Adam Kanigowski

Joanna Kułaga-Przymus

Mariusz Lemańczyk

Thierry De La Rue

On arithmetic functions orthogonal to deterministic sequences

We prove Veech's conjecture on the equivalence of Sarnak's conjecture on Möbius orthogonality with a Kolmogorov type property of Furstenberg systems of the Möbius function. This yields a combinatorial condition on the Möbius function itself which is equivalent to Sarnak's conjecture. As a matter of fact, our arguments remain valid in a larger context: we characterize all bounded arithmetic functions orthogonal to all topological systems whose all ergodic measures yield systems from a fixed characteristic class (zero entropy class is an example of such a characteristic class) with the characterization persisting in the logarithmic setup. As a corollary, we obtain that the logarithmic Sarnak's conjecture holds if and only if the logarithmic Möbius orthogonality is satisfied for all dynamical systems whose ergodic measures yield nilsystems.

.

2 In what follows, we replace CZE by C F , where F is a general characteristic class. Note that the argument u ¨v K C F whenever u K C F and v is an arbitrary F -sequence (cf. Def. 1.1) persists.

3 Not all invariant measures, even in a transitive subshift, have to be visible: let Xy Ă t0, 1, 2u N be given by y "

1 2 δ0 `1 2 δ2 (two fixed points). If A :" tx P Xy : x0 " 0 or x0 " 2u, then νpAq " 1 but no window in y has the property that the frequency of 0 jointly with 2 on it is close to the value 1.

Introduction

All transformations in this paper are assumed to be invertible. A topological dynamical system is a pair pX, T q, where T is a homeomorphism of a compact metric space X. A measure-theoretic dynamical system is a system of the form pZ, BpZq, κ, Rq where R is an automorphism (invertible measure-preserving transformation) of a standard Borel probability space pZ, BpZq, κq.

Sarnak's conjecture Given a topological dynamical system pX, T q and a bounded arithmetic function u : N Ñ C, we consider the corresponding problem of orthogonality: [START_REF] Abdalaoui | Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals[END_REF] lim

N Ñ8
1 N ÿ nďN f pT n xqupnq " 0 for all f P CpXq and x P X.

Once (1) holds for pX, T q, we say that the system pX, T q satisfies the Sarnak property with respect to u and write u K pX, T q. When a class C of topological dynamical systems is given, and u K pX, T q for each pX, T q P C then we write u K C .

The main motivation to study this orthogonality problem is Sarnak's conjecture [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF] on Möbius orthogonality in which u is the Möbius function µ (or, equivalently [START_REF] Ferenczi | Sarnak's Conjecture -what's new[END_REF], the Liouville function λ) and C is the class C ZE of zero topological entropy dynamical systems. Sequences of the form pf pT n xqq with pX, T q running over C ZE , f P CpXq and x P X are often called deterministic sequences. Focusing on a special u in [START_REF] Abdalaoui | Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals[END_REF], say, being multiplicative, is important if we count on applications in number theory -the main motivation of Sarnak himself for the Möbius orthogonality conjecture was to "attack" the celebrated Chowla conjecture on auto-correlations of the Möbius function dynamically (indeed the Chowla conjecture implies Sarnak's conjecture [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF], see also [START_REF] Abdalaoui | The Chowla and the Sarnak conjectures from ergodic theory point of view[END_REF][START_REF] Tao | The Chowla and the Sarnak conjecture, What's new[END_REF]). 1 However, if we aim at providing an internal characterization of u being orthogonal to C ZE , then dropping the assumption of multiplicativity of u seems to be reasonable. Let us give an argument for that. First, note that the class of deterministic sequences equipped with the coordinatewise multiplication and addition is a ring (this is an easy consequence of the fact that the zero entropy class is closed under taking joinings and factors). Moreover, the class of bounded sequences u orthogonal to C ZE is a module over this ring. In other words, even if our "starting" u displays some additional arithmetic property (like multiplicativity), the characterization which we aim at must still hold for u ¨v, where v is an arbitrary deterministic sequence. 2 Of course, properties like the multiplicativity of u ¨v changes dramatically if v is arbitrary, while the characterization we are looking for has to be stable under such a change of u.

Visible measures and Furstenberg systems Let us now briefly discuss the matter of the topological and measure-theoretic aspects of choosing a class C . Recall that M pXq stands for the set of probability (Borel) measures on X and M pX, T q is the (always non-empty) subset of T -invariant measures. Both spaces are compact in the weak-˚-topology. There is a third natural subspace V pX, T q Ă M pX, T q which is the set of (T -invariant) visible measures, i.e. measures possessing a quasi-generic point. Formally, ν P V pX, T q if, for some point x P X and some increasing sequence pN q, we have lim

Ñ8 1 N ÿ nďN δ T n x " ν. 3
We also write ν P V T pxq and say that x is generic for ν along pN q. Note that the set V pX, T q of visible measures contains the set M e pX, T q of all ergodic measures, but it may be strictly smaller than M pX, T q. To illustrate this, consider the automorphism px, yq Þ Ñ px, x `yq of T 2 in which each point is generic (i.e. generic along the whole sequence of natural numbers) and the corresponding measure-theoretic system is either a rational or an irrational rotation on the circle. Hence, each visible measure gives rise to a system with discrete spectrum, while there are many other invariant measures which yield partly continuous spectrum. Given a bounded u : N Ñ C, |u| ď L, we first extend it to Z by setting up´nq " upnq for n P N, up0q P D L :" tz P C : |z| ď Lu, and then take the closure X u :" tS k u : k P Zu of the orbit of u P D Z L under the left shift S : D Z L Ñ D Z L , Sppv k q kPZ q " pv k`1 q kPZ (recall that we only consider invertible systems). Hence, pX u , Sq is the subshift determined by u. By the very definition, u P X u . Each measure κ P V S puq yields a measure-theoretic system pX u , BpX u q, κ, Sq called a Furstenberg system of u (by some abuse of vocabulary, we may call κ itself a Furstenberg system of u). Fix x P X and consider the sequence p 1 N ř nďN δ pT n x,S n uq q. By compactness of M pX ˆXu q, we can select an increasing sequence pN q so that 1 N ÿ nďN δ pT n x,S n uq Ñ ρ, where ρ P V T ˆS ppx, uqq Ă V pX ˆXu , T ˆSq. The projections of ρ on X and X u are denoted by ν and κ, respectively, so that ρ is a joining of ν and κ. Since ρ is a visible measure, so are ν and κ:

(2)

ν P V T pxq, and κ yields a Furstenberg system of u. Now, setting

(3) π 0 :

# X u Ñ C z " pz n q nPZ Þ Ñ z 0 , we have lim Ñ8 1 N ÿ nďN f pT n xqupnq " lim Ñ8 ż f b π 0 d ˜1 N ÿ nďN δ pT n x,S n uq " ż f b π 0 dρ " ż E ρ pf |X u q ¨π0 dκ. (4)
So, the fact that the limit is 0 in (1), i.e. u K pX, T q, depends on the joinings of measures from V pX, T q with those from V S puq -the invariant measures from M pX, T qzV pX, T q are irrelevant in this context. More precisely, what will matter is the "geometric position" of the single continuous function π 0 in the L 2 -space of all Furstenberg systems κ P V S puq for such joinings. Namely, inside the L 2 pρq-space, we want π 0 to be orthogonal to L 2 pνq (in condition [START_REF]Sarnak's Conjecture[END_REF] we use all continuous functions f on X and we take into account all ν P V pX, T q). Of course, whether this orthogonality can be established without referring to the (visible) joinings ρ (and remaining at the level of κ P V S puq) is another question. A kind of surprise is that the answer to this question will turn out to be positive for C ZE and some other characteristic classes, see Theorem A, below.

Characteristic classes and Veech's conjecture With the above in mind, a knowledge of some fundamental facts on joinings, especially on disjointness in the sense of Furstenberg in ergodic theory, strongly suggests to consider only the situation when the measures ν appearing above yield measure-theoretic systems pX, BpXq, ν, T q belonging to one of so called characteristic classes of measure-theoretic dynamical systems.

Definition 1.1. A class F of measure-theoretic dynamical systems is called characteristic if it is closed under taking isomorphisms, factors and (countable) joinings. Given a characteristic class F , by an F -factor of a measuretheoretic dynamical system pZ, D, κ, Rq we call any factor sub-σ-algebra of D on which the action of R yields a system in the class F . We denote by C F the class of topological systems pX, T q for which we have pX, BpXq, ν, T q P F for each ν P V pX, T q. The sequences of the form pf pT n xqq nPZ with f P CpXq, x P X and pX, T q P C F are called F -sequences.

It follows from the above definition that:

(a) every measure-theoretic dynamical system pZ, D, κ, Rq has a largest Ffactor (in the sense of inclusion of sub-σ-algebras), which we denote by D F , (b) any joining of pZ, D, κ, Rq with a system from F is uniquely determined by its restriction to the joining of the largest F -factor D F of R with the given system from F (see Section 2 for details). The class ZE of zero (Kolmogorov-Sinai) entropy is of course characteristic. The largest zero entropy factor D ZE of pZ, D, κ, Rq is called the Pinsker factor and is denoted by ΠpRq or Πpκq. Note that (via the variational principle) the family C ZE is precisely the family of all topological systems whose all invariant measures yield systems in ZE. Returning to Sarnak's conjecture, in [START_REF] Veech | Möbius Dynamics[END_REF], Veech proves the following result: [START_REF] Veech | Möbius Dynamics[END_REF]). If

Theorem 1.1 ([
(5) π 0 K L 2 pΠpκqq for each Furstenberg system κ P V S pµq then Sarnak's conjecture holds, i.e. µ K C ZE . 4Then he formulates the following conjecture (Conjecture 24.3 page 88 in [START_REF] Veech | Möbius Dynamics[END_REF]):

Conjecture 1 (Veech's conjecture). Condition (5) is equivalent to Sarnak's conjecture.

One of motivations for the present work was to prove the above conjecture. Let us first formulate (5) in full generality. Definition 1.2. Given a characteristic class F we say that a (bounded) arithmetic function u : N Ñ C satisfies the Veech condition with respect to F if [START_REF] Bergelson | Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation[END_REF] π 0 K L 2 ppBpX u q, κq F q for each Furstenberg system κ P V S puq.

Given a characteristic class F , we denote by F ec the class of those automorphisms R of pZ, D, κq such that a.e. ergodic component of κ yields a system in F . Then (see Section 2.3), F ec is also a characteristic class, [START_REF] Bourgain | From Fourier and Number Theory to Radon Transforms and Geometry[END_REF] C F Ă C Fec and (8) C Fec " tpX, T q : pX, BpXq, ν, T q P F for each ergodic ν P M pX, T qu.

Our main result is the following:

Theorem A. Assume that F is a characteristic class. Let u : N Ñ C be a bounded arithmetic function. Then u K C Fec if and only if u satisfies the Veech condition (6) (with respect to F ec ).

Remark 1.2. A subsequence version of this result also holds: If pN q is an increasing sequence of integers, then u is pN q-orthogonal to C Fec (i.e.

1 N ř nďN f pT n xqupnq Ñ 0 for each pX, T q P C Fec and all f P CpXq, x P X) if and only if the Veech condition (6) holds for each measure κ for which u is generic along some subsequence of pN q. The reason for the validity of this "local" version is that all tools used in the proof of Theorem A work well on subsequences, see also Remarks 1.4 and 2.20 to cope with the strong u-MOMO property along subsequences.

See [START_REF] Chinis | Siegel zeros and Sarnak's conjecture[END_REF] for the validity of the alternative: either there are no Siegel zeros or there exists a (universal) subsequence along which Sarnak's conjecture holds and [START_REF] Gomilko | Möbius orthogonality in density for zero entropy dynamical systems[END_REF] for a density version of Sarnak's conjecture.

implicit in [START_REF] Abdalaoui | The Chowla and the Sarnak conjectures from ergodic theory point of view[END_REF], where the implication "Chowla conjecture ñ Sarnak's conjecture" has been proved using joinings. Veech cites [START_REF] Abdalaoui | The Chowla and the Sarnak conjectures from ergodic theory point of view[END_REF] (which was on arXiv two years before Veech's lecture notes [START_REF] Veech | Möbius Dynamics[END_REF] appeared) but instead he gives his own (slightly complicated) proof using the concept of quasi-factors of Glasner and Weiss.

Theorem A for u " µ and F " ZE (note that ZE ec " ZE) has the following consequence: Corollary 1.3. Veech's conjecture holds. Moreover, if Sarnak's conjecture holds then all Furstenberg systems κ P V S pµq have positive entropy.

We also generalize Veech's theorem (Theorem 1.1) to the setting of characteristic classes:

Theorem B. Assume that F is a characteristic class. Let u : N Ñ C be a bounded arithmetic function. If u satisfies the Veech condition (6) with respect to F then u K C F .
Let us now briefly discuss the methods involved in the proofs of the above theorems. The proof of Theorem B is provided in Section 4.1. It is a straightforward application of the above Property (b) of the largest F -factor (see Proposition 2.2 for more details). The proof of Theorem A which we provide in Section 4.2 relies on:

• so called Hansel models [START_REF] Hansel | Strict Uniformity in Ergodic Theory[END_REF] of possibly highly non-ergodic measurepreserving systems,

• our version of a lifting lemma by Conze, Downarowicz, Serafin [START_REF] Conze | Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles[END_REF] on the existence of generic points for joinings, valid in a general context and using the concept of quasi-generic sequences (Proposition 3.1),

• general joining techniques.

Our first proof of Theorem A was based on a different lifting lemma by Bergelson, Downarowicz and Vandehey (Theorem 5.16 in [START_REF] Bergelson | Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation[END_REF]) and a recent result by Downarowicz and Weiss showing the existence, for each zero-entropy measure-theoretic system, of a special Hansel model which is symbolic [START_REF] Downarowicz | Pure strictly uniform models of non-ergodic measure automorphisms[END_REF]. The proof we finally chose to present here does not require the use of this symbolic model, as our lifting lemma works for all topological systems. Its additional advantage is that it is also valid in the context of logarithmic averages (see Section 3.3). This makes all our results also true in the logarithmic set up: for example, the logarithmic Sarnak's conjecture (denoted as µ K log C ZE ) is equivalent to the Veech condition for µ for all logarithmic Furstenberg systems κ P V log S pµq, see Corollary 1.12 for more. In Theorem A, it is crucial that we deal with a characteristic class of the form F ec since, by Proposition 2.17, u K C Fec is equivalent to the so called strong u-MOMO property for systems in C Fec . This property, whose definition is recalled below, has been introduced in [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF][START_REF] Gomilko | Möbius orthogonality in density for zero entropy dynamical systems[END_REF]. We leave as an open problem whether Theorem A holds for an arbitrary characteristic class. Definition 1.3. A topological system pX, T q satisfies the strong u-MOMO property, if [START_REF] Conze | Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles[END_REF] lim

KÑ8 1 b K ÿ kăK › › › › › › ÿ b k ďnăb k`1 upnqf ˝T n › › › › › › CpXq " 0
for each f P CpXq and each increasing sequence

pb k q Ă N such that b k`1 bk Ñ 8.
Clearly, the strong u-MOMO property implies (1) uniformly in x P X. The concept of strong u-MOMO is formally stronger than the usual orthogonality. To see the difference between the usual orthogonality and strong MOMO, consider the system px, yq Þ Ñ px, x `yq on T 2 whose Möbius orhogonality follows easily from the DDKBSZ criterion,5 see e.g. [START_REF] Ferenczi | Sarnak's Conjecture -what's new[END_REF]) (in fact, the orthogonality holds even uniformly due to the Davenport estimate [START_REF] Davenport | On some infinite series involving arithmetical functions[END_REF]) , while the strong µ-MOMO property (apply the definition to f px, yq " e 2πiy ) yields

1 b K ÿ kăK sup xPT ˇˇˇˇˇÿ b k ďnăb k`1 µpnqe 2πinx ˇˇˇˇˇÑ 0 when K Ñ 8
which is an open problem,6 see Sections 5.4 and 5.5 for more details.

Remark 1.4. Note that in the definition of the strong u-MOMO property, convergence (9) can be replaced by [START_REF] Davenport | On some infinite series involving arithmetical functions[END_REF] lim

N Ñ8 1 N ¨ÿ kăK N › › › › › › ÿ b k ďnăb k`1 upnqf ˝T n › › › › › › CpXq `› › › › › › ÿ b K N ďnăN upnqf ˝T n › › › › › › CpXq ‹ '" 0,
where K N :" maxtk : b k ă N u. As a matter of fact, the definition is unchanged if we only restrict to sequences pb k q which further satisfy the condition

lim kÑ8 b k`1 ´bk b k " 0.
(If we have to consider a sequence pb k q which does not satisfy the above condition, we can always add more integers to the set tb k : k ě 1u so that this convergence holds for the new sequence. And the validity of ( 9) or [START_REF] Davenport | On some infinite series involving arithmetical functions[END_REF] for the new sequence is stronger than the same for the former sequence pb k q.) Then it is easy to define also the strong u-MOMO property along an increasing sequence pN q, by restricting convergence [START_REF] Davenport | On some infinite series involving arithmetical functions[END_REF] to the subsequence pN q.

Veech condition and combinatorics Given a characteristic class F , Theorems B and A determine a natural strategy to describe the arithmetic functions u orthogonal to all F -sequences. Namely, we need to describe the F -factors and understand the orthogonality to their L 2 -space (i.e. the Veech condition), hoping that this description can be expressed (for π 0 in X u and a Furstenberg system κ of u) by some asymptotics of the integrals of continuous functions on X u . The final step would be to use the definition of a Furstenberg system to obtain a combinatorial condition on u itself. The first part of the strategy should be seen as an extension of the theory of characteristic factors Z s pT q (given an automorphism T ) and the Gowers-Host-Kra (GHK in what follows) seminorms } ¨}u s for s ě 1 [START_REF] Host | Nilpotent structures in ergodic theory[END_REF]. In this perspective the Veech condition on π 0 is the counterpart of }π 0 } u s " 0 for each Furstenberg system κ P V S puq (and has its combinatorial translation in terms of the GHK seminorm of u). We will give more details on this shortly.

Let us discuss this strategy for the class ZE, see Section 5.1 for details. Here, the characteristic factor of a measure-preserving system is the Pinsker factor. The reader has certainly noticed that even though we study dynamical properties of Furstenberg systems, as a matter of fact, at the end we deal with a process pπ 0 ˝Sn q nPZ , stationary with respect to κ P V S puq (each such measure is invariant in the subshift pX u , Sq). Now, the Veech condition leads to the following concept. Definition 1.4. A centered stationary process X " pX n q taking finitely many values is called a Sarnak process if EpX 0 | σpX N , X N `1, . . .qq Ñ 0 in L 2 (or a.e.); equivalently EpX 0 | ΠpXqq " 0, where ΠpXq stands for the tail σ-algebra.

Understanding the structure of Sarnak processes seems to be a problem of an independent interest and it will be studied elsewhere. Now, when u takes finitely many values, our main result (Theorem A applied to ZE) can be reformulated in the following manner: Corollary 1.5. Let u : N Ñ C be an arithmetic function taking finitely many values. Then u K C ZE if and only if all stationary processes pπ 0 ˝Sn q nPZ determined by κ P V S puq are Sarnak.

From the ergodic theory point of view we are close to the concept of the relative Kolmogorov property (K-property) which is however perturbed by the fact that we need this property for a single function. But even though only π 0 is involved, the dynamical idea of the equivalence between the Kproperty and K-mixing (uniform mixing) works, and we can apply K-mixing of π 0 against the family of functions depending on finitely many non-negative coordinates. This leads to studying the asymptotics of integrals of some continuous functions and finally gives the following combinatorial characterization of the orthogonality of u to all deterministic sequences. In the following corollary, we use the fact that when u takes its values in a finite set, a subset A Ă X u depends on finitely many non-negative coordinates if and only if there exists ě 1 and a set C of blocks of length appearing in u such that 1 A pyq " 1 pyp0q,yp1q,...,yp ´1qqPC .

Corollary C. Let u : N Ñ C be an arithmetic function taking finitely many values. Then u K C ZE if and only if, for each subsequence pN k q defining a Furstenberg system of u and each subset A Ă X u depending on finitely many non-negative coordinates, we have the cancellation phenomenon of the values of u uniformly along sufficiently large shifts of the set of visits of u in A: for each ε ą 0, there exists M ě 1 such that for each ě 1 and each set C of blocks of length , we have for each m ě M [START_REF] Downarowicz | Almost Full Entropy Subshifts Uncorrelated to the Möbius Function[END_REF]

lim kÑ8 ˇˇˇˇˇˇˇ1 N k ÿ nďN k upnq 1 pupm`nq,upm`n`1q,...,upm`n` ´1qqPC loooooooooooooooooooooomoooooooooooooooooooooon 1 A pS m`n uq ˇˇˇˇˇˇˇď ε.
Note that M above depends on pN k q and ε. This combinatorial condition looks more attractive if we assume that u is generic: ˇˇˇˇ" 0 uniformly in ě 1 and in C, a set of blocks of length .

Corollary C'. Let u : N Ñ C
Remark 1.6. Note however that the above two corollaries do not say much if the (clopen) sets C are already of small measures (e.g., in the most interesting case of blocks of large length). In fact, condition [START_REF] Downarowicz | Almost Full Entropy Subshifts Uncorrelated to the Möbius Function[END_REF] in Corollary C is equivalent to the following: for each subsequence pN k q defining a Furstenberg system of u we have the conditional cancellation phenomenon of the values of u uniformly along sufficiently large shifts of the set of visits of u to "typical" blocks. More precisely, for each ε ą 0 there exists M ě 1 such that namely, the first one does not display any cancellation (that is, along the return times to a fixed block, we have no cancellation) and in the second one cancellations are possible and the fact that along further and further shifts of the set of return times we observe more and more cancellations, characterizes the Sarnak property.

Veech's and Sarnak's conjectures for other characteristic classes

The family of all characteristic classes is enormous, see Section 2 for natural examples. Here, let us just notice that the discrete spectrum automorphisms form a characteristic class and it contains uncountably many characteristic subclasses whose pairwise intersections are equal to the class of all identities (indeed, discrete spectrum automorphisms whose group of eigenvalues is contained in Zα, with α irrational, is a characteristic class). Moreover, there are the largest proper and the smallest non-trivial characteristic classes. Indeed, although our study of the zero entropy class was originally motivated by Sarnak's conjecture, yet, ZE plays a special role since it is the largest (proper) characteristic class. In fact, we have

(12) t˚u Ă ID Ă F Ă ZE Ă ALL
for each characteristic class F , where ID stands for the (characteristic) class of identities (of all standard Borel probability spaces) and ALL stands for the (characteristic) class of all systems. Note that ID " t˚u ec and ZE " ZE ec .

Clearly, the topological class C ALL consists of all topological systems and the only u orthogonal to all of them is upnq " 0 on a subset of n of full density which is compatible with the Veech condition (which in this setting means that π 0 equals (a.e.) zero for each Furstenberg system).

The topological class C t˚u consists of topological systems whose all visible measures are given by fixed points. The reader can check that the Veech condition here is just ş Xu π 0 dκ " 0 for each κ P V S puq. The combinatorial condition [START_REF] Downarowicz | Almost Full Entropy Subshifts Uncorrelated to the Möbius Function[END_REF] from Corollary C (equivalent to the Veech condition) in this setting reduces to lim N Ñ8

1 N ř nďN upnq " 0.
Clearly, in this setting, the latter is nothing but the Sarnak condition.

It is not hard to see that the topological class C ID consists of topological systems whose only ergodic measures are Dirac measures at fixed points. The Veech condition here is π 0 K L 2 pI κ q for each κ P V S puq, where I κ stands for the σ-algebra of invariant sets. Finally, the counterparts of Corollaries C and C' are the following.

Corollary 1.8. Let u : N Ñ C be a bounded arithmetic function. Then u K C ID if and only if for each ε ą 0 and each subsequence pN k q defining a Furstenberg system of u, there exists H 0 ě 1 such that for each H ě H 0 ,

(13) lim kÑ8 1 N k ÿ nďN k ˇˇˇˇ1 H ÿ hďH upn `hq ˇˇˇˇ2 ď ε.
Note that if, additionally, u is generic and it satisfies a non-quantitative version of the Matomäki-Radziwiłł [START_REF] Matomäki | Multiplicative functions in short intervals[END_REF] convergence on a typical short interval:

lim M,HÑ8,H"opM q 1 M ÿ nďM ˇˇˇˇ1 H ÿ hďH upn `hq ˇˇˇˇ2 " 0 then u K C ID .
Remark 1.9. It is also worth mentioning that the ID-sequences are precisely the mean slowly varying functions (see Proposition 5.1 in [START_REF] Gomilko | On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao[END_REF]), i.e. (bounded) arithmetic functions v : N Ñ C for which

lim N Ñ8 1 N ÿ năN |vpn `1q ´vpnq| " 0.
Therefore, sequences satisfying ( 13) are precisely those orthogonal to all mean slowly varying functions.

Notice that it follows from (12) that, for a non-trivial class F , the "zero mean condition on a typical short interval" (13) is a necessary condition for u K C F , whereas the condition given by Corollary C is sufficient for u K C F .

In Section 5.5, we discuss the case F " DISP ec , where DISP stands for the (characteristic) class of discrete spectrum automorphisms. In view of (8), C DISPec consists of homeomorphisms whose all ergodic measures yield systems with discrete spectrum. Let u : N Ñ C be bounded. For an increasing sequence of integers pN k q, we set ( 14)

}u} 2 u 1 ppN k qq :" lim HÑ8 1 H ÿ hďH ´lim kÑ8 1 N k ÿ nďN k upnqupn `hq ānd, for s ą 1, (15)
}u} 2 s`1 u s ppN k qq :" lim HÑ8 1 H ÿ hďH }up¨`hq ¨up¨q} 2 s u s ppN k qq ,
whenever all the above limits exist. If N k " k, we set }u} u s :" }u} u s ppN k qq .

Corollary 1.10. Let u : N Ñ C be bounded. Then u K C DISPec if and only if }u} u 2 ppN k qq " 0 for each subsequence pN k q along which u is generic. In particular, if u is generic then }u} u 2 " 0.

The main assertion of Corollary 1.10 is equivalent to }π 0 } u 2 pκq " 0 for each Furstenberg system κ P V S puq. The reason for the validity of this result is that given an automorphism pZ, D, ρ, Rq, we have the equality

D DISPec " Z 1 pR, ρq
which is a consequence of DISP ec " NIL 1 , where NIL s stands for the class of automorphisms whose a.a. ergodic components are inverse limits of s-step nil-automorphisms (see Section 2 for more details).

When we turn to the classes NIL s p" pNIL s q ec q and return to the original Sarnak's conjecture (for u) then clearly π 0 K L 2 pΠpκqq implies π 0 K L 2 pZ s pκqq for each s ě 1. We hence obtain one more necessary condition for u to be orthogonal to all deterministic sequences: Corollary 1.11. Let u : N Ñ C be bounded. If u K C ZE then }u} u s ppN k qq " 0 for each s P N (for each subsequence pN k q along which u is generic).

In Section 2.3.2, we prove that the Sarnak property of u for the fundamental (in ergodic theory) class of distal automorphisms is equivalent to the Veech property of u. We leave as an open problem whether the Veech property can be expressed combinatorially in the distal case.

The logarithmic Sarnak's conjecture As we have already noticed, our results also hold in the logarithmic case. In the corollary below we put together conditions which are equivalent to the logarithmic Sarnak's conjecture. 7Corollary 1.12. Let u " µ or λ. The following conditions are equivalent: (i) u K log C ZE (i.e. zero entropy systems satisfy the logarithmic Sarnak property with respect to u), (ii) u K log C NILs for each s ě 1, (iii) π 0 K L 2 pΠpκqq for each κ P V log S puq (i.e. u satisfies the Veech condition for each logarithmic Furstenberg system of u), (iv) π 0 K L 2 pZ s pκqq for each κ P V log S puq and s ě 1, (v) u satisfies the logarithmic Chowla conjecture.

The equivalence of (ii) and (iv) is due to Theorem A (in its logarithmic form), the equivalence of (iv) and (v) (based on the facts proved by Tao [START_REF] Tao | Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, Number Theory -Diophantine Problems, Uniform Distribution and Applications[END_REF] for the equivalence of the logarithmic Sarnak's and Chowla's conjectures for the Liouville function) is formally proved in [START_REF] Frantzikinakis | Ergodicity of the Liouville system implies the Chowla conjecture[END_REF] (implicit in Section 2.7 therein). Other (needed) implications are standard. Recall also that, by [START_REF] Gomilko | Möbius orthogonality in density for zero entropy dynamical systems[END_REF], (i) is equivalent to the logarithmic strong u-MOMO property of all zero entropy systems.

Note that the equivalence of (i) and (ii) in Corollary 1.12 yields immediately the following.

Corollary 1.13. The logarithmic Sarnak's conjecture holds if and only if for each s ě 1, µ is orthogonal to all systems whose all ergodic measures yield NIL s -systems.

Remark 1.14. Yet, in [START_REF] Huang | Polynomial mean complexity and logarithmic Sarnak conjecture[END_REF] another characterization of logarithmic Sarnak's conjecture is given, namely, the conjecture holds if and only if µ is orthogonal to all topological systems with polynomial mean complexity. It follows from [START_REF] Kanigowski | Slow Entropy of Some Parabolic Flows[END_REF] that the topological (and hence measure theoretic) mean complexity of any zero entropy homogeneous system is polynomial. However, the polynomial mean complexity in [START_REF] Huang | Polynomial mean complexity and logarithmic Sarnak conjecture[END_REF] is considered in the topological setting (for systems that need not be homogeneous), so it seems there is no direct relation between Corollary 1.13 and the main result in [START_REF] Huang | Polynomial mean complexity and logarithmic Sarnak conjecture[END_REF].

Remark 1.15. As Theorem A is true in a larger context, also Corollary 1.12 can be formulated for more general multiplicative functions bounded by 1, cf. Theorem 1.8 in [START_REF] Frantzikinakis | Ergodicity of the Liouville system implies the Chowla conjecture[END_REF].

Averaged Chowla property The "iff" assertion of Theorem A cannot be applied to the class C DISP . However, in Section 5.6, we will show that the Sarnak and Veech conditions are equivalent in this setting for each bounded u : N Ñ C such that [START_REF] Frantzikinakis | Ergodicity of the Liouville system implies the Chowla conjecture[END_REF] all circle rotations satisfy the strong u-MOMO property (strong u-MOMO property has been defined in Definition 1.3).

Corollary 1.16. Assume that u : N Ñ C is bounded by 1 and satisfies [START_REF] Frantzikinakis | Ergodicity of the Liouville system implies the Chowla conjecture[END_REF].

Then u satisfies the Veech condition for the F "DISP. In particular, the Sarnak and Veech conditions are equivalent for F "DISP. Moreover, for every sequence pN k q along which u is generic, u satisfies the averaged 2-Chowla property:

(17) lim HÑ8 1 H ÿ hďH lim kÑ8 1 N k ˇˇˇˇÿ nďN k upnqupn `hq ˇˇˇˇ" 0
and, for all sequences v 1 , . . . , v k (bounded by 1), we have

(18) lim HÑ8 1 H k ÿ h 1 ,...,h k ďH lim kÑ8 1 N k ˇˇˇˇÿ nďN k upnq k ź i"1 v i pn `hi q ˇˇˇˇ" 0.
Property [START_REF] Frantzikinakis | Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam[END_REF], called the averaged Chowla property (cf. [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF]), follows from ( 17) -this will be shown in Appendix A. For an alternative approach to obtain the assertions of the above corollary, see the method in [START_REF] Frantzikinakis | An averaged Chowla and Elliott conjecture along independent polynomials[END_REF]: for (17) cf. Thm. 4.1 and Prop. 4.3 [START_REF] Frantzikinakis | An averaged Chowla and Elliott conjecture along independent polynomials[END_REF], and for (18) cf. Thm. 2.1 and Prop. 5.1 therein.

Corollary 1.17. Let u : N Ñ C be a multiplicative function bounded by 1. If, for each Dirichlet character χ, u¨χ satisfies the short interval behaviour (13)8 then u satisfies the averaged Chowla property (18) (along each sequence pN k q for which u is generic).

The above result with pN k q being all positive integers has been proved by Matomäki, Radziwiłł and Tao in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF]. Note that to obtain Corollary 1.17 it is enough to show that irrational rotations satisfy the strong u-MOMO property. Via the DDKBSZ criterion, this follows from the ergodic property AOP introduced and proved to hold for totally ergodic rotations in [START_REF] Abdalaoui | Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals[END_REF]. Thus, we obtain an ergodic proof of the averaged Chowla property for each sequence pN k q as above for all multiplicative u enjoying the special short interval behavior. In particular, assuming that u is generic, we get a non-quantitative version of [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF].

In Section 6, we prove (see Theorem D) that for each u : N Ñ C taking finitely many values, if it satisfies the Sarnak property for the class C ZE , then no positive entropy system satisfies the strong u-MOMO property (this was previously known for the Liouville function assuming the Chowla conjecture [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF]).

Characteristic classes

Definition, examples, basic properties

Recall that a class F of measure-theoretic dynamical systems is characteristic if it is closed under taking isomorphisms, factors and (countable) joinings. Recall also the following classical result on such classes (see e.g. [START_REF] De La Rue | Notes on Austin's multiple ergodic theorem[END_REF]). The following result whose proof is based on a fundamental non-disjointness lemma from [START_REF] Lemańczyk | Gaussian automorphisms whose ergodic self-joinings are Gaussian[END_REF] will be crucial for us: [START_REF] De La Rue | Notes on Austin's multiple ergodic theorem[END_REF]). Let pX, B, ν, T q be a measure-theoretic dynamical system in the characteristic class F , and let pZ, D, κ, Rq be any measuretheoretic dynamical system. Then any joining of R and T is relatively independent over the largest F -factor D F of R. That is: if g P L 2 pZ, κq is such that E κ rg|D F s " 0, and if ρ is a joining of T and R, then for any f P L 2 pX, νq we have

Proposition 2.2 ([
(19) E ρ pf b gq " 0.
Examples of characteristic classes (some acronyms are used for those which will be used in the sequel):

• ALL: all automorphisms of standard Borel probability spaces;

• t˚u: the identity on the one-point space;

• ID: identity automorphisms (of all standard Borel probability spaces);

• DISP: discrete spectrum automorphisms;

• RDISP: rational discrete spectrum automorphisms;

• DISP(G): discrete spectrum automorphisms whose group of eigenvalues is contained in fixed countable subgroup G of the circle;

• NIL s : automorphisms whose a.a. ergodic components are inverse limits of s-step nilautomorphisms. The fact that ergodic joinings of nilsystems remain nil, see Proposition 15, page 186 in the book [START_REF] Host | Nilpotent structures in ergodic theory[END_REF], and the same holds for inverse limits (this is actually Lemma A.4 in [START_REF] Frantzikinakis | The logarithmic Sarnak conjecture for ergodic weights[END_REF]).

Regarding factors of ergodic nilsystems, see Theorem 11 in page 230 [START_REF] Host | Nilpotent structures in ergodic theory[END_REF]. Here s P N.

• DIST: distal automorphisms are those which are given as a transfinite (indexed by ordinals smaller than a fixed countable ordinal) sequence of consecutive extensions each of which either has relative discrete spectrum or (in case of a limit ordinal) is the corresponding inverse limit. The structural theorem Theorem 6.17 together with the concluding remark (for Z-actions) on page 139 [START_REF] Furstenberg | Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation[END_REF] tell us that each system has a largest distal factor, hence DIST is closed under countable joinings. In Lemma 2.15, we note that an automorphism is distal if and only if a.a. its ergodic components are distal. To see that this class is closed under taking factors, let us first recall that this fact holds for ergodic automorphisms (see Theorem 10.18 [START_REF] Glasner | Ergodic Theory via Joinings[END_REF]). If now A Ă D is a factor of a distal automorphism pZ, D, κ, Rq then A (as an R-invariant σ-algebra) is also a factor of a.e. of its ergodic components. So a double use of Lemma 2.15 together with Theorem 10.18 from [START_REF] Glasner | Ergodic Theory via Joinings[END_REF] gives that A is also distal.

• ZE: zero entropy automorphisms;

• RIG pqnq : automorphisms with a fixed sequence pq n q of rigidity;

• multipliers M pD K q of a class D K (D is any class of automorphisms and by D K we mean the set of automorphisms disjoint from all systems from D, and by M pD K q we mean the set of systems whose all joinings with any element of D K remain in D K ); interesting classes of multipliers arise e.g. for D=all weakly mixing (cf. Proposition 5.1 in [START_REF] Lemańczyk | Rokhlin extensions and lifting disjointness, Ergodic Theory Dynam[END_REF]) or all mixing automorphisms; see [START_REF] Glasner | On the multipliers of W K , Ergodic Theory Dynam[END_REF], [START_REF] Lemańczyk | Rokhlin extensions and lifting disjointness, Ergodic Theory Dynam[END_REF].

• the class of factors of all infinite self-joinings of a fixed automorphism R (the smallest characteristic class containing R); especially in case of MSJ and simple automorphisms (cf. [START_REF] Glasner | Ergodic Theory via Joinings[END_REF], Chapter 12). Characteristic classes of such type were used in [START_REF] Lesigne | Weak disjointness of measure preserving dynamical systems[END_REF].

Note also that the intersection of any family of characteristic classes yields again a characteristic class. In Section 2.3, we will show that each characteristic class F determines another characteristic class F ec consisting of those automorphisms whose ergodic components are in F .

The smallest nontrivial and the largest proper characteristic class

An obvious observation has been made already in the introduction that the family ALL of all automorphisms is the largest characteristic class, while the one-element t˚u family (which is the one-point space automorphism) is the smallest characteristic class. It is more interesting however that the smallest non-trivial and the largest proper characteristic classes exist.

Proposition 2.3. ID is the smallest non-trivial characteristic class.

Proof. Let us first notice that the system pr0, 1s, Leb, Idq has all other identities as factors. Indeed, any standard Borel probability space is determined by a sequence pt i q iě0 of non-negative numbers such that ř iě0 t i " 1 and t 0 corresponds to the mass of the continuous part and t 1 , t 2 , . . . correspond to the masses of atoms. Then, take the corresponding partition of r0, 1s into intervals I i of length t i and, for each i ě 1, the factor map will glue points in I i . Now, notice that any non-trivial characteristic class F contains a nonergodic automorphism. Indeed, suppose that T is ergodic, acting on a nontrivial space pY, νq. Since Y is non-trivial and so is T , the graph joinings ∆ Id and ∆ T are ergodic and different, so any non-trivial convex combination of them yields a non-ergodic member of F . It follows that by taking the factor I ν of (a.e.) T -invariant sets (which belongs to F ), we obtain the identity on a non-trivial standard Borel probability space pY , νq. But then the infinite Cartesian product pY ˆ8, ν b8 q is also in F and this infinite product is isomorphic to pr0, 1s, Lebq, which completes the proof.

In order to prove the existence of the largest characteristic (proper) class, we need to recall some results.

Theorem 2.4 (non-ergodic Sinai's factor theorem [START_REF] Kieffer | Selecting universal partitions in ergodic theory[END_REF][START_REF] Takens | Renyi entropies of aperiodic dynamical systems[END_REF]). Assume that R is an automorphism of pZ, D, ρq and let ρ " ş X{Iρ ρ x dmpxq stand for the ergodic decomposition of ρ. Assume that

m ´essinf xPX{Iρ hpρ x , Rq ě α ą 0.
Then a Bernoulli automorphism of entropy α is a factor of R.

In [START_REF] Takens | Renyi entropies of aperiodic dynamical systems[END_REF] (see Theorem 4.3 therein) the above result is attributed to Kieffer and Rahe [START_REF] Kieffer | Selecting universal partitions in ergodic theory[END_REF], see also [START_REF] Seward | Positive entropy actions of countable groups factor onto Bernoulli shifts[END_REF] p.2 (The non-ergodic factor theorem).

We also need the following well-known result (we include its proof for completeness).

Proposition 2.5. Each automorphism R is a factor of a self-joining of the (infinite entropy) Bernoulli automorphism pr0, 1s Z , Leb bZ , Sq.

Remark 2.6. Before we prove the above result, let us notice that any automorphism T of pX, B, µq has an isomorphic copy in the space pr0, 1s Z , κ, Sq. 9Consider first the aperiodic part of T which is realized on a standard Borel space. This space is isomorphic to r0, 1s, via a Borel isomorphism I. It follows that the distribution µ 1 of the process pI ˝Sk q kPZ yields a realization of the aperiodic part. Now, µ 1 takes measure zero on the set of periodic points for the shift. Moreover, the set of periodic points of S can be identified with a subset of r0, 1s, points of period 2 with a subset of r0, 1s 2 etc., and we can easily settle an isomorphism of the fixed point subspace for T with a subset of r0, 1s, period 2 points with a subset of r0, 1s 2 , etc. Thus, it suffices to take κ equal to the sum of µ 1 and the relevant atomic measures corresponding to the periodic points.

Proof of Proposition 2.5. Fix any automorphism R of pZ, D, ρq and take its isomorphic copy in the space pr0, 1s Z , κ, Sq. Take the product space pr0, 1s Z r0, 1s Z , Leb bZ b κq and consider the map ψ : r0, 1s Z ˆr0, 1s Z Ñ r0, 1s Z given by px n , y n q Þ Ñ px n `yn q.

Then ψ ˚pLeb bZ b κq " Leb bZ and clearly the join of the σ-algebra of the first coordinate and of ψ ´1pBpr0, 1s bZ qq is the product σ-algebra in r0, 1s bZ r0, 1s bZ . The result follows.

We now prove the following.

Lemma 2.7. Assume that F is a characteristic class such that F zZE ‰ H.

Then F " ALL.

Proof. Fix T P F zZE. Because of Proposition 2.5, we only need to prove that the infinite entropy Bernoulli automorphism is in F . The first step is to consider the factor of T that arises by gluing together the periodic part and the ergodic components from the aperiodic part whose entropy is smaller than α " hpT q. Clearly, this factor remains in F zZE. Moreover, in its ergodic decomposition we have a single point and the remaining part (which may still be non-ergodic) consists of ergodic components of entropy at least α. By Theorem 2.4, it follows that as a further factor R P F zZE we can obtain a non-ergodic automorphism with two ergodic components: one of them is a Bernoulli of entropy α and the other one is a fixed point. Finally, we take the infinite Cartesian product R ˆ8. It is not hard to see that a.e. ergodic component of this automorphism is a Bernoulli with infinite entropy. Using once more Sinai's theorem (Theorem 2.4), we obtain that a Bernoulli with infinite entropy belongs to F which completes the proof. Now, using the lemma we obtain the following.

Proposition 2.8. ZE is the largest proper characteristic class.

Characteristic classes given by ergodic components

Assume that F is a characteristic class. By F ec we denote the class of those automorphisms R such that (a.e.) ergodic components of R are in F (or more precisely, in F X Erg, where Erg stands for the family of all ergodic automorphisms). Note that we have

(20) F X Erg " F ec X Erg.
Lemma 2.9. F ec is a characteristic class.

Proof. The proof has two parts: we need to show that F ec is closed under taking factors and joinings.

Factors Let R acting on pZ, D, κq belong to F ec and fix a factor A Ă D of R. Let κ " ş κ x dP pxq denote the ergodic decomposition of κ. Since the ergodic components κ x are R-invariant measures, A (being an R-invariant sub-σ-algebra) is also a factor of the automorphism pZ, κ x , Rq and κ| A " ş κ x | A dP pxq is the ergodic decomposition of κ| A . It follows that the ergodic components of the factor are factors of ergodic components of R, and since R P F ec , pκ x , Rq P F , so also pκ x | A , R| A q P F for P -a.e. x.

Joinings Take pX, µ, T q and pY, ν, Sq from F ec and let ρ P JpT, Sq be their joining. Let

ρ " ż 1 0 ρ t dP ptq, µ " ż 1 0 µ t dQptq, ν " ż 1 0 ν t dRptq
be the relevant ergodic decompositions. Then

ż 1 0 µ t dQptq " µ " ρ| X " ż 1 0 ρ t | X dP ptq,
so since ρ t |X are also ergodic, these two decompositions are the same. So for a P -"typical" t P r0, 1s, the projection of ρ t on X is an ergodic component of T . The same argument applies on the coordinate Y and we see that the ergodic components of ρ are joinings of ergodic components of µ and ν. It follows that pX ˆY, ρ, T ˆSq P F ec . The argument extends to countable joinings.

Largest characteristic factor

ID, ZE, DISP and RIG pqnq Given a characteristic class F , according to Proposition 2.1, each automorphism R acting on pZ, D, κq has a largest F -factor D F Ă D. Often, its description is classical:

• the σ-algebra of invariant sets for F " ID,

• the Pinsker factor for F " ZE,

• the Kronecker factor for F " DISP,

• the largest factor for which pq n q is a rigidity time for F " RIG pqnq .

DISP ec We will comment now on D Fec when F " DISP, cf. Proposition 2.12 (ii) and its connections with the theory of nil-factors. Most of the material presented below is known to aficionados but not necessarily the material is explicitly present in the literature. Our discussion is based on [START_REF] Frantzikinakis | Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam[END_REF], [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF], [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF] and [START_REF] Host | Nilpotent structures in ergodic theory[END_REF]. We provide some details to explain clearly why the problem of whether µ K DISP ec is open, cf. Corollary 1.10, Corollary 2.11 and Remark 5.9.

Recall that according to the Furstenberg-Zimmer theory [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF], given R on pZ, D, κq and a factor C Ă D, there exists a certain intermediate factor there exists a dense set of functions F P L 2 pK, κ| K q such that for each δ ą 0 there is a finite set g 1 , . . . , g k P L 2 pK, κ| K q such that for each h P Z,

C Ă K " KpCq Ă D,
min 1ďjďk }F ˝Rh ´gj } L 2 pκyq ă δ (21)
for a.e. y P Z{C, where [START_REF] Furstenberg | Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation[END_REF] κ| K " ż Z{C κ y dκpyq.

Whenever condition ( 21) holds, we speak of relative compactness or of relatively discrete spectrum of the intermediate factor over C.

A particular situation arises when C " I κ , i.e. it is the σ-algebra of invariant sets. Then ( 22) is nothing but the ergodic decomposition of κ and the conditional measures κ y are also R-invariant. In this case condition [START_REF] Frantzikinakis | Furstenberg systems of bounded multiplicative functions and applications[END_REF] yields in a.e. fiber π ´1pyq (where π : Z{K Ñ Z{I κ stands for the factor map) a dense set of functions F | π ´1pyq in L 2 pK, κ y q whose orbits under the unitary action of R are relatively compact. It follows that the (ergodic) automorphism pR, κ y q has discrete spectrum for a.e. y P Z{I κ . In other words, KpI κ q Ă D DISPec

In fact, the opposite inclusion is also true, i.e.

(23) D DISPec " KpI κ q, that is, A :" D DISPec has relatively discrete spectrum over I κ . Indeed, by the definition of A, a.e. ergodic component of R| A has discrete spectrum. Fix F P L 8 pA, κ| A q. Fix also ε, δ ą 0 and k ě 1. Consider the set W k Ă Z{I κ of those y for which

min ´kďjďk › › F ˝Rn ´F ˝Rj › › L 2 pκyq ă ε
for each n P Z. Since on each fiber R is an ergodic automorphism with discrete spectrum, the measure of W k goes to 1, when k Ñ 8, so it will be greater than 1 ´δ for k large enough. It follows that the function F is compact as it has been defined in the proof of Theorem 6.15 [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF]. Therefore, F P L 2 pKpI κ qq, which (by [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF]) concludes the proof of (23).

Remark 2.10. As a matter of fact, in [START_REF] Austin | Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications[END_REF], the Furstenberg-Zimmer theory is developed without assuming ergodicity (cf. e.g. Proposition 5.7 therein to obtain the equality D DISPec " KpI κ q).

We will now see that D DISPec appears naturally in the classical theory of characteristic nil-factors [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF][START_REF] Host | Nilpotent structures in ergodic theory[END_REF]. 10 Recall that if R acting on pZ, D, κq is ergodic then for a function f P L 8 pZ, D, κq its u s norms (in fact, seminorms) are defined in the following way: [START_REF] Glasner | On the multipliers of W K , Ergodic Theory Dynam[END_REF] }f } u 1 :" ˇˇż f dκ ˇˇ,

(25) }f } 2 s`1 u s`1 :" lim HÑ8 1 H ÿ hďH }f ˝Rh ¨f } 2 s u s .
If R is non-ergodic then instead of ( 24), we put

}f } 2 u 1 :" lim HÑ8 1 H ÿ hďH ż f ˝Rh ¨f dκ
and (25) remains unchanged. Then, by [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF][START_REF] Host | Nilpotent structures in ergodic theory[END_REF], for each s ě 1 there is a special factor Z s " Z s pRq Ă B, namely, the largest factor whose [START_REF] Gomilko | Sarnak's conjecture implies the Chowla conjecture along a subsequence[END_REF] a.e. ergodic component is an inverse limit of s-step nil-systems.

In other words, Z s pRq is the largest (characteristic) NIL s -factor of R. Moreover (see Proposition 7 (page 138) and Proposition 13 (page 141) in [START_REF] Host | Nilpotent structures in ergodic theory[END_REF]), ( 27) }f } u s`1 " 0 ðñ f K L 2 pZ s q ðñ f K L 2 pZ s pR, κ y qq for κ-a.e. y.

A special case arises when our measure-preserving systems are Furstenberg systems of a bounded u : N Ñ C. As in (for example) [START_REF] Frantzikinakis | Ergodicity of the Liouville system implies the Chowla conjecture[END_REF], see Sections 2.4 and 2.5 therein, one can introduce the uniformity norms (along subsequences of intervals) for u. The definitions are given in ( 14) and [START_REF] Ferenczi | Sarnak's Conjecture -what's new[END_REF]. They are very similar to those (in the non-ergodic case) to the definitions for functions.

We will now show that (28) Z 1 pRq " KpI κ q.

If R is ergodic then the above means just that (29) Z 1 is the Kronecker factor of R.

To see that (29) indeed holds, notice that (27) for s " 1 yields

}f } u 2 " 0 ðñ f K L 2 pZ 1 q
and it remains to notice that (using the Wiener lemma)

}f } 4 u 2 " lim HÑ8 1 H ÿ hďH }f ˝Rh ¨f } 2 u 1 " lim HÑ8 1 H ÿ hďH ˇˇˇż f ˝Rh ¨f ˇˇˇ2 dκ Ñ ÿ zPT σ f ptzuq 2 ,
where σ f stands for the spectral measure of f . Let us return to a possibly non-ergodic R. The inclusion Z 1 pRq Ă K :" KpI κ q follows directly by Theorem 5.2 in [START_REF] Frantzikinakis | Weighted multiple ergodic averages and correlation sequences, Ergodic Theory Dynam[END_REF]. To obtain the opposite inclusion, one can argue in the following way. Suppose that f K L 2 pZ 1 pRqq and |f | ď 1. Take g P L 2 pKq, cf. ( 21) with F " g. We want to show that ş f g dκ " 0. Notice that

ż f g dκ " ż ˜1 N ÿ nďN ż f ˝T n ¨g ˝T n dκ y ¸dκpyq.
Let g j , 1 ď j ď k, be as in [START_REF] Frantzikinakis | Furstenberg systems of bounded multiplicative functions and applications[END_REF]. Then Each term in the average in the second summand is bounded by δ. Moreover,

N ÿ nďN ˇˇˇż f ˝T n ¨gj dκ y ˇˇˇ2 Ñ ÿ zPT σ f,g j ,κy ptzuq 2 , (30) 1
where σ f,g j ,κy stands for the spectral measure of the pair f, g j (on the ergodic component pπ ´1pyq, κ y q given by y). But by ( 27) and ( 29), we have f K L 2 pZ 1 q ðñ f K L 2 pZ 1 pR, κ y qq for a.e. y ðñ σ f,κy is continuous for a.e. y (σ f,κy stands for the spectral measure of f on the ergodic component pπ ´1pyq, κ y q given by y). Since f K L 2 pZ 1 q and σ f,g i ,κy ! σ f,κy , it remains to use the classical equivalence

1 N ÿ nďN a n Ñ 0 ðñ 1 N ÿ nďN a 2 n Ñ 0
for any bounded sequence pa n q Ă r0, 8q, to conclude that the limit in ( 30) is equal to zero. Thus f K L 2 pZ 1 q ùñ f K L 2 pKq.

Finally, let us compare the above with the notion of relative weak mixing. Recall that relative weak mixing over I κ for f means that

1 H ÿ hďH ż ˇˇEpf ˝Rh ¨f |I κ q ˇˇ2 dκ Ñ 0.
Moreover,

1 H ÿ hďH ż ˇˇEpf ˝Rh ¨f |I κ q ˇˇ2 dκ " ż ˜1 H ÿ hďH ˇˇˇż f ˝Rh ¨f dκ y ˇˇˇ2 ¸dκ,
and, once more by the Wiener lemma,

1 H ÿ hďH ˇˇˇż f ˝Rh ¨f dκ y ˇˇˇ2 Ñ ÿ zPT σ f,κy ptzuq 2 .
It follows immediately that σ f,κy is continuous for a.e. y if and only if f is relatively weakly mixing over I κ .

The above discussion can be summarized in the following statement.

Corollary 2.11. Let pZ, D, κ, Rq be a measure-theoretic dynamical system and let f P L 2 pZ, D, κq. The following conditions are equivalent:

(i) f K L 2 pZ 1 q, (ii) f K L 2 pD DISPec q, (iii) f K L 2 pKpI κ qq,
(iv) σ f,κy is continuous for κ-a.e. y, (v) f is relatively weakly mixing over I κ .

A class vs. its ec-class

Let us continue our observations on the relations between characteristic classes and the corresponding ec-classes. Note that in general there are no relations between F and F ec :

Proposition 2.12. We have: (i) ZE " ZE ec , ALL " ALL ec , ID " ID ec , NIL s " pNIL s q ec , t˚u Ĺ t˚u ec ;

(ii) DISP Ĺ DISP ec ;

(iii) RDISP " RDISP ec ;

(iv) DIST " DIST ec ; (v) ´RIG pqnq ¯ec Ĺ RIG pqnq .

Proof of (i)-(iii).

(i) The first claim follows from the fact that the entropy function is convex, the other claims are obvious.

(ii) If an automorphism has discrete spectrum then its L 2 -space is generated by eigenfunctions. The restrictions (if non-zero) of these (global) eigenfunctions yield orthonormal bases in L 2 -spaces of ergodic components. The inclusion is strict since px, yq Þ Ñ px, x `yq (on T 2 , considered with Lebesgue measure) does not have discrete spectrum while the ergodic components do.

(iii) We want to show that if each ergodic component has rational discrete spectrum then the whole automorphism has. Given p{q P Q and an ergodic component c, we choose f c a modulus 1 eigenfunction corresponding to the eigenvalue e 2πip{q . Since f c is unique up to a constant of modulus 1, this choice can be done measurably. In this way, we will create global eigefunctions. 11Before we give the proof of (iv), we need to recall some more notions and facts from the relative ergodic theory, e.g. [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF][START_REF] Glasner | Ergodic Theory via Joinings[END_REF]. Given an automorphism T of pX, B, µq and its factor S on pY, C, νq with the factor map π : X Ñ Y , we say that this extension is relatively ergodic (rel. erg.) if each f P L 1 pX, B, µq satisfying f ˝T " f (µ-a.e.) is π ´1pCq-measurable. It follows immediately from the definition that:

• any composition of relatively ergodic extensions remains relatively ergodic;

• an inverse limit of relatively ergodic extensions remains relatively ergodic (as the conditional expectation, with respect to a factor, of an invariant function remains invariant);

• π : Y Ñ Y :" Y {I ν , where pY , νq stands for the space of ergodic components (on which acts the identity map), is relatively ergodic. Note that it follows that the ergodic components of T have as their factors (via the relevant restriction of π) ergodic components of S, and that the spaces of ergodic components of T and S are the same (i.e. X " Y ). Proof. By Lemma 2.13, we see that the disintegration of an ergodic component π ´1pπ ´1pyqq over π ´1pyq (which is its factor) consists of the same conditional measures µ y as the total disintegration of µ over ν. We proceed now as in the proof of the equality KpI κ q " D DISPec (page 21), showing compactness.

Recall that an automorphism T is distal if it is a limit of a transfinite (indexed by countable ordinals) sequence of consecutive maximal Kronecker extensions (if an ordinal is not isolated, we pass to the corresponding inverse limit). Note that, by the very definition, the σ-algebra Inv is contained in the Kronecker factor of T , so in this transfinite chain of consecutive extensions, all but (perhaps) the first one are relatively ergodic. By applying Lemma 2.14 and transfinite induction, we obtain the following. Lemma 2.15. T is distal if and only if all its ergodic components are distal.

Proof of (iv)-(v).

(iv) This follows directly from Lemma 2.15. (v) It is clear that if pq n q is a rigidity time for an a.e. ergodic component, it is also a rigidity time for the whole automorphisms. Not vice versa however (for pq n q sufficiently sparsed). We will provide a relevant construction below.

RIG ec is a proper subclass of RIG Let us first notice that we only need to construct a continuous measure σ on the circle such that [START_REF] Host | Nilpotent structures in ergodic theory[END_REF] e 2πiqn¨Ñ 1 in measure σ but not σ ´a.e.

Indeed, suppose (31) holds, and consider on T 2 the automorphism T px, yq " px, y `xq with measure σ b Leb.

If F px, yq " f pxqe 2πi y then by [START_REF] Host | Nilpotent structures in ergodic theory[END_REF], ż |F pT qn px, yqq ´F px, yq| dσpxqdy " ż |f pxq||e 2πiqn x ´1| dσpxq Ñ 0 when n Ñ 8. On the other hand, the rotation by x on an ergodic component txu ˆT has pq n q as its rigidity time if and only if q n x Ñ 0 mod 1. This is not true for σ-a.e. x P T in view of [START_REF] Host | Nilpotent structures in ergodic theory[END_REF].

We now sketch how to construct such a measure assuming that pq n q is sufficiently sparsed. Fix 0 ă p n ă 1 so that p n is decreasing to zero and ř ně1 p n " 8. Set f n pxq " tq n xu. We intend to construct a Cantor set (together with a Cantor measure σ on it). Let

A n :" f ´1 n pr1{4, 3{4sq, B n " f ´1 n pr0, p n sq.
Our postulates are:

σpB n q " 1 ´pn , σpA n q " p n .

In fact, we need to be more precise in description of the measure at stage n to be able to continue its definition. So at stage n the circle is divided into intervals of the form r j qn , j`1 qn q (many of such intervals are of measure σ equal to zero). We now require that the conditional measures satisfy: [START_REF] Huang | Measure complexity and Möbius disjointness[END_REF] σ ˆBn |r j q n , j `1 q n q ˙" 1 ´pn , σ ˆAn |r j q n , j `1 q n q ˙" p n for each j " 0, . . . , q n ´1q. Passing to step n `1, we require that all the intervals r j qn , j`1 qn qq contain at least two intervals of the form r k q n`1 , k`1 q n`1 q, we choose two of such (of course only in those r j qn , j`1 qn q which are of positive measure σ) and apply the rule [START_REF] Huang | Measure complexity and Möbius disjointness[END_REF] to A n`1 , B n`1 with p n replaced with p n`1 .

Note that ş e 2πiqnx dσpxq " 1 `Opp n p1 `pn q `1 ¨pn q, so e 2πiqn¨Ñ 1 in measure σ. On the other hand σpA n q " p n and the sets A n are almost independent. Since ř ně1 p n " 8, for σ-a.e. x, we have x P A n for infinitely many n (by the Borel-Cantelli lemma), so (31) holds.

Strong u-MOMO property of systems whose visible measures yield systems in an ec-class

While we have seen rather unclear relations between F and F ec (cf. Proposition 2.12), on the topological level we always have the following. and x n " x 0 is a fixed arbitrary point of Y ˆA for n ď 0. By setting F :" f b Id on Y ˆA, it easily follows that (34) amounts to

1 b K ÿ kăK ÿ b k ďnăb k`1 F ˝π0 pT n xqupnq Ý ÝÝÝ Ñ KÑ8 0.
To prove the convergence above, we define the subspace X x as the closure of tT n x : n P Zu. By assumption on u, we only have to check that the system pX x , T q is in C Fec . So let µ be a visible measure in pX x , T q, and we first consider the case where x itself is generic for µ, along a sequence pN q. Set B :" pv j , a j q jPZ P X : pv 1 , a 1 q " pSv 0 , a 0 q ( Since b k`1 ´bk Ñ 8, we have

1 N ÿ năN δ T n x pBq Ý ÝÝ Ñ Ñ8 1,
and since the set B is closed, by the Portmanteau theorem, it must be of full measure µ in pX, T q. Moreover, such a measure µ must be T -invariant, hence,

(35) 1 " µ ˜č nPZ T n B ¸" µ
´ pv j , a j q jPZ P X : @j, pv j , a j q " pS j v 0 , a 0 q ( ¯.

Denote by µ p0q the restriction of µ to the zero-coordinate (that is, with the above notation, the distribution of pv 0 , a 0 q under µ). Since µ is T -invariant, it follows that µ p0q is pS ˆId A q-invariant. Moreover, Y ˆA consists of three copies of Y , each of them is invariant under S ˆId A . Thus,

µ p0q " α 0 µ p0q 1 `α1 µ p0q 2 `α2 µ p0q 3 ,
where α 0 `α1 `α2 " 1, α j ě 0 and µ p0q j pY ˆte 2πij{3 uq " 1 for j " 0, 1, 2. It follows that the ergodic components of pY ˆA, µ p0q , S ˆId A q yield measuretheoretic systems isomorphic to ergodic measures on pY, Sq, hence in F since pY, Sq P C Fec (this is the moment in our proof where we use the fact that we deal with a characteristic ec-class and not a general characteristic class F ). Thus pY ˆA, µ p0q , S ˆId A q P F ec . Now, using [START_REF] Kanigowski | On disjointness of some parabolic flows[END_REF], we see that pX x , µ, T q is isomorphic to pY ˆA, µ p0q , S ˆId A q, thus is also in F ec . Now, suppose that µ P V T px 1 q for some point x 1 in the orbit closure of x, say x 1 " lim rÑ8 T nr x.

If x 1 " T n x for some n P Z, then µ P V T pxq and we already know that pX x , µ, T q P t˚u Ă F ec in this case.

If n r Ñ ´8, then x 1 " p. . . , x 0 , x 0 , x 0 , . . .q is a fixed point, and µ " δ x 1 . In this case, pX x , µ, T q P t˚u Ă F ec .

If n r Ñ `8, and if we write x 1 " pv j , a j q jPZ " lim rÑ8 T nr x, then as b k`1 ´bk Ñ 8, there exists at most one j P Z such that pv j`1 , a j q ‰ pSv j , a j q. We can then use the same arguments as for x to show that a measure µ for which x is quasi-generic satisfies pX x , µ, T q P F ec .

We conclude that pX x , T q is in C Fec .

Remark 2.19. In general, when instead of F ec we consider F , u K C F implies the strong u-MOMO property for each for pY, Sq in which all invariant measures yield systems in F (in particular, if pY, Sq P C F and each invariant measure is visible).

Question 1. Is Proposition 2.17 true for each characteristic class?

Remark 2.20. A straightforward adaptation of the proof shows that the subsequence version of Proposition 2.17 also holds: for each characteristic class F and each increasing sequence of integers pN q, u is pN q-orthogonal to C Fec if and only if each element in C Fec satisfies the strong u-MOMO property along pN q. See Remarks 1.2 and 1.4.

Lifting lemma

The purpose of this section is to prove Proposition 3.1, which is an alternative version of Conze-Downarowicz-Serafin lifting lemma from [START_REF] Conze | Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles[END_REF] and seems to be of independent interest. It may seem weaker than the original where the genericity was lifted to a single orbit, but the main advantage here is that we do not need assumptions on the nature of the second topological space: it does not have to be a full shift. The second advantage is that the result has its extension to the logarithmic case, see Appendix 3.3, while the lifting lemma of Conze-Downarowicz-Serafin and other results of that type so far have been proved for Cesàro averages.

Proposition 3.1. Let pY, Sq and pX, T q be two topological systems and u P Y generic along an increasing sequence pN m q for some S-invariant measure κ on Y . Let ρ be a joining of κ with a T -invariant measure ν on X. Then there exist a sequence px n q Ă X and a subsequence pN m q such that:

• the sequence pS n u, x n q is generic for ρ along pN m q:

1 N m ÿ 0ďnăNm δ pS n u,xnq Ý ÝÝ Ñ Ñ8 ρ,
• The sequence px n q is constituted of longer and longer pieces of orbits. More precisely, tn ě 0 :

x n`1 ‰ T x n u is of the form tb 1 ă b 2 ă ¨¨¨ă b k ă b k`1 ă ¨¨¨u, where b k`1 ´bk Ñ 8.

Good sequences of partitions

We need a convenient tool to estimate the weak*-convergence of a sequence of probability measures to a given measure.

Definition 3.1. Let pE, dq be a compact metric space, and let ν be a Borel probability measure on E, i.e. ν P M pEq. We consider a sequence pP q of finite partitions of E into Borel subsets. The sequence pP q is said to be good for pE, νq if the following conditions hold:

• for each , P `1 refines P ,

• diampP q :" max P atom of P diampP q Ý ÝÝ Ñ Ñ8 0,

• for each and each atom P of P , νpBP q " 0.

The motivation for introducing this definition comes from the following result.

Lemma 3.2. If pP q is a good sequence of partitions for pE, νq, then a sequence pν n q Ă M pEq converges to ν in the weak*-topology if and only if, for each and each atom P of P , we have

ν n pP q ÝÝÝÑ nÑ8 νpP q. Proof. If ν n wÝ ÝÝÑ nÑ8
ν, then by the Portmanteau theorem, for each P Ă E such that νpBP q " 0, we have ν n pP q Ñ νpP q.

Conversely, assume that for each and each atom P of P we have ν n pP q Ñ νpP q. Then any weak*-limit µ of a subsequence of pν n q satisfies (again by the Portmanteau theorem) µpP q " νpP q for each atom P of P . But since diampP q Ñ 0, the sequence pP q separates points in E, hence it generates the Borel σ-algebra of E. Thus we have µ " ν, and using the compactness of M pEq for the weak* topology, we get that ν n wÝ ÝÝÑ nÑ8 ν.

Lemma 3.3. For each ν P M pEq of a compact metric space pE, dq, there exists a good sequence of partitions for pE, νq.

Proof. We first show that, for each ě 1, there exists a finite partition Q in which each atom Q satisfies

• diampQq ă 1{ , • νpBQq " 0.
Indeed, by compactness, there exists a finite set tx 1 , . . . , x k u Ă E such that

E Ă ď 1ďiďk B `xi , 1 3 ˘.
Then, for each 1 ď i ď k, there exist at most countably many r ą 0 such that ν pBBpx i , rqq ą 0.

Therefore, we can find r P `1 3 , 1 2 ˘such that @1 ď i ď k, ν pBBpx i , rqq " 0.

Then the partition Q generated by the open balls Bpx i , rq, 1 ď i ď k, satisfies the required conditions. Once we have Q for each ě 1, we set

P :" Q 1 _ ¨¨¨_ Q ,
and we get a good sequence pP q for pE, νq.

Lemma 3.4. Let pP q be a good sequence of partitions for pE 1 , ν 1 q, and let pQ q be a good sequence of partitions for pE 2 , ν 2 q. Then for each coupling ρ of ν 1 and ν 2 , pP ˆQ q is a good sequence of partitions for pE 1 ˆE2 , ρq.

Proof. This is obvious, since for each atom P of P and each atom Q of Q , BpP ˆQq Ă pBP ˆE2 q Y pE 1 ˆBQq, and the marginals of ρ are ν 1 and ν 2 .

Proof of Proposition 3.1

Without loss of generality, we can (and we do) assume that the measuretheoretic dynamical system pY, κ, Sq is aperiodic. Indeed, if this is not the case, we consider any uniquely ergodic topological system pY 1 , S 1 q whose unique invariant measure κ 1 is such that pY 1 , κ 1 , S 1 q is aperiodic. Then we take any point u 1 P Y 1 , and we replace Y by Y ˆY 1 , S by S ˆS1 , and u by pu, u 1 q. We also replace pN m q by a subsequence of pN m q along which pu, u 1 q is generic, for some measure κ whose marginals have to be κ and κ 1 . But then the system pY ˆY 1 , κ, S ˆS1 q is aperiodic, because it is an extension of the aperiodic system pY 1 , κ 1 , S 1 q.

We fix a good sequence of partitions pQ q for pY, κq and a good sequence of partitions pP q for pX, νq. Then by Lemma 3.4, pQ ˆP q is a good sequence of partitions for pY ˆX, ρq.

Definition 3.2. Let M ą 0. A subset E of N is said to be M -separated if for each integers n ‰ m, n, m P E ùñ |n ´m| ě M .
The main argument to prove Proposition 3.1 stands in the following proposition.

Proposition 3.5. Under the assumptions of Proposition 3.1, and assuming also that pY, κ, Sq is aperiodic (see above), given 0 ě 1 and ε P p0, 1 2 q, there exists a sequence px n q of points in X such that:

• tn ě 0 : x n`1 ‰ T x n u is 1 ε -separated, • for each atom A of Q 0 ˆP 0 , we have [START_REF] Kanigowski | Slow Entropy of Some Parabolic Flows[END_REF] ρpAq ´ε ă lim inf

mÑ8 1 N m ÿ 0ďnăNm 1 A pS n u, x n q, and (37)
lim sup mÑ8 1 N m ÿ 0ďnăNm 1 A pS n u, x n q ă ρpAq `ε.
Proof. Let h be a natural number such that 1 h ă ε. We claim that for large enough, we can find a set B Ă Y which is measurable with respect to Ž 0ďjďh´1 S j Q , and such that • B, SB, . . . , S h´1 B are pairwise disjoint,

• κ ´Ť0ďjďh´1 S j B ¯ą 1 ´ε.

Indeed, since pY, κ, Sq is assumed to be aperiodic, we can use the Rokhlin lemma to find a Borel subset B Ă Y such that B, S B, . . . , S h´1 B are pairwise disjoint, and such that

κ ¨ď 0ďjďh´1 S j B' ą 1 ´ε 2 .
Then we use the fact that the good sequence of partitions pQ q generates the Borel σ-algebra: it follows that for large enough, we can find a Qmeasurable set B 1 such that

κpB 1 Bq ă ε 8h 2 .
For each 1 ď j ď h ´1, we have

B 1 X S j B 1 Ă pB 1 z Bq Y pS j B 1 zS j Bq, hence (38) κpB 1 X S j B 1 q ď ε 4h 2 .
It remains to define B by

B :" B 1 z ¨ď 1ďjďh´1 S j B 1 '.
Then, by construction, B is disjoint from S j B for each 1 ď j ď h ´1, thus B, SB, . . . , S h´1 B are pairwise disjoint. Moreover, from [START_REF] Kieffer | Selecting universal partitions in ergodic theory[END_REF], we have κpBq ě κpB 1 q ´ε 4h ě κp Bq ´ε 2h , which implies

κ ¨ď 0ďjďh´1 S j B '" hκpBq ě hκp Bq ´ε 2 ą 1 ´ε,
and our first claim is proved. Since u is generic for κ along pN m q, and since the set Ť 0ďjďh´1 S j B is measurable with respect to Ž 0ďjď2h S j Q (in particular, the κ-measure of its boundary vanishes), we have [START_REF] Kułaga-Przymus | Sarnak's conjecture from the ergodic theory point of view[END_REF] 1

N m ÿ 0ďnăNm 1 Ť 0ďjďh´1 S j B pS n uq Ý ÝÝÝ Ñ mÑ8 κ ¨ď 0ďjďh´1 S j B 'ą 1 ´ε.
This implies in particular that the set P B puq :" tn ě 0 : S n u P Bu is infinite. We number in order the elements of this set:

P B puq " tb 1 ă b 2 ă ¨¨¨ă b k ă ¨¨¨u
The integers pb k q will correspond to the times when we will be allowed to change the orbit of the desired sequence. As B is disjoint from S j B for each 1 ď j ď h ´1, the set P B puq is h-separated, hence 1 ε -separated. We consider the partition Ž 0ďjďh´1 pS ˆT q ´j pQ 0 ˆP 0 q of Y ˆX. Any atom of this partition is of the form Q ˆP , where Q (respectively P ) is an atom of Ž 0ďjďh´1 S ´j Q 0 (respectively of Ž 0ďjďh´1 T ´j P 0 ). For such atoms Q and P , we can write

(40) Q " Q 0 X S ´1Q 1 X ¨¨¨X S ´ph´1q Q h´1 , each Q j being an atom of Q 0 , and (41)
P " P 0 X S ´1P 1 X ¨¨¨X S ´ph´1q P h´1 , each P j being an atom of P 0 . Since the κ-measure of the boundary of each involved set is always 0, we again have for each atom

Q of Ž 0ďjďh´1 S ´j Q 0 (42) 1 N m ÿ b k ăNm 1 QpS b k uq " 1 N m ÿ 0ďnăNm 1 BX QpS n uq Ý ÝÝÝ Ñ mÑ8 κpB X Qq.
If C is a measurable subset of Y with κpCq ą 0, we denote by ρ Y C the marginal on X of the conditional probability measure ρp ¨|C ˆXq. Then, for each measurable A Ă X, we have ρpC ˆAq " ρ `pC ˆXq X pY ˆAq "

ρpC ˆXq ρ `Y ˆA|C ˆX" κpCq ρ Y C pAq. (43)
On an appropriate probability space, we construct a sequence pξ k q of independent random variables, taking values in X, such that for each k, ξ k is distributed according to ρ Y BX Q, where Q is the atom of

Ž 0ďjďh´1 S ´j Q 0 containing S b k u.
For each atom Q of Ž 0ďjďh´1 S ´j Q 0 and each atom P of Ž 0ďjďh´1 T ´j P 0 , by [START_REF] Lesigne | Weak disjointness of measure preserving dynamical systems[END_REF], the law of large numbers and ( 43), with probability 1, we have [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] 1

N m ÿ b k ăNm 1 QpS b k uq1 P pξ k q Ý ÝÝÝ Ñ mÑ8 κpB X Qqρ Y BX Qp P q " ρ `pB X Qq ˆP ˘.
Let us fix a realization of pξ k q which satisfies (44) for each atom Q ˆP of Ž 0ďjďh´1 pS ˆT q ´j pQ 0 ˆP 0 q. Then, for each n ě 0, we define the point x n P X as follows:

x n :"

# T n´b 1 ξ 1 if n ă b 1 , T n´b k ξ k if b k ď n ă b k`1 for some k ě 1.
The set of integers n such that x n`1 ‰ T x n is contained in P B puq, therefore, it is 1 ε -separated. Now, let A " Q ˆP be a fixed atom of Q 0 ˆP 0 . We set R :" ď 0ďjďh´1 S j B ˆX, and we observe that [START_REF] Matomäki | Fourier uniformity of bounded multiplicative functions in short intervals on average[END_REF] ρpRq " κ ¨ď 0ďjďh´1 S j B 'ą 1 ´ε.

We also note that for each n ě b 1 , pS n u, x n q P R if and only if there exists k and 0 ď j ď h ´1 such that n " b k `j. In this case, pS n u, x n q P A X R if and only if the atom Q ˆP of Ž 0ďjďh´1 pS ˆT q ´j pQ 0 ˆP 0 q containing pS b k u, ξ k q satisfies Q j " Q and P j " P (using the notations given in [START_REF] Lemańczyk | Rokhlin extensions and lifting disjointness, Ergodic Theory Dynam[END_REF] and [START_REF] Lemańczyk | Gaussian automorphisms whose ergodic self-joinings are Gaussian[END_REF], and remembering that A " Q ˆP ). validity of the analog inequalities for each A which is a finite union of atoms of Q `1 ˆP `1 (in particular, for each A which is an atom of the previous partitions), but with ε instead of ε `1.

The sequence px n q ně0 of points in X and the subsequence pN m q we construct with the above inductive procedure then satisfy the conditions announced in Proposition 3.1.

Logarithmic case

We would like to study the logarithmic version of Proposition 3.1, in which we replace each arithmetic average of the form (Here we use the notation LpN q :" 1 `1 2 `¨¨¨`1 N .) In fact, this logarithmic version, whose statement is written below, is also valid, and the arguments to prove it are exactly the same as in the arithmetic average case. We just point out below the few technical changes that need to be made in the proof for the logarithmic case.

Proposition 3.6. Let pY, Sq and pX, T q be two topological systems and u P Y , logarithmically generic along an increasing sequence pN m q for some Sinvariant measure κ on Y . Let ρ be a joining of κ with a T -invariant measure ν on X. Then there exist a sequence px n q Ă X and a subsequence pN m q such that:

• the sequence pS n u, x n q is logarithmically generic for ρ along pN m q:

1 LpN m q ÿ 1ďnďNm 1 n δ pS n u,xnq Ý ÝÝ Ñ Ñ8 ρ,
• the set tn ě 0 :

x n`1 ‰ T x n u is of the form tb 1 ă b 2 ă ¨¨¨ă b k ă b k`1 ă ¨¨¨u, where b k`1 ´bk Ñ 8.
The changes that need to be made to the proof are almost all quite obvious, they consist in formally replacing the arithmetic average by the logarithmic average. One point maybe needs some explanations, namely when we arrive at the proof of the logarithmic analog of [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF]. We put these explanations in the form of a lemma, which we will apply in the following context: pd k q is the ordered sequence of positive integers n such that S n u P B X Q, and the sequence pρ k q is defined by ρ k :" 1 P pξ k q. Lemma 3.7. Let pd k q be an increasing sequence of positive integers such that

1 LpN m q ÿ d k ďNm 1 d k Ý ÝÝÝ Ñ mÑ8 κ P r0, 1s,
and let pρ k q be a sequence of real numbers in r0, 1s such that

1 K ÿ 1ďkďK ρ k Ý ÝÝÝ Ñ KÑ8 ρ P r0, 1s.
Then we have

1 LpN m q ÿ d k ďNm ρ k d k Ý ÝÝÝ Ñ mÑ8 κρ.
Proof. For each m, let us denote by k m the largest k such that d k ď N m . We use the classical identity

ÿ d k ďNm ρ k d k " ÿ 1ďkăkm ˆ1 d k ´1 d k`1 ˙pρ 1 `¨¨¨`ρ k q `1 d km pρ 1 `¨¨¨`ρ km q.
Given ε ą 0, let K ε be such that

K ě K ε ùñ ˇˇˇˇ1 K ÿ 1ďkďK ρ k ´ρˇˇˇˇˇă ε.
We can then write

ˇˇˇˇ1 LpN m q ÿ d k ďNm ρ k d k ´1 LpN m q ÿ d k ďNm ρ d k ˇˇˇ" ˇˇˇˇ1 LpN m q ÿ Kεďkăkm k ˆ1 d k ´1 d k`1 ˙ˆ1 k pρ 1 `¨¨¨`ρ k q ´ρ˙ˇˇˇˇˇ`O ˆ1 LpN m q ă ε `O ˆ1 LpN m q ˙.
But by assumption, we have

1 LpN m q ÿ d k ďNm ρ d k Ý ÝÝÝ Ñ mÑ8 κρ, hence we get 1 LpN m q ÿ d k ďNm ρ k d k Ý ÝÝÝ Ñ mÑ8 κρ.
5 Combinatorics We need the following interpretation of the Veech condition in terms of relative uniform mixing (K-mixing) of the function π 0 . For n P N, let π n :" π 0 ˝Sn .

Proposition 5.1. Let κ P V S puq and ş π 0 dκ " 0. Then the following conditions are equivalent:

(a) π 0 K L 2 pΠpκqq, (b) π 0 is relatively K-mixing, i.e. for each ε ą 0, there exists N such that

ˇˇˇż π 0 1 C dκ ´ż π 0 dκ ż 1 C dκ ˇˇˇ" ˇˇˇż π 0 1 C dκ ˇˇˇă ε
for each set C P σpπ n , π n`1 , ...q and n ě N .

If we additionally assume that u takes values in a finite set E Ă C and pM k q is a sequence along which we have a Furstenberg system κ then the above conditions are equivalent to (c) for each ε ą 0 there exists N ě 1 such that for any s ě 1 and any function f depending on coordinates N ď n, n`1, . . . , n`s, }f } CpXuq ď 1, we have

lim sup kÑ8 ˇˇˇˇ1 M k ÿ mďM k upmqf pS m uq ˇˇˇˇă ε.
Proof. (a) ùñ (b). Assume that Epπ 0 |Πpκqq " 0. Let C P σpπ n , π n`1 , . . .q.

We have

ˇˇˇż π 0 1 C dκ ˇˇˇ" ˇˇˇż 1 C Epπ 0 |σpπ n , π n`1 , ...q dκ ˇˇď ż |Epπ 0 |σpπ n , π n`1 , . . .qq| dκ.
Hence, we have an upper bound which does not depend on C. Since Epπ 0 |σpπ n , π n`1 , ...qq Ñ Epπ 0 |Πpκqq " 0 κ-a.e. and thus also in L 1 , which is precisely the relative K-mixing for π 0 . (b) ùñ (a). Suppose that π 0 is relatively K-mixing. Then, in particular, we have (5.1) for each C P Πpκq. In fact, since ε ą 0 is arbitrary, ş π 0 1 C dκ " 0 for each C P Πpκq. Whence Epπ 0 |Πpκqq " 0. (c) ùñ (b). Suppose that ˇˇş Xu π 0 f dκ ˇˇă ε, whenever f depending on coordinates n, n `1, . . . , n `s with n ě N is bounded by 1. Consider all blocks on coordinates n, n `1, . . . , n `s that is all B " tx P X u : x n " b 0 , . . . , x n`s " b s u with b j P E. Let C be any union of such blocks. Then 1 C is a (continuous) function depending on coordinates n, . . . , n `s and is bounded by 1 and, by assumption, ˇˇˇż Xu π 0 1 C dκ ˇˇˇă ε.

Note that with N fixed and s arbitrary, the family of C defined above is dense in the σ-algebra σpπ N , π N `1, . . .q. Hence, given D P σpπ N , π N `1, . . .q and ε ą 0, we first find s ě 0 and then C as above (a union of blocks "sitting" on coordinates N, . . . , N `s) such that κpC Dq ă ε and find that ˇˇˇż π 0 1 D dκ ˇˇˇď ˇˇˇż π 0 1 C dκ ˇˇˇ`κ pC Dq ă 2ε. Now, since each clopen set is a finite union of cylinders of a fixed length, Corollary C' follows directly by the above proposition. Corollary C' is a special case of Corollary C.

Conditional cancellations. Remark 1.6

The "cancellation law" of the values of u along large shifts of the return times to a block (for most of the blocks) claimed in Remark 1.6 is a consequence of a refinement of Proposition 5.1. Proposition 5.2. Let κ P V S puq and ş π 0 dκ " 0. Then the following conditions are equivalent: (a) π 0 K L 2 pΠpκqq, (d) for each ε ą 0 there exist N ě 1 and L ě 1 such that for each ě L there is a family of "good" -blocks C P σpπ N , π N `1, . . .q, i.e. of blocks satisfying ˇˇˇż 1 C ¨π0 dκ ˇˇˇď κpCqε, whose measure is ą 1 ´ε. In other words, for a "good" -block C, ˇˇş π 0 dκ C ˇˇă ε, where κ C stands for the conditional measure on C.

Proof. (a) ùñ (d). Fix ε ą 0 and note that E κ pπ 0 |σpπ N , π N `1, . . .qq Ñ 0 κ-a.e. This implies convergence in measure, i.e., we can find a set A ε of measure at least 1 ´ε such that for N large enough,

|E κ pπ 0 |σpπ N , π N `1, . . .qqpxq| ă ε for all x P A ε .
Fix such an N . There is M ě 1 large enough such that

κpA ε A pM q ε q ă ε,
where A pM q ε P σpπ ´M , π ´M `1, . . .q and note that S N `M A pM q ε P σpπ N , π N `1, . . .q. Now, for large enough, we can approximate S N `M A pM q ε by a (disjoint) union of -blocks belonging to σpπ N , π N `1, . . .q, κpS N `M A pM q ε ď jPJ C p q j q ă ε.

But κpS

N `M A pM q ε A ε q ă 2ε, so κp ď jPJ C p q j zA ε q ď κpA ε ď jPJ C p q j q ă 3ε.
Consider those j P J for which κpC p q j zA ε q ě ? εκpC p q j q. Then the measure m of the union of such blocks has to satisfy ? εm ă 3ε, so m ă 3 ? ε. In other words "most" (in measure) of the C p q j 's are "good", i.e. they satisfy κpC p q j X A ε q ą p1 ´3? εqκpC p q j q. Take such a "good" C. We have

ˇˇˇż 1 C ¨π0 dκ ˇˇˇď ż 1 C |E κ pπ 0 |σpπ N , π N `1, . . .qq| dκ " ż Aε 1 C |E κ pπ 0 |σpπ N , π N `1, . . .qq| dκ `żA c ε 1 C |E κ pπ 0 |σpπ N , π N `1, . . .qq| dκ ď εκpCq `3? εκpCq.
(d) ùñ (a). Fix A P Πpκq of positive measure κ. Then for ε ą 0, we can find N such that for all large enough "most" of the -blocks in σpπ N , π N `1 ă . . .q is "good" in the sense that ˇˇˇż 1 C ¨π0 dκ ˇˇˇď κpCqε.

Since A P σpπ N , π N `1, . . .q, we can approximate it by unions of -blocks (for sufficiently large) and most of the used blocks is "good". Whence ˇˇˇż 1 A ¨π0 dκ ˇˇˇď 2ε, and since ε ą 0 was arbitrary, π 0 K L 2 pΠpκqq.

cannot appear (up to isomorphism) as a Furstenberg system of u (because π 0 is orthogonal to the L 2 pKpI κ qq but for the unipotent system (58) the whole system is relative Kronecker over the σ-algebra of invariant sets).

In particular, if }λ} u 2 " 0 holds for the Liouville function then (58) is not its Furstenberg system -this would answer a question by N. Frantzikinakis asked in 2016 (it is an official Problem 3.1 in [START_REF]Sarnak's Conjecture[END_REF]). However, the problem of whether }λ} u 2 " 0 (or more generally }λ} u s " 0) seems to be difficult. The best known results [START_REF] Matomäki | Fourier uniformity of bounded multiplicative functions in short intervals on average[END_REF][START_REF] Matomäki | Higher uniformity of bounded multiplicative functions in short intervals on average[END_REF] require a quantitative dependence between the parameters M and N , i.e. M " N θ , for arbitrary small, but fixed θ ą 0.

If }λ} u 2 " 0 holds then Sarnak's conjecture holds for all (zero entropy) systems whose ergodic measures yield discrete spectrum. So far it is only known that Sarnak's conjecture holds for systems whose all invariant measure yield discrete spectrum [START_REF] Ferenczi | Sarnak's Conjecture -what's new[END_REF][START_REF] Huang | Measure complexity and Möbius disjointness[END_REF][START_REF] Huang | Möbius disjointness for topological models of ergodic systems with discrete spectrum[END_REF].

Ruling out (58) (or, more generally, nilpotent type systems) from the list of potential Furstenberg systems of λ is important in view of Frantzikinakis and Host's results [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] concerning the structure of Furstenberg systems of multiplicative functions (although, for the moment, this structure is known only for the logarithmic case).

In the light of [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF], it would be also interesting to know whether }u} u 2 " 0 holds for some classical multiplicative functions. Note that this is not the case for the class of aperiodic multiplicative functions defined in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] since as shown in [START_REF] Gomilko | On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao[END_REF] they have the unipotent system as a Furstenberg system14 (see also Remark 5.4).

Orthogonality to C DISP . Averaged Chowla property for multiplicative functions

The assertion "iff" of Theorem A cannot be applied to the class C DISP . In this section we will show however that the assertion of this theorem holds whenever u : N Ñ C satisfies the following additional property:

(˚) all rotations on the circle satisfy the strong u-MOMO property.

We will need the following fact (see, e.g., [START_REF] Edeko | On the isomorphism problem for non-minimal transformations with discrete spectrum[END_REF]):

(59) each discrete spectrum automorphism is a factor of R α ˆId r0,1s , for some ergodic rotation by α P G on a compact (Abelian) metric group G.

Our key tool will be the following lemma.

Lemma 5.11. Suppose that p˚q holds. Then R α ˆId r0,1s satisfies the strong u-MOMO property. 1 Xn"x j ˜ÿ 1ďiďj´1 P i,n `Vn P j,n ¸.

Informally, if X n " x j , we pick U n uniformly at random (using V n ) inside I j,n (see Figure 1). Therefore, L pU n | pX m q mďn´1 , pV m q mďn´1 q " U r0,1s , i.e., it is uniform on r0, 1s. But all U m , m ď n ´1, are measurable with respect to pX m q mďn´1 and pV m q mďn´1 , thus we also have (62) L pU n | pU m q mďn´1 q " U r0,1s and L pU n q " U r0,1s .

Indeed, this is just the application of the tower property of conditional expectations: to obtain the left equality, notice that for any measurable A Ă r0, 1s, we have P `Un P A | pU m q mďn´1 " E " P `Un P A | pX m q mďn´1 , pV m q mďn´1 looooooooooooooooooooooomooooooooooooooooooooooon LebpAq | pU m q mďn´1 ı " LebpAq.

Moreover, it also follows from (62) that U is i.i.d.

Note that by construction, U n is a measurable function of V n , X n and pX m q mďn´1 , which we abusively write as U n " U n pV n , X n , pX m q mďn´1 q .

Moreover, whenever we fix realizations ξ of pX m q mďn and v of V n then U n as a function of its second argument is increasing:

(63)
U n pv, x j 1 , ξq ă U n pv, x j 2 , ξq, whenever x j 1 ă x j 2 .

Step 2: process Y as a function of U

We want to define Y n for a given n P Z. We use another partition of r0, 1r into subintervals, according to the probability distribution β intended for Y n : for 1 ď k ď s, set β k :" βpy k q and define the interval J k :" " β 1 `¨¨¨β k´1 ; β 1 `¨¨¨`β k " . Then we simply define Y n as a function of U n by setting

Y n :" s ÿ k"1 y k 1 J k pU n q.
It follows by the choice of the intervals J k and by LpU n q " U r0,1s that Y n is distributed according to β. Moreover, by the independence of U , we have the independence of Y . Thus, Y is a Bernoulli process with distribution β bZ . It remains to prove the announced inequality (61). Observe that Y n is, like U n , constructed as a measurable function of V n , X n and pX m q mďn´1 , which we also abusively write as Y n " Y n pV n , X n , pX m q mďn´1 q .

Since Y n is a non-decreasing function of U n , we get from (63) that for a fixed realization ξ of pX m q mďn´1 and v of V n , we have for 1 ď j 1 ă j 2 ď r (64)

Y n pv, x j 1 , ξq ă Y n pv, x j 2 , ξq and it follows that the map

x P A Þ Ñ E " Y n | pX m q mďn´1 " ξ, X n " x ‰
is non-decreasing. Moreover, by the construction of Y , we have L `Yn | pX m q mďn´1 ˘" LpY n q " β, whence (65) E " Y n | pX m q mďn´1 ‰ " ErY n s " 0.

Thus there exists j 0 P t1, . . . , ru (depending on ξ) such that E " Y n | pX m q mďn´1 " ξ, X n " x j s ď 0 for 1 ď j ď j 0 , E " Y n | pX m q mďn´1 " ξ, X n " x j s ą 0 for j 0 `1 ď j ď r. (66)

We then have, using (65) and (66), E " X n Y n | pX m q mďn´1 " ξ ‰ " E " pX n ´xj 0 qY n | pX m q mďn´1 " ξ ‰ " j 0 ÿ j"1 px j ´xj 0 q E " Y n | pX m q mďn´1 " ξ, X n " x j s `r ÿ j"j 0 `1px j ´xj 0 q E " Y n | pX m q mďn´1 " ξ, X n " x j s (67) ě 0. Now, we claim that the announced result is a consequence of the following lemma. Lemma 6.6. If the realization ξ of pX m q mďn´1 is such that the conditional distribution LpX n | pX m q mďn´1 " ξq is non-trivial, then

E " X n Y n | pX m q mďn´1 " ξ ‰ ą 0.
Indeed, since X has positive entropy, LpX n | pX m q mďn´1 q is non-trivial with positive probability, and thus we can conclude that

E " X n Y n ‰ " E " E " X n Y n | pX m q mďn´1 ‰ ı ą 0.
Proof of Lemma 6.6. We fix a realization ξ of pX m q mďn´1 such that the conditional distribution L `Xn | pX m q mďn´1 " ξ ˘is non-trivial. Then the random variables P j,n and the intervals I j,n are fixed, because their values only depend on ξ. Setting j 1 :" min j P t1, . . . , ru : P j,n ą 0 ( , and j 2 :" max j P t1, . . . , ru : P j,n ą 0 ( , we have j 1 ă j 2 . Moreover the intervals I j 1 ,n and I j 2 ,n are respectively of the form r0, P j 1 ,n r and r1 ´Pj 2 ,n , 1r, with 0 ă P j 1 ,n ď 1 ´Pj 2 ,n ă 1. We now discuss according to the relative position of the interval I j 2 ,n with respect to the interval J 1 (used to define Y n ).

Case 1: J 1 X I j 2 ,n " H (see Figure 2). Then we have (68) P `Yn " y 1 | pX m q mďn´1 " ξ, X n " x j 2 ˘" 0, whereas (69) P `Yn " y 1 | pX m q mďn´1 " ξ, X n " x j 1 ˘ą 0.

Moreover, notice that (68) is equivalent to (70) P `Yn ą y 1 | pX m q mďn´1 " ξ, X n " x j 2 ˘" 1, It follows from (69) and (70) that there exists a V n -measurable event A of positive probability such that, on A, Y n `Vn , x j 2 , ξ ˘ą y 1 " Y n `Vn , x j 1 , ξ ˘.

Remembering (64), we get (71)

E " Y n | pX m q mďn´1 " ξ, X n " x j 2 ‰ ą E " Y n | pX m q mďn´1 " ξ, X n " x j 1 ‰ .
Case 2: J 1 X I j 2 ,n ‰ H (see Figure 3). Then I j 1 ,n Ă J 1 and I j 1 ,n X J s " H. It follows that P `Yn " y s | pX m q mďn´1 " ξ, X n " x j 1 ˘" 0, whereas P `Yn " y s | pX m q mďn´1 " ξ, X n " x j 2 ˘ą 0. In this case, we get a V n -measurable event A of positive probability such that, on A, Y n `Vn , x j 2 , ξ ˘" y s ą Y n `Vn , x j 1 , ξ ˘, and as before we conclude that (71) holds. Now, since (71) always holds, and since E " Y n | pX m q mďn´1 " ξ ‰ " 0 " r ÿ j"1 P j,n E " Y n | pX m q mďn´1 " ξ, X n " x j ‰ , we deduce that E " Y n | pX m q mďn´1 " ξ, X n " x j 2 ‰ ą 0.

It follows that in the sum (67), at least the term corresponding to j " j 2 is positive, and this yields

E " X n Y n | pX m q mďn´1 " ξ ‰ ą 0.

Proposition 2 . 1 .

 21 Given a characteristic class F , each automorphism R on pZ, D, κq has a largest F -factor, denoted by D F .

  called the relative Kronecker factor (with respect to C). It is the largest intermediate factor with the following property (see condition C 2 in [23], p. 131):

  n pg ˝T n ´gi q| dκ y .

  Let µ " ş Y µ y dνpyq stand for the disintegration of µ over ν and let ν " ż Y ν y dν denote the ergodic decomposition of ν (which is precisely the disintegration of ν over ν). Then the ergodic components of S acting on Y are of the form pπ ´1pyq, ν y , Sq (the measures ν y are S-invariant). Therefore, we have the following lemma. Lemma 2.13. If T is relatively ergodic over S then the ergodic components of T are of the form ˜π´1 pπ ´1pyqq, ż π ´1pyq µ y dν y pyq ¸.

Lemma 2 . 14 .

 214 Let T be relatively ergodic over S. Then the following are equivalent: (a) T over S has relatively discrete spectrum. (b) The ergodic components of T have relatively discrete spectrum over the ergodic components of S being their relevant factors.

5. 1

 1 Orthogonality to zero entropy systems 5.1.1 Cancellations. Proof of Corollaries C and C'

  qpS m uq Ñ ż Xu π 0 f dκ, we can repeat the same argument as was used to prove (a) ùñ (b) (replacing 1 C by f ).

Figure 1 :

 1 Figure 1: Definition of U n

Figure 2 :Figure 3 :

 23 Figure 2: Case 1 (J 1 X I j 2 ,n " H)

be an arithmetic function taking finitely many values. If u is generic then u K C ZE if and only if lim mÑ8 lim

		ˇˇˇˇ1
			ÿ
	N Ñ8	N	nďN	upnq1 pupm`nq,upm`n`1q,...,upm`n` ´1qqPC

An ergodic proof (which goes back to a suggestion of Sarnak in[START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF]) of this result is

DDKBSZ stands for Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler[START_REF] Bourgain | From Fourier and Number Theory to Radon Transforms and Geometry[END_REF],[START_REF] Kátai | A remark on a theorem of H. Daboussi[END_REF].

In fact, it is open whether a non-periodic, zero entropy, continuous, algebraic automorphism of T 2 satisfies the strong µ-MOMO property.

Our thanks go to N. Frantizkinakis who pointed out to us one of crucial equivalences: (iv) ô (v).

This is equivalent to saying that along arithmetic sequences the averages on a typical short interval vanish.

The same arguments apply to D " tz P C : |z| ď 1u instead of r0, 1s.

We would like to thank Bryna Kra and Nikos Frantzikinakis for fruitful discussions and useful references on this subject.

The same argument works if we consider the characteristic class of automorphisms having discrete spectrum contained in a fixed countable subgroup of the circle.

In fact, for such functions u we have already }u} u 1 ppN k qq ą 0 (for some pN k q), see Corollary 6.5 in[START_REF] Gomilko | On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao[END_REF].

Kpκq stands for the Kronecker factor of pXu, κ, Sq.

Acknowledgments: We would like to thank Tomasz Downarowicz, Nikos Frantzikinakis and Krzysztof Fraczek for useful discussions on the paper. Research of the second and third authors supported by Narodowe Centrum Nauki grant UMO-2019/33/B/ST1/00364.

Proposition 2.16. Let F be a characteristic class. Then C F Ă C Fec .

Proof. This follows immediately from the fact that a homeomorphism T (acting on a compact metric space X) belongs to C Fec if and only if for each κ P M e pX, T q, pX, BpXq, κ, T q P F .

Note that in view of Proposition 2.16 and Proposition 2.12, [START_REF] Huang | Polynomial mean complexity and logarithmic Sarnak conjecture[END_REF] C RIG pqnq " C pRIG pqnq qec .

The special role of ec-classes stands in the next proposition.

Proposition 2.17. Let F be a characteristic class. Then u K C Fec if and only if each element in C Fec satisfies the strong u-MOMO property.

The below proof of Proposition 2.17 is an adaptation of the proof of Corollary 9 in [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF]. It uses the following elementary result (see Lemma 18 in [3]).

Lemma 2.18. Assume that pc n q Ă C and pm n q Ă N. Then if the sequence pc n q is contained in a closed convex cone which is not a half-plane then

Proof of Proposition 2.17. Only one implication needs to be proved. Suppose that u K C Fec , and let pY, Sq P C Fec . We fix f P CpY q, an increasing sequence pb k q in N, with b 1 " 1 and b k`1 ´bk Ñ 8, and a sequence py k q of points in Y . We introduce the finite set A :" t1, e 2πi{3 , e 4πi{3 u, and for each k ě 1, we define e k P A such that the complex number e k ´ÿ b k ďnăb k`1 f pS n´b k y k qupnq īs in the closed convex cone t0u Y tz P C ˚: argpzq P r´π{3, π{3su. Then, by Lemma 2.18, the convergence that we need to prove, i.e.

Consider the dynamical system pX, T q, where X :" pY ˆAq Z , and T is the left shift. Let x P X be such that

x n :" pS n´b k y k , e k q, whenever b k ď n ă b k`1 ,

We can then use [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] to get

p Q, P q: Q j "Q and P j "P 1 QˆP pS b k u, ξ k q Ý ÝÝÝ Ñ mÑ8 ÿ 0ďjďh´1 ÿ p Q, P q: Q j "Q and P j "P ρ `pB X Qq ˆP ˘.

(

But, on the other hand, we can write ρpA X Rq " ÿ 0ďjďh´1 ρ `A X pS j B ˆXq " ÿ 0ďjďh´1 ρ `pS ´j Q ˆT ´j P q X pB ˆXq " ÿ 0ďjďh´1 ÿ p Q, P q: Q j "Q and P j "P ρ `pB X Qq ˆP ˘.

(

From ( 46) and [START_REF] De La Rue | Notes on Austin's multiple ergodic theorem[END_REF], it follows that (48)

From [START_REF] Kułaga-Przymus | Sarnak's conjecture from the ergodic theory point of view[END_REF], we get that

1 pY ˆXqzR pS n u, x n q ă ε, and since 1 A ď 1 AXR `1Y ˆXzR , this yields by [START_REF] Sarnak | Three lectures on the Möbius function, randomness and dynamics[END_REF],

and we have [START_REF] Kátai | A remark on a theorem of H. Daboussi[END_REF]. On the other hand, using 1 A ě 1 AXR , we get by ( 45)

and we have [START_REF] Kanigowski | Slow Entropy of Some Parabolic Flows[END_REF].

We can now give the proof of Proposition 3.1, in which we use the following obvious fact: if we modify the sequence px n q given by Proposition 3.5 on a finite number of terms, we still get [START_REF] Kanigowski | Slow Entropy of Some Parabolic Flows[END_REF] and [START_REF] Kátai | A remark on a theorem of H. Daboussi[END_REF].

Proof of Proposition (3.1). We fix a sequence pε q ě1 of numbers in p0, 1 2 q, decreasing to 0, and we construct inductively the desired sequence px n q and the subsequence pN m q by a repeated use of Proposition 3.5.

We start by applying Proposition 3.5 with ε :" ε 1 and 0 :" 1. It provides us with an integer m 1 , and a finite sequence px n q 0ďnăNm 1 of points in X such that

Now, assume that for some ě 1 we have already constructed m 1 ă ¨¨¨ă m and the sequence px n q 0ďnăNm of points in X such that • for each 1 ď j ă , the set of integers n P tN m j´1 , . . . , N m j ´2u such that x n`1 ‰ T x n is 1 ε j -separated (with the convention that N m 0 " 0),

• for each atom A of Q ˆP , we have

Then we apply again Proposition 3.5, with ε :" ε `1 and 0 :" `1. It provides us with an integer m `1 and a finite sequence of points px n q Nm ďnăNm `1 in X which satisfy:

(We keep the points px n q 0ďnăNm already provided by the induction hypothesis, refering to the obvious fact stated before the proof.)

Moreover, we can assume that the sequence pε q decreases sufficiently fast so that the validity of (49) for each atom A of Q `1 ˆP `1 ensures the The last place in the proof where a (very easy) correction should be made in the logarithmic case is to get the analog of [START_REF] Matomäki | Higher uniformity of bounded multiplicative functions in short intervals on average[END_REF]: at some point we have to replace some coefficients 1 b k `j by 1 b k , which is of no consequence since j remains bounded between 0 and h´1 here. To be more precise, [START_REF] Matomäki | Higher uniformity of bounded multiplicative functions in short intervals on average[END_REF] becomes

4 Proofs of main theorems

Proof of Theorem B

Proof of Theorem B. Take any topological system pX, T q P C F and fix f P CpXq, x P X. Take any increasing sequence pN k q for which, with no loss of generality, we can assume that 1

But ρ is a joining of some T -invariant measure ν P V pX, T q for which x is generic along pN k q, and some Furstenberg system κ of u. Since pX, T q P C F , the system pX, ν, T q is in F , and the integral on the right-hand side above vanishes by the Veech condition and Proposition 2.2.

Proof of Theorem A

Before we begin the proof, let us make the following remark concerning topological models. Given an automorphism pZ, D, κ, Rq, and a fixed subset of full measure of ergodic components of κ, recall that by a Hansel model of R, we mean a topological system pX, T q which has a T -invariant measure ν such that, as dynamical systems, κ and ν are isomorphic and such that each point x P X is generic for one of these chosen ergodic components. In [START_REF] Hansel | Strict Uniformity in Ergodic Theory[END_REF], it is proved that each automorphism has a Hansel model.

We assume that u K C Fec for some characteristic class F . Take κ P V puq and fix pN m q so that 1

Denote by Apκq Ă BpX u q the largest F ec -factor of pX u , κ, Sq, i.e. Apκq " BpX u q Fec . Consider the factor pX u {Apκq, Apκq, κ| Apκq , Sq and take a Hansel model pX, ν, T q of it (by choosing in the ergodic decomposition of κ| Apκq only ergodic measures in F ). By definition, [START_REF] Tao | The Chowla and the Sarnak conjecture, What's new[END_REF] pX, T q P C Fec .

Fix a measure-theoretic factor map J : pX u , κ, Sq Ñ pX, ν, T q such that J ´1pBpX qq " Apκq, and let ν J denote the corresponding graph joining (of ν and κ| Apκq ). Let p ν J be the relatively independent extension of ν J to a joining of ν and κ: for f P L 2 pνq and g P L 2 pκq, we have

Now, by applying Proposition 3.1, we can find px n q Ă X such that [START_REF] Tao | The logarithmically averaged and non-logarithmically averaged Chowla conjectures[END_REF] ppx n q, uq is generic for p ν J along some subsequence pN m q, and the set tn ě 0 :

Since u K C Fec , (51) and Proposition 2.17 ensure that the system pX, T q satisfies the strong u-MOMO property. Therefore, for each f P CpXq we have lim

and it follows from (53) that ş f b π 0 dp ν J " 0. Using (52), we get

But tf ˝J : f P CpXqu is dense in L 2 pApκqq and therefore π 0 K L 2 pApκqq, which is the Veech condition for u with respect to the characteristic class F ec .

Orthogonality to C ID . Proof of Corollary 1.8

We recall that (Proposition 2.12)

Since the characteristic factor is represented by the σ-algebra of invariant sets, by Theorems B and A, we obtain immediately that:

Let us now pass to a combinatorial characterization of the Veech condition. Assume that κ is given as the limit of 1

In view of Corollary 5.3, we need to decipher Epπ 0 |I κ q " 0. By the von Neumann theorem, we have

The latter is equivalent to

The proof of Corollary 1.8 follows immediately.

Remark 5.4. Hence, the Matomäki-Radziwiłł theorem [START_REF] Matomäki | Multiplicative functions in short intervals[END_REF] on the behaviour of a strongly aperiodic multiplicative function u on a typical short interval implies u K C ID . However, as shown in [START_REF] Gomilko | On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł and Tao[END_REF], the aperiodic multiplicative functions defined in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] do not satisfy the assertion of Corollary 1.8.

In Corollary 5.3, the Veech condition (for u) equivalent to u K C ID is written as π 0 K L 2 pI κ q. If we look at it more spectrally, we obtain immediately that [START_REF] Veech | Möbius Dynamics[END_REF] u K C ID if and only if σ π 0 ,κ pt1uq " 0 for all κ P V S puq, i.e. the spectral measure of π 0 (with respect to each Furstenberg system) has no atom at 1. Classically (by a simple computation), we have:

Hence, the Veech condition is equivalent to

Combinatorially, we obtain

for each sequence pN k q defining a Furstenberg system κ.

It follows that (55) is equivalent to the short interval behaviour [START_REF] Downarowicz | Pure strictly uniform models of non-ergodic measure automorphisms[END_REF]. In other words, condition

Orthogonality to C DISPpGq with G countable

Let G Ă S 1 be a countable subgroup and recall that DISPpGq stands for the (characteristic) class of discrete spectrum automorphisms whose groups of eigenvalues are contained in G. Since z P S 1 is an eigenvalue of pZ, D, κ, Rq if and only if it is an eigenvalues of a subset of positive measure of ergodic components, it is not hard to see that

It follows that u K C F DISPpGq if and only if σ π 0 ,κ pGq " 0, i.e. the spectral measure of π 0 has no atoms belonging to G (for each Furstenberg system κ P V S puq).

Suppose that e 2πiα P G. Consider vpnq :" e 2πinα upnq for n P N. Note that

So, if we have a subsequence pN k q along which both 1 12 then σ π 0 ,κ " δ e 2πiα ˚σπ 0 ,κ 1 , whence σ π 0 ,κ pte 2πiα uq " 0 if and only if σ π 0 ,κ 1 pt1uq.

By our previous subsection it follows that the latter condition is equivalent to:

which is the strong u-MOMO condition for the irrational rotation by α. 13

Furstenberg systems and the strong u-MOMO property

The following proposition helps us to exclude some measure-theoretic systems from the list of Furstenberg systems of an arithmetic function.

Proposition 5.6. Let u : N Ñ C be a bounded arithmetic function. Then no Furstenberg system κ P V S puq has a topological model which is strongly u-MOMO.

12 Note that these common sequences yield all Furstenberg systems for both u and v. 13 Note that if f ptq " e 2πit then 1 bK

Proof. Suppose pX u , κ, Sq has a topological model pZ, ν, Rq which satisfies the strong u-MOMO property. Let J : Z Ñ X u settles a measure-theoretic isomorphism and let ν J be the corresponding graph joining. We assume that

From Proposition 3.1 we can find a sequence pz n q Ă Z consisting of pieces of orbits of different points: tn : Rz n ‰ z n`1 u " tb k : k ě 1u with b k`1 ´bk Ñ 8, and a subsequence pN j q such that

Then, by the strong u-MOMO property of pZ, Rq,

Hence, ş E ν J pf |X u qπ 0 dκ " 0 for each continuous f on Z, and we obtain a contradiction since E ν J pL 2 pνq|X u q " L 2 pκq.

Corollary 5.7. Assume that for each pb k q with b k`1 ´bk Ñ 8,

Then the unipotent system px, yq Þ Ñ px, y `xq (on T 2 ) is not a Furstenberg system of u.

Proof. Since condition (56) is the strong u-MOMO property of the unipotent system, the result follows from Proposition 5.6.

Remark 5.8. Corollary 5.7 brings a better understanding of Problem 3.1 (due to Frantzikinakis) of the workshop [START_REF]Sarnak's Conjecture[END_REF]:

The system px, yq Þ Ñ px, y `xq is not a Furstenberg system of the Liouville function (see also slide no 6 in [17]). We recall that in [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF] (see Theorem 1.3 therein), it is proved that In view of Corollary 2.11 (see also [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF]) and Theorem A, in order to obtain u K C DISPec it is sufficient and necessary to have u K L 2 pKpI κ qq for each Furstenberg system κ of u. By our previous results, for the class of all topological systems whose all ergodic measures yield discrete spectra, Sarnak and Veech conditions are equivalent. We now write the Veech condition combinatorially, i.e., we provide the proof of Corollary 1.10.

Proof of Corollary 1.10. By Corollary 2.11, we need to show that for each κ being a Furstenberg system of u, we have (57)

By the von Neumann theorem,

1 N ÿ nďN ż pπ 0 ˝Sh ¨π0 q ˝Sn ¨π0 ˝Sh ¨π0 dκ.

Therefore, (57) is equivalent to

1 N ÿ nďN ż pπ 0 ˝Sh ¨π0 q ˝Sn ¨π0 ˝Sh ¨π0 dκ " 0.

Let pM k q be such that κ " lim kÑ8

which is precisely }u} u 2 ppM k qq " 0. Now, it suffices to use [START_REF] Gomilko | Möbius orthogonality in density for zero entropy dynamical systems[END_REF]. Remark 5.9. In fact, already Frantzikinakis [17] (see slide no 10) showed that if u is generic then }u} u 2 " 0 if and only if

We recall that this condition is equivalent to the strong u-MOMO property of the unipotent system px, yq Þ Ñ px, y `xq.

Remark 5.10. Note that for each (bounded) u : N Ñ C satisfying }u} u 2 " 0 the system (58) px, yq Þ Ñ px, x `yq on pT 2 , Leb b Lebq Proof. It is enough to check the strong u-MOMO for functions F of the form χ b f , where χ P p G and f P Cpr0, 1sq. We have

Our claim follows from p˚q.

Theorem 5.12. Assume that u enjoys the property p˚q. Then u K C DISP if and only if π 0 K L 2 pKpκqq for each κ P V S puq (iff the spectral measure σ π 0 is continuous for each Furstenberg system κ).

Proof. We only need to show that u K C DISP implies π 0 K L 2 pKpκqq for each κ P V S puq. 15 Using (59), let p settle a factor map from R α ˆId r0,1s acting on pG ˆr0, 1s, m G b Lebq and pX u {Kpκq, Kpκq, κ| Kpκq q. Let pm G b Lebq p stand for the corresponding graph joining and ρ for the relatively independent extension of it to a joining of pG ˆr0, 1s, m G b Leb, R α ˆIdq with pX u , κ, Sq. Now, by Proposition 3.1, the integral ş F bπ 0 dρ can be computed using a quasi-generic sequence ppg n q, pS n uqq. Since, by Lemma 5.11, our topological system R α ˆId satisfies the strong u-MOMO property, this integral vanishes. On the other hand, for each F P CpG ˆr0, 1sq,

and since E ρ pCpG ˆr0, 1sq|X u q is dense in L 2 pK, κ| K q (in view of the definition of ρ), it follows that π 0 K L 2 pKpκqq.

Proof of Corollary 1.16. Note that in the proof of Theorem 5.12, we have shown that our original assumption p˚q already implies the Veech condition.

In particular, the Sarnak and the Veech properties are equivalent. Condition ( 17) is just rewriting the Wiener condition combinatorially. Finally, the last part ( 18) is proved in Appendix A.

Proof of Corollary 1.17. By Corollary 1.16, we only need to show that irrational rotations satisfy the strong u-MOMO property. This follows from the fact that irrational rotations satisfy the AOP property [START_REF] Abdalaoui | Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals[END_REF] and that the AOP property implies the strong u-MOMO property [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF].

No strong u-MOMO in positive entropy

In this section we discuss the problem of orthogonality to C ZE and the reversed problem of the absence of orthogonality to an arbitrary positive entropy systems, following some ideas from [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF]. Recall that the following has been proved in [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF].

Proposition 6.1 ([3]

). Let u : N Ñ C be a bounded arithmetic function.

The following are equivalent:

(b) For each pX, T q of zero entropy and f P CpXq, (1) holds uniformly in x P X.

(c) Each zero entropy pX, T q satisfies the strong u-MOMO property.

On the other hand, it follows from the results of Downarowicz and Serafin [START_REF] Downarowicz | Almost Full Entropy Subshifts Uncorrelated to the Möbius Function[END_REF][START_REF] Downarowicz | A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Möbius function[END_REF] that for each u K C ZE there exists pX, T q such that (60) u K pX, T q and pX, T q R ZE.

In fact, one can get a positive entropy system pX, T q in which for every f P CpXq (1) holds uniformly in x P X. We prove however that (60) fails if orthogonality is replaced by the strong u-MOMO property. To avoid technical details, we restrict ourselves to the case of an arithmetic function u taking finitely many values.

Theorem D. Let u : N Ñ C be an arithmetic function taking finitely many values. Assume that u K C ZE . Then no positive entropy topological dynamical system satisfies the strong u-MOMO property.

Proof of Theorem D

We fix a bounded arithmetic function u : N Ñ C. We need a series of results from [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF] in some modified forms. In [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF], the equivalence of certain three properties (P1), (P2) and (P3) of an ergodic measure-theoretic dynamical system pZ, BpZq, κ, Rq was proved. Condition (P1) was nothing but the strong u-MOMO for some topological system being a model of the system given by κ. Instead of recalling (P2), let us formulate red its subsequence version: (P2') Assume that pX, T q is any topological system and let x P X. If x is generic along pN k q for a measure which is isomorphic (as dynamical systems) to κ then lim kÑ8

The proof of the implication (P1) ùñ (P2') is a repetition of the proof of (P1) implies (P2). In Lemma 17 in [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF], we need to consider the sequence pN k q instead of N and start with lim sup along this sequence. As a consequence of the above, we obtain the following version of Corollary 12 from [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF]. Corollary 6.2. Assume that κ is an ergodic shift-invariant measure on D Z L , and that there exists y P D Z L , generic along pN k q for κ, correlating with u along pN k q, i.e. the sequence p 1 N k ř nďN k ypnqupnqq does not go to zero. Then the strong u-MOMO property fails for any uniquely ergodic model of pD Z L , κ, Sq.

Then, by repeating the proof from [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF], we obtain the following form of Corollary 14 in [START_REF] Abdalaoui | Möbius disjointness for models of an ergodic system and beyond[END_REF]. Corollary 6.3. Assume that y is generic along pN k q for a Bernoulli measure ν, and that y and u correlate along pN k q. Then the strong u-MOMO property fails for any pX, T q with hpX, T q ą hpνq.

We also need the following crucial probabilistic lemma whose proof we postpone to the next subsection. Lemma 6.4. Assume that X " pX n q nPZ is a a stationary process of positive entropy, taking finitely many complex values. Then for any non-trivial probability distribution β concentrated on a finite subset of R, there exists a stationary coupling of X with a Bernoulli process Y " pY n q nPZ of distribution β bZ such that ErX 0 Y 0 s ‰ ErX 0 sErY 0 s.

We now assume that u takes finitely many values and satisfies the Veech condition: π 0 K L 2 pΠpκqq for each Furstenberg system κ of u. Lemma 6.5. For each h ą 0 there exists a sequence y, generic for a Bernoulli measure of entropy h along some increasing sequence pN k q, and correlating with u along pN k q.

Proof. Let κ be a Furstenberg system of u, and pM q such that u is generic for κ along pM q. By assumption, the entropy of the stationary process defined by π 0 under κ is positive. Take a real-valued Bernoulli shift of entropy h (Bernoulli measure denoted by ν). Using Lemma 6.4, find a joining of κ and ν for which π 0 (in L 2 pX u , κq) is not orthogonal to π 0 in L 2 pνq: ş π 0 b π 0 dρ ‰ 0. Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [START_REF] Bergelson | Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation[END_REF]) to find y in the subshift defining the Bernoulli automorphism such that pu, yq is generic, along a subsequence pN k q " pM k q, for ρ. Then

which means that u and y correlate along pN k q.

Now the proof of Theorem D is a straightforward consequence of Lemma 6.5 and Corollary 6.3.

Proof of Lemma 6.4

Let X " pX n q nPZ be a positive entropy stationary process as in the statement of the lemma. Without loss of generality (considering its real or imaginary part), we can assume that this process takes its values in a finite subset tx 1 ă x 2 ă ¨¨¨ă x r u of R. We also consider a given probability measure β supported on a possibly different finite subset of R ty 1 ă y 2 ă ¨¨¨ă y s u, which is supposed to be non trivial (i.e. not reduced to a Dirac measure). Thus we can assume that s ě 2, and βpy j q ą 0 for each 1 ď j ď s. The purpose of this section is to show how we can construct a stationary coupling of X with a Bernoulli process Y whose distribution is β bZ , in such a way that for each n P Z, (61)

ErX n Y n s ą ErX n s ErY n s.

We observe that the validity of the preceding inequality is unchanged if we replace Y n by Y n `C for a fixed C. Thus we can and we do assume without loss of generality that the probability β is such that ErY n s " 0.

To construct the announced coupling, we just assume that, on the probability space where the process X is defined, we also have an i.i.d. process V " pV n q nPZ such that

The construction will be divided into two steps: first we construct an auxiliary (uniform i.i.d.) process U and then we use it to construct Y which satisfies the assertion of Lemma 6.4.

Step 1: uniform i.i.d. process U For n P Z and j P t1, . . . , ru, we consider the random variable P j,n defined by P j,n :" P `Xn " x j | pX m q mďn´1 ˘.

When j is fixed, pP j,n q nPZ is a stationary process. On the other hand, if we fix n, then pP 1,n , . . . , P r,n q is the conditional distribution of X n given pX m q mďn´1 , in particular we have almost surely 0 ď P j,n ď 1, and r ÿ j"1

This allows us to define a random partition of r0, 1r into disjoint subintervals I 1,n , . . . , I r,n where for each j, I j,n is the interval of length P j,n defined by

Appendix

A From averaged double to averaged multiple correlations

This section follows some arguments from [START_REF] Matomäki | An averaged form of Chowla's conjecture[END_REF].

Remark A.1. In the proof below we will use the following standard fact: let pxpnqq be a sequence of complex number bounded by 1. Then

The little "o" is uniform with respect to M . 16 We have the following general lemma:

Lemma A.2. Let pN q PN be a sequence of natural numbers. b i pn `hi q ˇˇ" 0.

Proof. Notice first that (73) can be rewritten as the following: for every ε ą 0, there exists H ε such that for H ą H ε and all sufficiently large (depending on H), we have

Now, notice that for any H, N , H 1 and any h 1 ď H 1 , by shifting the summation over n ď N by h 1 (for every fixed choice of h 1 , . . . h k ), we have

and OpH k ¨h1 q " OpH k ¨H1 q. Notice that as h i is taken from r0, Hs, then h i `h1 is taken from rh 1 , H `h1 s (which is a small shift of r0, Hs if h 1 is much smaller than H). So putting h 1 to the summation over h i , we get

Putting the two displayed equations together we get that for every h 1 ď H

Averaging the above equation over all h 1 ď H 1 , we get that

where Gpnq " ś k i"1 b i pn `hi q. We will now estimate apn `h1 qapn 1 `h1 q ¯GpnqGpn 1 q, which will be easier to handle than the above expression for A (and then use Remark A. apn `h1 qapn 1 `h1 qapn `h2 q ¨apn 1 `h2 q "

apn `h1 qapn `h2 qapn 1 `h2 qapn 1 `h1 q.

The sum in the last term by exchanging the order of summation is equal to Therefore, (77) (and, thus, also (76)) is of the order of op1q `O ´pH 1 q 2 N 2

¯.

Using again Remark A.1, we conclude that also (75) is of the same order. It follows immediately that also the order of (74) is the same. Thus, we have proved that