra. 4 A prominent example of such a situation is the automorphisms T 2 px 1 , x 2 q " px 1 , x 1 `x2 q on T 2 whose ergodic components are (all) irrational rotations (and Z 0 is given by f 2 : px 1 , x 2 q Þ Ñ e i2πx2 ).

( 17 )

 17 P d pnq :" π d pn, n `1, . . . , n `dq, and Q d pnq :" πd pn, n `1, . . . , n `dq.

  `1qP d pnq ´Pd pn `1qQ d pnq " `Pd pn `1q `Rd pn `1q ˘Pd pnq ´Pd pn `1q `Pd pnq `Rd pnq " R d pn `1qP d pnq ´Rd pnqP d pn `1q.

nÑ8 0 ,

 0 Archimedean characters fall into the category of slowly varying arithmetic functions, that are bounded arithmetic functions u satisfying upn `1q ´upnq ÝÝÝÑ nÑ8 0.

Remark 5 . 4 .

 54 (a) Note that the subshift generated by a mean slowly varying function u : N Ñ D can be of infinite entropy: we can modify a slowly varying function on a subset of density zero to get a mean slowly varying function generating the full subshift pD N , Sq.(b) If u is mean slowly varying and if, for each ℓ ě 1, the limit lim exists then there is only one Furstenberg system of u: there exists a shiftinvariant measure ν such that E N puq Ý ÝÝÝ Ñ N Ñ8

Proposition 6 . 4 .

 64 concerning strongly aperiodic multiplicative functions, we say that u : N Ñ D has zero mean on typical short interval if lim Assume that lim inf ℓÑ8 1 ℓ ř 1ďjďℓ |upjq| ": α ą 0. If u has identity as a Furstenberg system then u has no zero mean on short intervals.

Proposition 7 . 4 .

 74 For each d ě 1, T d P Erg K . Moreover, the stationary process pf ˝T m d q mPN is d-independent, i.e. for all m 1 ă . . . ă m d the variables f T m1 d , . . . , f ˝T m d d are independent.

  pγq be its ergodic decomposition. Then, for P -a.e. γ, the projection ρ γ | Y of ρ γ on Y equals ν since ν is ergodic. Moreover, (35) Leb T bd " ρ| T d " ż ρ γ | T d dP pγq is a decomposition of Leb T bd into ergodic measures. By the uniqueness of ergodic decomposition, it is (35) which is the ergodic decomposition, and (by disjointness) we obtain that ρ " ż T pδ x b pLeb T q bpd´1q q b ν dLeb T pxq and the first claim easily follows. It is not hard to check that by elementary properties of Pascal triangle:(36) T r d px 1 , . . . , x d q " ´x1 , . . . ,

  Our purpose is to apply Proposition 4.3 to u, therefore we have to evaluate the expressions of the form Note that, by Proposition 4.4, for d ě 1, the d terms of highest degrees in P d and Q d coincide. In particular, we have

	(19)		1 N m	1ďnďNm ÿ	φ ℓ d `upnq, upn `1q, . . . , upn `dq ˘.
	By Proposition 3.4, we have
		1 N m	1ďnďNm ÿ	φ ℓ d `upnq, . . . , upn `dq "
			1 N m	1ďnďNm ÿ	φ ℓ d	`nism`1 , . . . , pn `dq ism`1 ˘`op1q
	(20)	"	1 N m	1ďnďNm ÿ	e	iℓsm`1 log	P d pnq Q d pnq `op1q.
	Therefore, in view of applying Proposition 4.1, we can replace an expression of
	the form (19) by
						1 N m	1ďnďNm ÿ	e iℓsm`1f d pnq ,
	where f d is defined by
	(21)							f d pxq :" log	P d pxq Q d pxq	.
								lim xÑ8	P d pxq Q d pxq	" 1,

so f d pxq is well defined for x P Ê large enough. We will use the following results on the asymptotic behaviour of f d . Lemma 4.6. For each d ě 1, there exists K d ‰ 0 such that

[START_REF] Matomäki | Multiplicative functions in short intervals[END_REF]

  1 x and the result concerning f 1 0 is obvious. We consider now the case d ě 1. From Proposition 4.4, we can write where deg P d " 2 d´1 and deg R d " 2 d´1 ´d, which yields[START_REF] Matomäki | Multiplicative functions in short intervals[END_REF].

	f d pnq " log	P d pnq P d pnq `Rd pnq	" ´log ˆ1	P d pnq `Rd pnq	˙,
	Deriving f d gives
	f 1 d "

  the degrees of P d and R d are different, the degree of the numerator is equal to deg P d `deg R d ´1 " 2 d ´pd `1q. But the degree of the denominator is deg P d `deg Q d " 2 d , and this gives

Their functions are totally aperiodic, that is, all powers u k , k ě 1 are also aperiodic.

Cf. this equality with a remark after Theorem 10 in[START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF] as a topological instance of validity of Chowla conjecture.

That is, we identify the space of ergodic components with pT, Leb T q.

Note that by Proposition 7.1 it follows that Z 1 " cZ 0 , whence (34) is satisfied wheneverř k i"1 ℓ i " ř k i"1 ℓ 1 i .

Acknowledgments Research of the first and second authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364. The second and third authors would like to thank the American Institute of Mathematics for hosting a workshop Sarnak's Conjecture, where this work was first discussed.