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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))


An overview of the last 7 years results concerning Sarnak’s conjecture on Möbius disjointness is presented, focusing on ergodic theory aspects of the conjecture.

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  1. 1.

    Most often, however not always, T will be a homeomorphism.

  2. 2.

    μ stands for the arithmetic Möbius function, see next sections for explanations of notions that appear in Introduction.

  3. 3.

    To be compared with Möbius Randomness Law by Iwaniec and Kowalski [99], p. 338, that any “reasonable” sequence of complex numbers is orthogonal to μ.

  4. 4.

    In order to establish Möbius disjointness, we need to show convergence (11.1) (for all x ∈ X) only for a set of functions linearly dense in C(X), so, for the rotations on the (additive) circle \({\mathbb {T}}=[0,1)\), we only need to consider characters. Note also that if the topological system (X, T) is uniquely ergodic then we need to check (11.1) (for all x ∈ X) only for a subset of C(X) which is linearly dense in L1.

    In what follows, for inequalities (as (11.3)), we will also use notation ≪ or O(⋅), or ≪ A or O A (⋅) if we need to emphasize a role of A > 0.

  5. 5.

    For a presentation of a part of it, see [37].

  6. 6.

    For a detailed account of these results, we refer the reader to [153].

  7. 7.

    As proved by Tao [156], the logarithmic averages version of the Chowla conjecture is equivalent to the logarithmic version of Sarnak’s conjecture. We will see later in Sect. 11.4 that once the logarithmic Chowla conjecture holds for the Liouville function λ, we have that all configurations of ± 1s appear in λ (infinitely often).

  8. 8.

    The same argument applied to the Liouville function λ implies that the subshift X λ generated by λ is uncountable, see Sect. 11.4.

  9. 9.

    σ f stands for the spectral measure of f.

  10. 10.

    We recall that Bourgain in [16,17,18], proved that for each \(\alpha \geqslant (1+\sqrt {3})/2\), each automorphism T of a probability standard Borel space \({(X,{\mathcal {B}},\mu )}\) and each \(f\in L^{\alpha {(X,{\mathcal {B}},\mu )}}\) the sums in (11.5) converge for a.e. x ∈ X. The result has been extended by Wierdl in [170] for all α > 1.

  11. 11.

    Indeed, we have

    $$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left| \frac 1N\sum_{p\leqslant N}f(T^px)\log p - \frac 1N\sum_{p\leqslant N}g(T^px)\log p\right|\\ &\displaystyle &\displaystyle \quad \leqslant \frac 1N\sum_{p\leqslant N}|\,f(T^px)-g(T^px)|\log p\leqslant\|\,f-g\| \frac 1N\sum_{p\leqslant N}\log p={\mathrm{O}}(\|\,f-g\|), \end{array} \end{aligned} $$

    as condition \(\frac 1N\sum _{n\leqslant N}\boldsymbol {\Lambda }(n)\xrightarrow [N\to \infty ]{} 1\) is equivalent to \(\frac 1N\sum _{p\leqslant N}\log p\xrightarrow [N\to \infty ]{} 1\).

  12. 12.

    We recall that if \((Z,{\mathcal {D}},\kappa ,R)\) is a measure-preserving system then by its uniquely ergodic model we mean a uniquely ergodic system (X, T) with the unique (Borel) T-invariant measure μ such that \((Z,{\mathcal {D}},\kappa ,R)\) is measure-theoretically isomorphic to \((X,{{\mathcal {B}}}(X),\mu ,T)\).

  13. 13.

    The Möbius Inversion Formula is given by .

  14. 14.

    We will not discuss here the problem of convergence of Dirichlet series, see [146].

  15. 15.

    An analytic continuation of ζ yields a meromorphic function on \({{\mathbb {C}}}\) (with one pole at s = 1) satisfying the functional equation

    $$\displaystyle \begin{aligned} \zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\varGamma(1-s)\zeta(1-s).\end{aligned} $$

    Because of the sine, ζ(−2k) = 0 for all integers \(k\geqslant 1\)—these are so called trivial zeros of ζ (ζ(2k) ≠ 0 since Γ has simple poles at 0, −1, −2, …). In Re s > 1 there are no zeros of ζ (ζ is represented by a convergent infinite product), so except of − 2k, k⩾1, there are no zeros for \(s\in {{\mathbb {C}}}\), Re s < 0 (as Re(1 − s) > 1). The Riemann Hypothesis asserts that all nontrivial zeros of ζ are on the line \(x=\frac 12\). See [146].

  16. 16.

    This proof of (11.15) has been shown to us by G. Tenenbaum.

  17. 17.

    For \(N\in {{\mathbb {N}}}\) we write [N] for the set {1,  2, …, N}. Given \(h,N\in {{\mathbb {N}}}\) and \(f\colon {{\mathbb {N}}}\rightarrow {{\mathbb {C}}}\), we let S h f(n) = f(n + h) and . For \(s\in {{\mathbb {N}}}\), the Gowers uniformity seminorm [80] \(\|.\|{ }_{U^s_{[N]}}\) is defined in the following way:

    $$\displaystyle \begin{aligned} \|\,f\|{}_{U^1_{[N]}}:= \left|\frac{1}{N}\sum_{n=1}^N f_N(n)\right| \end{aligned}$$

    and for \(s\geqslant 1\)

    $$\displaystyle \begin{aligned} \|\,f\|{}_{U^{s+1}_{[N]}}^{2^{s+1}}:= \frac{1}{N}\sum_{h=1}^N \left\|\,f_N S_h\, \overline{f_N}\right\|{}_{U^s_{[N]}}^{2^s}. \end{aligned}$$

    A bounded function \(f\colon {{\mathbb {N}}}\rightarrow {{\mathbb {C}}}\) is called uniform if \(\|\,f \|{ }_{U^s_{[N]}}\) converges to zero as N → for each s⩾1.

  18. 18.

    An arithmetic function u is rational if for each ε > 0 there is a periodic function v such that \(\limsup _{N\to \infty }\frac 1N\sum _{n\leqslant N}|\boldsymbol {u}(n)-\boldsymbol {v}(n)|<{\varepsilon }\). Note that since μ is aperiodic, whence orthogonal to all periodic sequences, it will also be orthogonal to each rational u [12]. An example of rational sequence is given by μ2. For more examples, see the sets of \(\mathcal {B}\)-free numbers in the Erdös case in Sect. 11.7.

  19. 19.

    To be compared with the estimates (11.3), where we drop the \(\sup \) requirement.

  20. 20.

    The main ideas for this result appeared in [30] and [134]. It was first established in a slightly different form in [106] and then in [21], see also [88] for a proof. The criterion has its origin in the bilinear method of Vinogradov [164] which is a technique to study sums of a over primes in terms of sums over progressions \(\sum _{n\leqslant N} a_{dn}\) and sums \(\sum _{n\leqslant N} a_{d_1n}a_{d_2n}\). If a n  = f(Tnx) then these sums are Birkhoff sums for powers of T and their joinings.

    In what follows we will refer to Theorem 11.15 as to the KBSZ criterion.

  21. 21.

    We can easily see that when Tx = x + α is an irrational rotation on \({\mathbb {T}}=[0,1)\), then, by the Weyl criterion on uniform distribution, (11.21) is satisfied for all characters (for all \(x\in {\mathbb {T}}\)), but there are continuous zero mean functions for which (11.21) fails [112].

  22. 22.

    As a matter of fact, in [27], it is formulated for the Liouville function. We follow [150]. For a discussion on an equivalence of the Chowla conjecture with μ and λ, we invite the reader to [143]. As shown in [127], there are non-pretentious (completely) multiplicative functions for which Chowla conjecture fails. For more information, see the discussion on Elliot’s conjecture in [127].

  23. 23.

    The Chowla conjecture is rather “close” in spirit to the Twin Number Conjecture in the sense that the latter is expressed by \((\ast )\;\sum _{n\leqslant x}\boldsymbol {\Lambda }(n)\boldsymbol {\Lambda }(n+2)=(2\varPi _2)\cdot x+{\mathrm {o}}(x)\), where \(\varPi _2=\prod _{p>2}(1-\frac {1}{(p-1)^2})=0,66016\ldots \) which can be compared with \(\sum _{n\leqslant x}\boldsymbol {\mu }(n)\boldsymbol {\mu }(n+2)={\mathrm {o}}(x)\) which is “close” to the Chowla conjecture, see e.g. [158]. A recent development shows that it is realistic to claim that the Chowla conjecture with an error term of the form \({\mathrm {o}}((\log N)^{-A})\) for some A large enough (A depending on the number of shifts of μ that are considered) implies (∗). (Of course, everywhere Λ is a good approximation of .)

    See also [138] for a (conditional) equivalence of (∗) with \(\sum _{n\leqslant N}\boldsymbol {\Lambda }(n)\boldsymbol {\mu }(n+2)={\mathrm {o}}(N)\).

  24. 24.

    We recall that either x is generic or Q-gen(x) is a connected uncountable set, see Proposition 3.8 in [40].

  25. 25.

    The point μ2 is recurrent, so there is a “completion” of μ2 to a two-sided sequence generating the same subshift.

  26. 26.

    Consider Bernoulli measure B(1∕2,  1∕2) on \(\{-1,1\}^{{{\mathbb {Z}}}}\) and Mirsky measure \(\nu _{\boldsymbol {\mu }^2}\) on \(\{0,1\}^{{{\mathbb {Z}}}}\). Measure \(\widehat {\nu }_{\boldsymbol {\mu }^2}\) is the image of the product measure \(B(1/2,1/2){\otimes }\nu _{\boldsymbol {\mu }^2}\) via the map

    $$\displaystyle \begin{aligned}(x,y)\mapsto ((x(n)\cdot y(n)))_{n\in{{\mathbb{Z}}}}\in\{-1,0,1\}^{{{\mathbb{Z}}}}.\end{aligned}$$
  27. 27.

    Recall that if R i is an automorphism of a probability standard Borel space \((Z_i,{\mathcal {D}}_i, \nu _i)\), i = 1,  2, then each R1 × R2-invariant measure λ on \((Z_1\times Z_2,{\mathcal {D}}_1{\otimes }{\mathcal {D}}_2)\) having the projections ν1 and ν2, respectively is called a joining of R1 and R2: we write λ ∈ J(R1, R2). If R1, R2 are ergodic then the set Je(R1, R2) of ergodic joinings between R1 and R2 is non-empty. A fundamental notion here is the disjointness (in sense of Furstenberg) [73]: R1 and R2 are disjoint if J(R1, R2) = {ν1ν2}: we write R1 ⊥ R2. For example, zero entropy automorphisms are disjoint with automorphisms having completely positive entropy (Kolmogorov automorphisms) and also a relativized version of this assertion holds.

  28. 28.

    We will see later that some special cases of validity of convergence in (11.25) also have their ergodic interpretations and they imply Möbius disjointness for restricted classes of dynamical systems of zero entropy; in particular, see Corollaries 11.37 and 11.42.

  29. 29.

    The above proof was already suggested by Sarnak in [150].

  30. 30.

    If Möbius disjointness in a dynamical system is shown through the KBSZ criterion then we obtain orthogonality with respect to all multiplicative functions.

  31. 31.

    If we consider general sequences \(z\in \{-1,0,1\}^{{{\mathbb {N}}}}\) then we can speak about the Sarnak and Chowla properties on a more abstract level: for example the Chowla property of z means (11.25) with μ replaced by z. See Example 5.1 and Remark 5.3 in [57] for sequences orthogonal to all deterministic sequences but not satisfying the Chowla property. However, arithmetic functions in these examples are not multiplicative.

    However, an analogy between disjointness results in ergodic theory and disjointness of sequences is sometimes accurate. For example, a measure-theoretic dynamical system has zero entropy if and only if it is disjoint with all Bernoulli automorphisms. As pointed out in [57] (Prop. 5.21), a sequence \(t\in \{-1,1\}^{{\mathbb {N}}}\) is completely deterministic if and only if it is disjoint with any sequence \(z\in \{-1,0,1\}^{{{\mathbb {N}}}}\) satisfying the Chowla property.

  32. 32.

    See Remark 1.9. Also, in [156] the Liouville function λ is considered, see page 2 in [156] how to replace λ by μ.

  33. 33.

    In fact, \(|\sum _{n\leqslant N}\boldsymbol {\mu }(n)/n|\leqslant 1\).

  34. 34.

    Assume that (a n ) is a bounded sequence and set A n  = a1 + … + a n . Then, we have by summation by parts

    $$\displaystyle \begin{aligned} \frac{1}{\log N}\sum_{n\leqslant N}\frac{a_n}{n}&=\frac{1}{\log N}\sum_{n\leqslant N}(A_{n+1}-A_n)\frac 1n \\ &=\frac{1}{\log N}\sum_{n\leqslant N}A_n \left(\frac 1n-\frac{1}{n+1}\right)+{\mathrm{o}}(1)= \frac{1}{\log N}\sum_{n\leqslant N}\frac{A_n}{n}\frac{1}{n+1}+{\mathrm{o}}(1).\end{aligned} $$

    It follows that:

    • If the Cesàro averages of (a n ) converge, so do the logarithmic averages of (a n ).

    • The converse does not hold (see e.g. [14] in \(\mathcal {B}\)-free case, Sect. 11.7.1).

    • If the Cesàro averages converge along a subsequence (N k ) then not necessarily the logarithmic averages do the same. Indeed, by (11.31), \(\frac {1}{\log N_k}\sum _{n\leqslant N_k}\frac {a_n}{n}\) is (up to a small error) a convex combination of the Cesàro averages for all \(n\leqslant N_k\).

  35. 35.

    The result has been observed in [71], cf. also [95].

  36. 36.

    We use here the standard result in the theory of unitary operators that mutual singularity of spectral measures implies orthogonality. Recall also the classical result in ergodic theory that spectral disjointness implies disjointness.

  37. 37.

    Consider \({X_1=X_2={\mathbb {T}}^2}\) with \({\mu _1=\mu _2=Leb_{{\mathbb {T}}^2}}\), the diagonal joining Δ on X1 × X2 and f(x, y) = θ(x, y)¯ with θ(x, y) = e2πiy. The spectral measure of θ is Lebesgue, and all ergodic components of the measure μ1 have discrete spectra.

  38. 38.

    I.e., we assume the existence of the limits of sequences \(\left (\frac 1N\sum _{n\leqslant N} a'(n)a'(n+k_1)\ldots a'(n+k_r)\right )_{N\geqslant 1}\) for every \(r\in {{\mathbb {N}}}\) and \(k_1,\ldots , k_r\in {{\mathbb {N}}}\) (not necessarily distinct) with a′ = a or \(\overline {a}\). It is not hard to see that a admits correlations if and only if it is generic, cf. Sect. 11.4.1.

  39. 39.

    We have \(\|\boldsymbol {\lambda }\|{ }_{U^1({{\mathbb {N}}})}=0\) by Matomäki and Radziwiłł [126], moreover \(\|\boldsymbol {\lambda }\|{ }_{U^2({{\mathbb {N}}})}=0\) is equivalent to \(\lim _{N\to \infty }{{\mathbb {E}}}_{m\in {{\mathbb {N}}}}\sup _{\alpha \in [0,1)} \left |{{\mathbb {E}}}_{n\in [m,m+N]}\boldsymbol {\lambda }(n)e^{2\pi in\alpha }\right |=0\) (cf. Conjecture C) and remains open. For a subsequence version of Theorem 11.40 for logarithmic averages, see [156].

  40. 40.

    By that we mean a(n) = f(gnΓ) for some continuous f ∈ C(GΓ) and g ∈ G.

  41. 41.

    Given a measure-theoretic dynamical system \((Z,\mathcal {D},\rho ,R)\), its system of arithmetic progressions with prime steps is of the form \((Z^{{\mathbb {Z}}},{{\mathcal {B}}}(Z^{{{\mathbb {Z}}}}),\widetilde {\rho },S)\), where S is the shift and the (shift invariant) measure \(\widetilde {\rho }\) is determined by

    $$\displaystyle \begin{aligned} \int_{Z^{{{\mathbb{Z}}}}}\prod_{j=-m}^mf_j(z_j)\,d\widetilde{\rho}(z)= \lim_{N\to\infty}\frac{\log N}{N}\sum_{p\leqslant N}\int_Z\prod_{j=-m}^m f_j\circ R^{pj}\,d\rho\end{aligned}$$

    for all m⩾0, fm, …, f m  ∈ L(Z, ρ) (here z = (z j )). It is proved that such shift systems have no irrational spectrum. One of key observations is that each Furstenberg system of the Liouville function is a factor of the associated system of arithmetic progressions with prime steps.

  42. 42.

    The product decomposition depends on the component.

  43. 43.

    They are new even for irrational rotations. Cf. the notions of (S)-strong and (S0)-strong and their equivalence to the Chowla type condition in [57].

  44. 44.

    Note that the answer is positive in all uniquely ergodic models of the one-point system: each such a model has a unique fixed point that attracts each orbit on a subset of density 1, cf. the map \(e^{2\pi ix}\mapsto e^{2\pi ix^2}\), x ∈ [0,  1). This argument is however insufficient already for uniquely ergodic models of the exchange of two points: in this case we have a density 1 attracting 2-periodic orbit {a, b}, but we do not control to which point a or b the orbit returns first. Quite surprisingly, it seems that already in this case we need [126] to obtain Möbius disjointness of all uniquely ergodic models.

  45. 45.

    If all uniquely ergodic systems were Möbius disjoint, then as noticed by T. Downarowicz, we would get that the Chowla conjecture fails in view of the result of B. Weiss [169] Thm. 4.4’ on approximation of generic points of ergodic measures by uniquely ergodic sequences.

  46. 46.

    Topological mixing for example excludes the possibility of having eigenfunctions continuous.

  47. 47.

    Our objective is of course the Möbius function μ, however the whole approach can be developed for an arithmetic function satisfying some additional properties.

  48. 48.

    The acronym comes from Möbius Orthogonality of Moving Orbits.

  49. 49.

    Inv stands here for the σ-algebra of T × S-invariant sets modulo ρ.

  50. 50.

    That is, Sarnak’s conjecture and the strong MOMO property (relatively to μ) for all deterministic systems are equivalent statements.

  51. 51.

    It is not hard to see that the MOMO property implies the relevant uniform convergence. As a matter of fact, the strong MOMO property is equivalent to the uniform convergence (in x, for a fixed f ∈ C(X)) on short intervals: \(\frac 1M\sum _{1\leqslant m<M}\left |\frac 1H\sum _{m\leqslant h<m+H}f(T^hx)\boldsymbol {\mu }(n)\right |\to 0\) (when H, M → and H = o(M)). It follows that we have equivalence of: Sarnak’s conjecture (11.2), Sarnak’s conjecture in its uniform form, Sarnak’s conjecture in its short interval uniform form and the strong MOMO property. Moreover, each of these conditions is implied by the Chowla conjecture.

  52. 52.

    This result means that there must be an observable sequence in (X, T) which significantly correlates with u.

  53. 53.

    This means that each measure-preserving transformation commuting with R must be invertible. Finite multiplicity of the Koopman operator associated to R guarantees coalescence. In particular, all ergodic rotations are coalescent.

  54. 54.

    Sarnak in [150] mentions that Bourgain has constructed a positive entropy system which is Möbius disjoint but this construction has never been published.

  55. 55.

    This is a “typical” property of an automorphism of a probability standard Borel space [38]. Möbius disjointness for uniquely ergodic models for this case is already noticed in [21].

  56. 56.

    Take an ergodic rotation with the group of eigenvalues \(\{e^{2\pi i\alpha m/n} : m,n\in {{\mathbb {Z}}}, n\neq 0, \alpha \notin {\mathbb {Q}}\}\).

  57. 57.

    By the suspension flow of R we mean the special flow over R under the constant function (equal to 1).

  58. 58.

    The same argument shows that if Sarnak’s conjecture holds then (11.42) holds for each zero entropy (X, T), \(a,b\in {{\mathbb {N}}}\), f ∈ C(X) uniformly in x ∈ X.

  59. 59.

    Horocycle flows are mixing of all orders, see [123].

  60. 60.

    In case of horocycle flows (Bourgain, Sarnak and Ziegler [21]) Ratner’s theorems on joinings are used and these provide no rate.

  61. 61.

    We will tacitly assume that Γ is cocompact, so that the homogenous space \(\varGamma \backslash PSL_2({\mathbb {R}})\) is compact and the system is uniquely ergodic by Furstenberg in [74]; otherwise, as in the modular case when \(\varGamma =PSL_2({{\mathbb {Z}}})\) we need to compactify our space. The proof of Theorem 11.62 in the modular case is slightly different than what we describe below.

  62. 62.

    We have \(h_t(\varGamma x)=\varGamma \cdot \left (x\cdot \left [\begin {array}{ll}1&t\\ 0&1\end {array}\right ]\right )\) and \(g_s(\varGamma x)=\varGamma \cdot \left (x\cdot \left [\begin {array}{ll} e^{-s}&0\\0&e^s\end {array}\right ]\right )\); we identify g s and h t with the relevant matrices.

  63. 63.

    The measure ρ depends on p, q and x and it is so called algebraic measure, i.e. a Haar measure.

  64. 64.

    To be compared with Remark 11.59; the difference however is that when the ratio of p and q is close to 1, we can choose graph joinings in a compact set.

  65. 65.

    I.e. \(p(n)=a_1^{p_1(n)}\ldots a_k^{p_k(n)}\), where \(p_j\colon {{\mathbb {N}}}\to {{\mathbb {N}}}\) is a polynomial, j = 1, …, k. See, Section 6 in [84] for the equivalence with the classical definition of polynomials sequences in nilpotent Lie groups.

  66. 66.

    We assume that G is connected.

  67. 67.

    For degree 1 polynomials, the result is already in [127].

  68. 68.

    More precisely, given an automorphism T of a probability standard Borel space \({(X,{\mathcal {B}},\mu )}\), we consider

    $$\displaystyle \begin{aligned} a_n=\int_X g_1\circ T^{p_1(n)}\cdot\ldots\cdot g_k\circ T^{p_k(n)}\,d\mu,\end{aligned} $$

    where g i  ∈ L(X, μ), \( p_i\in {{\mathbb {Z}}}[x] \), i = 1, …, k (k⩾1).

  69. 69.

    In [31, 98, 129] it is proved that the sequence (−1)u(n), n⩾1 is orthogonal to μ.

  70. 70.

    As noted in [13], this leads to dynamical systems given by rational sequences and such systems are Möbius disjoint. Note also that for the synchronized case, once the system is uniquely ergodic, it is automatically a uniquely ergodic model of an automorphism with discrete spectrum, cf. Corollary 11.37 and Remark 11.51.

  71. 71.

    This example has partly continuous spectrum.

  72. 72.

    One can also ask about Möbius disjointness of related systems as tiling systems.

  73. 73.

    Moreover, Möbius disjointness is established for some other famous classes of rank one transformations such as: Katok’s α-weak mixing class (these are a special case of three interval exchange maps) or rigid generalized Chacon’s maps.

  74. 74.

    Hence, Tp and Tq are disjoint in Furstenberg’s sense, and, in fact, we even have AOP.

  75. 75.

    This has been proved, e.g. in an unpublished preprint of El Abadalaoui, Kułaga-Przymus, Lemańczyk and de la Rue.

  76. 76.

    If \({\mathcal {S}}\) preserves a measure ν then \({T_{f,{\mathcal {S}}}}\) preserves measure μν, the AOP property is considered with respect to this measure.

  77. 77.

    Such a sequence (a n ) is of the form (φ(n)(x)) with φ(n)(x) = φ(x) + φ(Tx) + … + φ(Tn−1x), n⩾0.

  78. 78.

    See the most prominent example of such a reduction, namely, Furstenberg’s ergodic proof of Szemerédi theorem on the existence of arbitrarily long arithmetic progressions in subsets of integers of positive upper Banach density [76].

  79. 79.

    In general, there is no uniquely ergodic model (X, T) of R with T topologically distal.

  80. 80.

    As a matter of fact, such a question remains open even for 2-point extensions of irrational rotations.

  81. 81.

    As a matter of fact, in [58] it is proved that if a uniquely ergodic homeomorphism T satisfies the strong MOMO property (see Definition 1 on page 36) and (continuous) φ: X → G (G is a compact Abelian group) satisfies φ := ξ − ξ ∘ T has a measurable solution ξ : X → G, then the homeomorphism T φ of X × G is Möbius disjoint. This applies if (11.45) has a measurable solution for k = 1. It is however an open question whether we have Möbius disjointness when there is no measurable solution for k = 1 but there is such a solution for some k⩾2.

  82. 82.

    It follows from a subsequent paper [113] that the Anzai skew products considered in [112] enjoy the AOP property.

  83. 83.

    For the latter two classes all invariant measures yield discrete spectrum.

  84. 84.

    If B ∈{0, 1}k and C = C(0)C(1)…C( − 1) ∈{0, 1} then we define B × C := (B + C(0))(B + C(1))…(B + C( − 1)).

  85. 85.

    A sequence \(x\in A^{{{\mathbb {N}}}}\) is called Toeplitz if for each \(n\in {{\mathbb {N}}}\) there is \(q_n\in {{\mathbb {N}}}\) such that x(n + jq n ) = x(n) for each j = 0,  1, …

  86. 86.

    So called regular Toeplitz sequences are treated in [56] and [44], these are however uniquely ergodic models of odometers.

  87. 87.

    This is clearly a refinement of the fact that the asymptotic density of square-free integers exists (it is given by 6∕π2 = 1∕ζ(2)). It follows that μ2 is a completely deterministic point.

  88. 88.

    The frequencies of blocks on μ2 were first studied by Mirsky [132, 133] and that is why we refer to \(\nu _{\boldsymbol {\mu }^2}\) (and the analogous measure in case of general \(\mathcal {B}\)-free systems) as the Mirsky measure.

  89. 89.

    More precisely, it is isomorphic to \((H,{\mathbb {P}},T)\), where H is the closure of \({\{(n\text{ mod }b_k)_{k\geqslant 1} : n\in {{\mathbb {Z}}}\}}\) in \(\prod _{k\geqslant 1}{{\mathbb {Z}}}/b_k{{\mathbb {Z}}}\) and Tg = g + (1,  1, … ), cf. (11.49).

  90. 90.

    This has been recently improved in [105] and by A. Bartnicka: X η is minimal if and only if η is Toeplitz.

  91. 91.

    Mentzen’s result is extended in [7] to every hereditary \(\mathcal {B}\)-free subshift.


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The research resulting in this survey was carried out during the Research in Pairs Program of CIRM, Luminy, France, 15-19.05.2017. J. Kułaga-Przymus and M. Lemańczyk also acknowledge the support of Narodowe Centrum Nauki grant UMO-2014/15/B/ST1/03736. J. Kułaga-Przymus was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 647133 (ICHAOS)) and by the Foundation for Polish Science (FNP).

The authors special thanks go to N. Frantzikinakis and P. Sarnak for a careful reading of the manuscript, numerous remarks and suggestions to improve presentation. We also thank M. Baake, V. Bergelson, B. Green, D. Kwietniak, C. Mauduit and M. Radziwiłł for some useful comments on the subject.

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Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (2018). Sarnak’s Conjecture: What’s New. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham.

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