Oscillating Sequences, Gowers Norms and Sarnak's Conjecture

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Here, we will apply same classical ingredi- We remind that the Möbius-Liouville ran- ents to establish that the higher order oscil- domness law [34] assert that for any “reason- lating sequences are orthogonal to the wide able" sequence of complex numbers (an) we class of nilsequences and to the affine linear have maps on the Abelian group. Indeed, our proof yields that the higher order oscillating 1 N ∑ λ(n)an 0, (3) sequences are orthogonal to any dynamical se- N −−−−→N +∞ n n=1 → quence ( f (T x)) provided that T has a quasi- discrete spectrum. where λ is the Liouville function given by At this point one may ask if the previous result can be extended to the all nilsequences. 1 if n = 1; Since, as pointed out by W. parry [46], ”the λ(n) = r ( 1) if n is the product of r nilflows and nilmanifold unipotent affines  −  not necessarily distinct prime numbers. should be viewed as models generalizing the models defining quasi-discrete spectra” . Applying Chowla-Batman trick [10], the Liouville function can be replaced in (3) by We answer this question by establishing that the Möbius function µ. We remind that the there is an almost nilsequence which is higher Möbius function is defined by order oscillating. It follows that there is a higher order oscillating sequence with high λ(n) if n is not divisible by Gowers norms. We thus get that the notion of higher order oscillating sequence is not µ(n) = the square of any prime;  adapted to generalize the spirit of the Möbius- 0 if not. Liouville randomness law and Sarnak’s conjecture. In his seminal paper [50], P. Sarnak consider Although, the orthogonality of the higher the Möbius-Liouville randomness law for a order oscillating sequences and the quasi- class of deterministic sequences which arise discrete spectrum is in the spirit of Liu & P. from topological dynamical system with topo- Sarnak’s result, since the Möbius function is logical entropy zero. Precisely, the sequence n higher order oscillating sequence by Hua’s the- (an) is given by an = f (T x), for any n 1, ≥ orem [31]. Of Course, in the particular case of where T is homeomorphism acting on a G = Rd and Γ = Zd, the proof yields that the compact space X with topological entropy oscillating sequence of order d are orthogonal zero, f is a continuous function on X and x a to the standard homogeneous space (Td, T), point in X. where T is an affine map. We thus get the result of Jiang [35]. This is nowadays known as Sarnak’s conjecture. At now, as far as the author is aware, We remind that the dynamical system (X, , µ, T) is said to have a measurable this conjecture was established only for many B particular case of zero topological entropy quasi-discrete spectra if the closed linear dynamical systems (see [51] and the reference subspace spanned by H = n 0 Hn is all L2(X µ) H ≥ therein, see also [6]). , , where 0 is the set ofS the constant complex valued function of modulus 1, and 2 In particular, Liu & P. Sarnak proved that for any n 1, Hn = f L (X, µ) : f = ≥f T ∈ | | Sarnak’s conjecture holds for an affine linear 1 a.e.and ◦ Hn 1 . If for some d 1, f ∈ − ≥ map of nilmanifold [41] by applying a slightly Hd = Hd+1 we say that T has a discrete- strengthen version of Green-tao’s theorem spectrum of order d. [24] combined with a classical result from [15]. This class was defined and studied by L.

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M. Abramov [7]. Subsequently, F. Hahn & W. of all quasi-eigenvectors. By putting Hn = Parry introduced and studied the notion of Λ(Gn+1), we see that the elements of Gn are quasi-discrete spectrum in the topological dy- precisely the unimodular solutions f of the namics for a homeomorphism T of a compact equation Λ( f ) = g, where g Gn 1. The el- ∈ − set X that is assumed to be completely min- ements of the subgroup Hn are called a quasi- imal (that is, all its powers are minimal) [47], eigenvalue of order n 1 and − [48]. The quasi-eigen-functions are assumed to be continuous and separate the points of X. H = Hn, n 0 Therefore, by the Stone-Weierstrass theorem, [≥ the subalgebra generated by the quasi-eigen- is the group of all quasi-eigenvalues. Ob- functions is dense in C(X). Ten years later, R. viously, ι : f G f (x ) T where ∈ 1 7→ 0 ∈ J. Zimmer shows that a totally ergodic system x X is an isomorphism of groups. A triple 0 ∈ (X, , T, µ) has quasi-discrete spectrum if (H, Λ, ι) is called the signature of the dynami- B and only if it is distal and isomorphic to a cal system (X, T). According to Hahn-Parry’s totally ergodic affine transformation on a Theorem [29], if (X, T) is totally minimal compact connected Abelian group (G, S), that topological system with quasi-discrete spec- is, S : G G has the form Sx = x Ax, trum and signature (H, Λ, ι), then (X, T) → 0 where A : G G is an automorphism of is isomorphic to the affine automorphism → the group G and x G [55]. system (H, φ , η) where H is the dual group 0 ∈ ∗ of H, φ(h) = hΛ(h), for h H, and η denotes ∈ Applying some algebraic arguments, one any homomorphicb extensionb of η : H T 1 −→ can define for any n 1 the subgroup Gn = to all of H. ≥ ker(Λn) where λ is the derived homomorphism on the multiplicative group C(X, T) = The popular example of maps with quasi- f C(X) : f = 1 given by λ( f ) = discrete spectrum is given by the following ∈ | | f T transformation of the d-dimensional torus of ◦f = f T. f and G0 = 1 . Therefore ◦ { } the form G = n 1 Gn is an Abelian group and Λ is a quasi-nilpotent≥ homomorphism on it. We re- S T(x1,..., xd)=(x1 + α, x2 + x1,..., xd + xd 1). mind that Λ is called nilpotent if G = Gn for − some n and quasi-nilpotent if G = n 1 Gn. This transformation is an affine transfor- ≥ Notice also that the subspace of the invariantS mation, it can be written as x Ax + b d 7→ continuous functions is the subspace of the where A = [aij]i,j=1 is the matrix defined C constant functions .G0 by minimality. We fur- by a1,1 := 1, ai 1,i = aii := 1, i = 2, , d 2 − ··· ther have by the binomial theorem and all other coefficients equal to zero, and n b := (α,0,...,0). Taking again α irrational, (n) f Tn = ∏ Λj( f ) j (4) (Td, T) is a uniquely ergodic dynamical ◦ j=0 system, and it is totally ergodic with respect to the Haar measure on Td, which is the for each f C(X, T), where the binomial co- ∈ unique invariant measure [21]. More gener- efficients (n) are defined by j ally, H. Hoare and W. Parry established that

n(n 1) (n j+1) if T is a minimal afﬁne transformation of n − ··· − if 0 j n = j! ≤ ≤ a compact connected abelian group X, that j (0 if not is, T(x) = a.A(x), x X, where A is an ∈ automorphism of X and a X then T has ∈ The elements of Gn are called quasi- quasi-discrete spectrum [30]. For a recent eigenvectors of order n 1 and G is the group − exposition and analysis of the subject, we 2There is an analogy between this formula and Hall- refer the reader to [28]. Petresco identity for the nilpotent groups (see [52, p.118].

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Let us further remind that the authors in Table 1: the Möbius-Liouville randomness law [2] proved that Sarnak’s conjecture holds for vs Sarnak’s conjecture. any uniquely ergodic model of a dynamical Z 1,1 , S is a shift map, system with quasi-discrete spectrum. The {− } proof is based on the joining property of the (X, T), htop(T) = 0 Y = O(λ) the powers called Asymptotical Orthogonal x = Tnx y = Snλ ? Powers (AOP). This later property insure n n ⊥ f C(X) and π (y) = y that Katai-Bourgain-Sarnak-Ziegler criterion ∈ 0 0 holds. is the 1th projection, O(λ) is the orbit closure.

We end this section by pointing out that the cn = Xn(ω) is almost surely higher order oscillat- Möbius-Liouville randomness law can be seen ing sequence. as a weaker version of the following notion of independence introduced by Rauzy in [49]. For the proof of Theorem 2, we need the following classical inequalities due to J. Let X, Y be two metric spaces, we say that Marcinkiewicz and A. Zygmund. the sequence (xn) X and (yn) X are ⊂ ⊂ Theorem 4. [42] If Xn, n = 0, N, are indepen- independent if for any continuous functions ··· dent C-valued random variables with mean zero f C(X) and g C(Y) we have and ﬁnite Lp-norm, p 1. Then ∈ ∈ ≥ 1 N 1 N N N − f (x )g(y ) ∑ n n Ap ∑ Xj ∑ Xj Bp ∑ Xj , N n=0 2 ≤ p ≤ 2 n=1 n=1 n=1 N 1 N 1 1 − 1 − ∑ f (xn) ∑ g(yn) 0. for positive constants Ap and Bp depending only − N N −−−−→N +∞ n=0 n=0 → on p.

the main results Marcinkiewicz-Zygmund inequalities gen- II. eralize the well-known Khintchine inequalities which assert that the Lp-norms are equivalent We start by stating our ﬁrst main result. for the Rademacher variables. The proof Theorem 1. There exist a dynamical system given in [42] is in French language. For the (X, T) with topological entropy zero and a higher more recent proof and for its extension to the order oscillating sequence which is not orthogonal martingale setting, we refer to [23, Chap. 3. to (X, T) p.73] and [14, Chap.11. p.412].

Our second main result is the following We are now able to proof Theorem 2.

Theorem 2. Let (Xn) be any sequence of indepen- Proof of Theorem 2. Let us assume, without E dent random variables (Xj) such that (Xj) = 0 2 any loss of generality that supn 0 E( Xn ) E p < ∞ > ≥ | | ≤ and supj 0 ( Xj ) + , for some p 2. 1. Then, by our assumption combined with ≥ | | Then the sequence cn = Xn(ω) is almost surely Marcinkiewicz-Zygmund inequalities it fol- higher order oscillating sequence. lows that for any p > 1,

It follows that if Xn is a subnormal random 1 N 1 p 1 2 E − 2πiP(n) λ ∑ Xne Bp p . E λ.Xn 2 variable for each n 1, that is, (e ) e , N ≤ N 2 ≥ ≤ n=0 for any λ R. Then, we have ∈ Hence, for p > 2, we have Corollary 3 ([18]). Let (Xn) be any sequence N 1 p of independent random variables such that Xn is 1 − 2πiP(n) E ∑ ∑ Xne < ∞. subnormal for each n 1. Then, the sequence N 1 N n=0 ≥ ≥

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A standard argument yields the desired prop- The proof of (5) follows as a consequence of erty. the fact that the Koksma general metric criterion is valid for the short interval. Indeed, ap- The proof of Corollary 3 follows from the plying Cauchy-Schwarz inequality it suffices following well-know fact. to establish the following key inequality. For Fact 5. Let X be a random variable such that any 1 < a < b, E(X) = 0. Then the following are equivalent b 2 1 2πiℓ f (h) 1. There exist c > 0 such that for any λ 0, ∑ e α,β dβ ≥ a H < P X λ 2exp( λ2c). Z m h m+H | | ≥ ≤ − ≤ b a log(3H) 2. There exist c > 0 such that for any p 1, − + C. , ′ ≥ ≤ H ℓ .H X c √p. | | p ≤ ′ for same absolute constant C > 0. 3. There exist c > 0 such that for any t R, ′′ ∈ E(exp(tX)) exp c .t2/2 . ≤ ′′ This inequality is valid by the same argu- Remark 6. Applying Koksma classical re- ments as in the proof of Theorem 4.3 in [38]. sult combined with the well-know criterion of van der Corput, S. Akiyama & Y. Jiang established Before stating our third main result, let us in [8] that for any positive real valued 2-times remind the notion of nilsequences and some continuously differentiable function g on (1, +∞) tools. (that is, g(x) > 0, g (x), g (x) 0), for any ′ ′′ ≥ α R , for almost all real numbers β > 1 and ∈ ∗ I. Nilsequences and nilsystems for any real polynomials Q, the sequences A sequence (b ) is said to be a k-step basic f (n) = αβng(β) + Q(n) n α,β nilsequence if there is a nilpotent Lie group is higher order oscillating. Let us notice that the G of order k and a discrete co-compact n proof of the previous result can be obtained in the subgroup H of G such that bn = F(Tg xΓ) same spirit as the proof of Theorem 2 by the general where T : x.Γ G/H (gx).Γ, g G, g ∈ 7→ ∈ principle stated in [38, Theorem 4.2,p.33]. F is continuous function on X = G/H. The homogeneous space X = G/Γ equipped Applying further the Koksma general met- with the Haar measure h and the canonical ric criterion (Theorem 4.3 in [38]), one can eas- X complete σ-algebra c. the dynamical system ily seen for any α R∗, for almost all real B ∈ (X, c, h , Tg) is called a k-step nilsystem and numbers β > 1 and for any real polynomials B X X is a k-step nilmanifold. For a nice account Q, for any ℓ = 0, we have 6 of the theory of the homogeneous space we 1 1 ℓ refer the reader to [15]. Let us notice that ∑ ∑ e2πi f α,β(h) M H < any affine linear map on X has zero entropy M m 2M m h m+H ≤ ≤ ≤ if and only if if it is quasi-unipotent. So 0. (5) −−−−−−−−−−→H +∞, H/M 0 we assume here that the affine linear maps → → are quasi-unipotent. We further assume that or, equivalently [1], for any α R∗, for al- ∈ G most all real numbers β > 1, for any real poly- is connected and simply-connected by Leibman’s arguments [40]. If X is Abelian we nomials Q and for any increasing sequence (b ) of integers 0 = b < b < b < with say that the nilsequence n is an Abelian 0 1 2 ··· b b ∞, for almost all β > 1, for any nilsequence. k+1 − k → ℓ = 0, we have 6 Let us further point out that (bn) is any 1 πiℓ f (n) ℓ∞ Z ∑ ∑ e2 α,β 0. (6) element of ( ), the space of bounded bK < < −−−→K ∞ sequences, equipped with uniform norm k K bk n bk+1 → ≤

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d (an) ∞ = sup an .A k-step nilsequence, is and for any x T put Tx = Ax + b, k k n Z | | ∈ ∈ where b = (bd, , b1) and the matrix A = a uniform limit of basic k-step nilsequences. d ··· A[i, j] is defined by A[1,1] := 1, A[i We remind that X is said to be a k-step i,j=1 − 1, i] = A[i, i] := 1, for i = 2, , d and all other nilmanifold. ··· coefficients equal to zero. Then, a straightfor- Let G be a nilpotent Lie group with a co- ward computation by induction on n yields compact lattice Γ and a Γ-rational filtration G • n 2πiP(n) of length l, so that Γ = Γ G is a cocompact χd(T (x0)) = e , i ∩ i lattice in Gi for each i = 1, , l. 2πix ··· where x0 =(0, ,0, b0) and χd(x) = e d. We remind that the sequence of subgroups ··· (G ) of G is a filtration if G = G, G G , n 1 n+1 ⊂ n Furthermore, by Bergelson-Leibman’s ar- and [G , G ] G + , where [G , G ] denotes n p ⊂ n p n p gument [40], the map T can be viewed as the subgroup of G generated by the commu- a nilrotation. Indeed, Let G be the group 1 1 tators [x, y] = xyx y with x Gn and d+1 − − ∈ of upper trianglar matrix T = T[i, j] y Gp. The lower central filtration is given i,j=1 ∈ with T[i, i] = 1, i = 1, , d + 1, T[i, j] Z, by G1 = G and Gn+1 = [G, Gn]. It is well ··· ∈ 1 i < j d and T[i, d + 1] R, for know that the lower central filtration allows ≤ ≤ ∈ i = 1, , d. Consider Γ the subgroup of to construct a Lie algebra gr(G) over the ··· ring Z of integers. gr(G) is called a graded G consisting of the matrices with integer Lie algebra associated to G [12, p.38]. The entries. Then G is a nilpotent non-connected Tk filtration is said to be of length l if G = e , Lie group with X = G/Γ , and define l+1 { } ≃ where e is the identity of G. the nilrotation Tg on X by Tg(x) = gx where g[i, i] = 1, for i = 1, , d + 1, g[i 1, i] = 1, ··· − i = 2, , d, g[j, d + 1] = b , for j = 1, , d An example of k-nilsequence is given ··· j ··· by a continuous function F which satisfy and all other coefficients equal to zero. We thus get that the nilrotation Tg is isomorphic F(gly) = χ(glΓl)F(y), for any y G, gl Gl ∈ ∈ to the skew product T defined on Td. and χ G\/Γ where G\/Γ is the dual group ∈ l l l l of the Abelian group G /Γ . The function F is l l We can thus consider the dynamical se- called a vertical nilcharater. It follows that the n 2 quence F(T x), where F is a continuous Hilbert space L (X, hX) can be decomposed and T is a skew product on the d-torus as a into a sum of G-invariant orthogonal Hilbert d-nilsequence up to isomorphism. subspaces. Let us also point out that the quasi-unipotent case can be reduced to the Following [25], the 1-bounded sequence unipotent case, and for more details on the (a(n)) is said to be an almost nilsequence of Fourier analysis theory on the nilspaces we degree s with complexity O (1), where M > refer to [52]. M 1 is a given complexity parameter, if for any ε > 0 there is a nilsequence (aε(n)) with complexity Os,ε,M(1) such that A. Furstenberg’s argument and Bergel- son-Leibman’s observation. Acoording to 1 N ∑ a(n) aε(n) < ε. Furstenberg’s argument [22, p.23], if P(n) N − ∈ n=1 R [X], d 1, then the sequence e2πiP(n) is a d ≥ dynamical sequence. Indeed, write Roughly speaking, the L1 closure of the space of the nilsequences of degree s is the space of P(n) = a + a n + a n2 + + a nd the almost nilsequences of degree s. In [25], 0 1 2 ··· d n n the authors gives various examples of almost = b + b n + b + + b , 0 1 2 2 ··· d d nilsequences of degree s 3. ≤

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Applying the fundamental tools of this the- This achieve the proof of the theorem. ory combined with some ingredients and results due to A. Liebman, V. Bergelson & A. At this point let us mention the result of A. Liebman [40], we can establish the following Fan [19] and gives its proof. Theorem 7. There exist a higher order oscillating Theorem 11 ([19]). The sequence (cn) of os- sequence wich is an almost nilsequence. cillating order d = t.s, t, s N is or- ∈ ∗ However, by Lemma 3.4 from [25], it is easy thogonal to any dynamical sequence of the form to check that the oscillating sequences of order F Tq1(n)x, , Tqk(n)x , where T is a homeomor- ··· 1 are orthogonal to the almost nilsequence of phic map on a compact set X with quasi-discret degree 1. Moreover, we can easily check the spectrum of order t, F is a continuous function on following. Xk and q (n), i = 1, , k are a polynomials of i ··· degeree at most s. Theorem 8. The sequence (cn) of oscillating order d is orthogonal to any nilsequence of order d aris- Proof. By density argument it is sufﬁces to ing from skew product on the d-dimensional torus check the orthogonality for the function F of Td. the form Proof. A straightforward by Furstenberg’s argument (see subsection A.). F(x , x , , x ) = f (x ). f (x ) f (x ), 1 2 ··· k 1 1 2 2 ··· k k We thus get, by applying the classical den- where f , i = 1, , k are a eigen-functions. In i ··· sity argument, the following the same manner as before, we apply (4) to get

Corollary 9. The sequence (cn) of oscillating or- N 1 der d is orthogonal to any affine transformation on 1 − c F Tq1(n)x, , Tqk(n)x Td ∑ n the d-dimensional torus . N n=0 ··· N 1 Applying the same reasoning, we have 1 − 2πiQ(n) = F(x, , x) ∑ cne , N Theorem 10. The sequence (cn) of oscillating or- ··· n=0 der d is orthogonal to any quasi-discret system of order d. where Q(n) is a polynomials with at most d degree. We thus conclude that Proof. By the density argument, it suffices to establish the orthogonality for a functions f N 1 ∈ 1 − q (n) q (n) G , k 0. Let f G G , then, by (4), for ∑ cnF T 1 x, , T k x 0. k ≥ ∈ k+1 \ k N ··· −−−−→N +∞ n k, x X, we have n=1 → ≥ ∈ k n ( j ) The proof of the theorem is complete. f (Tnx) = ∏ Λj f (x) j=0 The poof of (5) can be adapted to obtain the k (n) j following = ∏ Λj f (x) j=0 Theorem 12. For any dynamical flow (X, T), for ( ) = f (x)e2πiP n , any continuous function f , for any x X, for any > ∈ k α = 0 and for almost all β 1, we have n R 6 Where P(n) = ∑ θj and θj is such j ∈ 1 1 ( ) j=1 ∑ ∑ f (Thx)e2πi fα,β h j 2πiθj that Λ f (x) = e . Therefore, by (2), it M M m 2M H m h

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Corollary 13. For any nilsequence (b ) C, for Matomäki, Radziwiłł and Tao [43]: for any in- n ⊂ any α = 0 and for almost all β > 1, we have teger N, L 10 , for any ǫ > 0, we have 6 ≥ 1 1 N L 2πi f α,β(h) 1 1 ∑ ∑ bhe sup ∑ ∑ µ(l + n)e2πiβl M H < M m 2M m h m+H β T N n=1 L l=1 ≤ ≤ ≤ ∈ 0. −−−−−−−−−−→H +∞, H/M 0 1 log(log(L)) → → Cǫ ǫ + , ≤ log(N) log(L) Notice that the tools applied here allows one to obtain a generalization of a results where Cǫ is an absolutely constant which of [18], [35] and [33] to the case of quasi- depend only on ǫ. discrete spectrum. Let us notice further that in [4] the authors established a criterion of Here in the same spirit as in [3] we establish Möbius disjointness for uniquely ergodic sys- the following tems. This criterion yields that Sarnak’s conjecture holds for any topological model for a class of uniquely ergodic dynamical systems Theorem 14. For any integers N, L 2, k 1 > ≥ ≥ including quasi-unipotent nilsystems. As in and for any ǫ 0, we have [33], the fondamental ingredients is based on N L 1 1 k the short intervall orthogonality of the Möbius sup ∑ ∑ µ(l + n)e2πiβl β T N n=1 L l=1 function to the rotation on the circle. This ∈ later result extends Theorem of Davenport- Cǫ 1 Hua [16], [31, Theorem 10.] (see also [41]) ǫ L + N , ≤ log(L + N) r NL which say that, for any k 1, for any ǫ > 0, ≥ where Cǫ is an absolutely constant which depend N 1 k C sup ∑ µ(n)e2πin θ ǫ . only on ǫ. N ≤ log(N)ǫ θ n=1

For the recent proof of the short intervall Notice that in the particular case N = L we orthogonality of the Möbius function to the get rotation on the circle, we refer to [32] and N L 1 1 k the references therein. Let us notice that sup ∑ ∑ µ(l + n)e2πiβl in the spirit of Davenport-hua’s theorem, β T N n=1 L l=1 ∈ E. H. El Abdalaoui and X. Ye proved in [3] Cǫ 2 L 2.Cǫ that for any For any integer N 2 and for ǫ ǫ ≥ ≤ log(L) L ≤ log(2L) any ǫ > 0, we have where Cǫ is an absolutely constant which 1 N 1 N sup ∑ ∑ µ(l + n)a depend only on ǫ. The proof of Theorem 14 is N N l ℓ∞ n=1 l=1 based on Bourgain’s observation [13] and the a , a ∞ 1 ∈ ≤ classical Bessel-Parseval inequality.

Cǫ ǫ , ≤ log(2N) Before given the proof of Theorem 14, let us present the proof of Theorem 7 . where Cǫ is an absolutely constant which depend only on ǫ. The proof is based on Proof of Theorem 7. Let α and β be a ratio- Bourgain’s observation and Parseval-Bessel nally independent irrational numbers and con- inequality. sider the sequence u(n) = nα[nβ] mod 1. Then, by Lemma 3.6 from [25], (u(n)) is an al- We remind that the authors in [33] used es- most nilsequence. Furthermore, by Bergelson- sentially the the following estimation due to Leibman’s result [40], for any polynomial P ∈

8 Oscillating sequences and Gowers norms November 2016 •

R[x], we have Whence, once again by Cauchy-Shwarz inequality, we have 1 N ∑ exp 2πi u(n) P(n) 0. N − −−−−→N +∞ N L n=1 → 1 1 2πiβlk ∑ ∑ µ(l + n)e bn Therefore the sequence (u(n)) mod 1 is a N L n=1 l=1 higher oscillating sequence. The proof of the L N 1 1 1 2 2 ∑ ∑ µ(l + n)bn . theorem is complete. ≤ L N l=1 n=1 Applying the same method as in [40], one can exhibit more large class of almost We further have nilsequences which are a higher oscillating L 1 N 2 sequences. ∑ ∑ µ(l + n)bn N l=1 n=1 L L+N N 2 Proof of Theorem 14. Let N, L 2, k 1 1 m n l > ≥ ≥ ∑ ∑ µ(m)z ∑ bnz− z− dz and ǫ 0. Then, by Cauchy-Schwarz inequal- ≤ T N l=1 Z m=1 n=1 ity, we have

N L Consequently, by Bessel-Parseval inequality, 1 1 k ∑ ∑ µ(l + n)e2πiβl we obtain N n=1 L l=1 L N N L 1 1 2 1 1 k 2 2 µ(l + n)b ∑ ∑ µ(l + n)e2πiβl . ∑ ∑ n l=1 N n=1 ≤ N n=1 L l=1 L+N 2 N 2 1 m n Therefore it sufﬁces to estimate the RHS. For ∑ µ(m)z ∑ bnz− dz ≤ N that, observe that by the classical Hilbert space Z m=1 n=1 analysis trick we have L+N 2 N 1 m 2 sup ∑ µ(m)z . ∑ bn N ≤ z T N m=1 n=1 | | 1 ∈ sup ∑ anbn = a 2, L+N 2 b =1 N n=1 k k 1 m k k2 sup ∑ µ(m)z .N

N N ≤ z T N m=1 where b = (b ) , a = (a ) , and . is ∈ n n=1 n n=1 k k2 the classical euclidienne norm on CN. Let b ∈ Furthermore, by applying Davenport-Hua the- CN such that b = 1. We thus need only to k k2 orem, we get esitmate the following L N N L 1 2 1 1 2πiβlk ∑ ∑ µ(l + n)bn ∑ ∑ µ(l + n)e bn . l=1 N n=1 N n=1 L l=1 2 2 Cǫ L + N But . .N, ≤ 2ǫ N N L log(L + N) 1 1 2πiβlk ∑ ∑ µ(l + n)e bn N L where Cǫ is an absolutely constant which de- n=1 l=1 L N pend only on ǫ. We thus conclude that we 1 1 2πiβlk = ∑ ∑ µ(l + n)bn .e , have L l=1 N n=1 N L 1 1 k and by the triangle inequality sup ∑ ∑ µ(l + n)e2πiβl β T N n=1 L l=1 N L ∈ 1 1 2πiβlk ∑ ∑ µ(l + n)e bn N L Cǫ 1 n=1 l=1 ǫ . L + N . . ≤ log(L + N) NL L N r 1 1 ∑ ∑ µ(l + n)bn ≤ L N This ﬁnish the proof of the Theorem. l=1 n=1

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Remark. Let us notice that our proof yields Proposition 15. (Cauchy-Bunyakovskii-Gowers- that for any integer N, L 2 and for any ǫ > 0, Schwarz inequality) ≥ we have

fc ∏ fc d . N L Ud(G) ≤ U (G) 1 1 c Cq sup ∑ ∑ µ(l + n)al D E ∈ N L ℓ∞ n=1 l=1 a , a ∞ 1 The proof of Cauchy-Bunyakovskii-Gowers- ∈ ≤ Schwarz inequality can be obtain easily by Cǫ 1 applying inductively Cauchy-Bunyakovskii- ǫ . L + N . , ≤ log(L + N) r NL Schwarz inequality. Indeed, it easy to check that we have where Cǫ is an absolutely constant which 1 depend only on ǫ. 2 fc ∏ fπ (c) , Ud(G) ≤ h i,j Ud(G) D E j=0,1 E For all i = 1, , d 1, where π (c) C is II. Gowers norms. ··· − i,j ∈ d formed from c by replacing the ith coordinates The notion of nilsequences is closely related with j. Iterated this, we obtain the complete to the notion of Gowers norms. These proof of Proposition 15. norms were introduced by T. Gowers in [27] . Therein T. Gowers produced a new proof Combining Cauchy-Bunyakovskii-Gowers- of Szemerédi’s theorem. Nowadays, it is Schwarz inequality with the binomial formula turn out that Gowers uniform norms are and the multilinearity of the Gowers inner tools of great use in additive number theory, product one can easily check that the triangle arithmetic combinatorics and ergodic theory. inequality for . Ud(G) holds. We further have d Let d 1 and Cd = 0,1 . Let (G, +) be f f , ≥ { } Ud(G) Ud+1(G) a local compact Abelian group. If h Gd and ≤ d ∈ c Cd, then c.h = ∑i=1 cihi. Let fc be by applying Cauchy-Bunyakovskii-Gowers- ∈ c Cd ∈ Schwarz inequality with f = 1 if c = 0 and a family of bounded functions that are com- c d pactly supported, that is, for each c Cd, fc fc = 1 if cd = 1. The Gowers norms are also ∞ ∈ is in Lc (G) the subspace of functions that are invariant under the shift and conjugacy. compactly supported. The Gowers inner product is given by If G is a finite discrete Abelian group, we define the discrete derivative of function f : c G C by putting fc = ∏ | | fc(g + c.h)dhdg, d d+1 → U (G) ZG c C C D E ∈ d ∂h( f ) = f (x + h). f (x), where c = c.1, 1 =(1,1, ,1) C and is | | ··· ∈ d C for all h, x G. We can thus write the Gowers the conjugacy anti-linear operator. If all fc are ∈ norm of f as follows the same function f then the Gowers uniform norms of f is defined by 2d f = ∂ ∂ ∂ ( f )(x)dhdx. d( ) h1 h2 hd U G Gd+1 ··· 2d Z f d = f . U (G) Ud(G) If further f take values on R/Z D E and ∂ ∂ ∂ ( f )(x) = 0, for all h1 h2 ··· hd+1 The fact that . d is a norm for d 2 h , , h , x G, then f is said to be U (G) ≥ 1 ··· d+1 ∈ follows from the following generalization of a polynomial function of degree at most d. Cauchy-Bunyakovski-Schwarz inequality for The degree of f is denoted by d◦( f ). the Gowers inner product.

10 Oscillating sequences and Gowers norms November 2016 •

According to this it is easy to see that for Theorem 16 ([39]). The Gowers uniform norm of any function f and any polynomial function φ Thue-Morse sequence t = (tn) ( respectively the of degree at most d, we have Rudin-Shapiro sequence r = (rn)) satisfy: for any d N∗, there exists c = c(d) > 0 such that e2πiφ(x) f (x) = f . ∈ c Ud(G) Ud(G) t Ud[N] = O(N− ) ( respectively r Ud[N] = c O ( N− )). Therefore

2πiφ(x) We thus have sup e f (x)dx f Ud(G). (7) φ, d (φ) d ZG ≤ ◦ ≤ Corollary 17. The Thue-Morse and Rudin-

In application and here we need to define Shapiro sequences are a higher order oscillating se- the Gowers norms for a bounded function quences. defined on 1, , N . For that, if f is a ··· We are now able to outline the proof of our bounded function defined on 1, , N 1 , ··· − main result 1. Following [26], we put Consider the dynamical system generated f Ud(Z/2d.NZ) f = , by the Thue-Morse sequence or Rudin-Shapiro Ud[N] I [N ] Ud(Z/2d.NZ) sequence and denote it respectively by (X , S ) e t t and (Xr, Sr). Therefore, obviously, there is a I Z d Z I where f = f (x). [N], x /2 .N , [N] is the continuous function f C(X ) and a point ∈ ∈ i indicator function of 1, , N . For more = ··· y Xi, i t, r such that detailse on Gowers norms, we refer to [52],[53]. ∈ N 1 n > lim ∑ f (Si y)i(n) 0, for i = t, r. The sequence f is said to have a small N +∞ N Gowers norms if for any d 1, −→ n=1 ≥ Notice that Sarnak’s conjecture holds for f d 0. U [N] −−−−→N +∞ the Thue-Morse sequence and Rudin-Shapiro → sequence [54], [2], [20],[56] . An example of sequences of small Gowers norms that we shall need here is given by We end this paper by pointing out Thue-Morse and Rudin-Shapiro sequences. that, by applying inductively the standard This result is due J. Konieczny [39]. van der corput lemma [38, p.25] or (7), on can easily obtain the following. The Thue-Morse and Rudin-Shapiro sequences are a classical sequences both arise Theorem 18. Assume that the Gowers norm of from a primitive substitution dynamical sys- order k of the bounded sequence (cn) is zero. Then, tems. The Thue-Morse and Rudin-Shapiro se- (cn) is an oscillating sequence of order k. quences are defined respectively by This allows us to ask the following question. t(n) = e2πis2(n), Question 1 ([3]). Do we have that for any multi- and plicative function (a(n)) with small Gowers norm, r(n)=( 1)u11(n), − for any dynamical flow on a compact set (X, T) where s2(n) is the sum of digits of n in base 2 with topological entropy zero, for any continuous and u (n) is the number of “11” in the 2-adic function, for all x X, 11 ∈ representation of n. N 1 n ∑ an f (T x) 0? Precisely, J. Konieczny proved the following N −−−−→N +∞ n=1 →

11 Oscillating sequences and Gowers norms November 2016 •

Notice that by our assumption, it is obvious SFT. But again by D. Karagulyan’s result [37] that for any nilsystem (X, T), for any continu- the Möbius function is not orthogonal to the ous function f , for any x X, we have soﬁc symbolic dynamical systems. However, ∈ this does not answer our question. N 1 n ∑ an f (T x) 0. N −−−−→N +∞ n=1 → Finally, in a very recent work under progress, P. Kurlberg and the author proved Let us further point out that by the recent re- that the answer to the question 2 is nega- sult of L. Matthiesen [44] there is a class of not tive. They further established that the Möbius- necessarily bounded multiplicative functions Liouville randomness law does not hold for with a small Gowers norms. any expansive map [5]. Precisely, P. Kurlberg Question 2 (β-shift, p maps and the and the author proved that for given an inte- × ger b 2, there exists x [0,1) and c > 0 Möbius-Liouville randomness law.). One may ≥ ∈ ask further on the validity of the Möbius-Liouville such that randomness law for the map T x = px mod 1, p p ∑ λ(n) sin(2πbnx) c N prime and for the beta-shift, that is, do we have n N ≥ · ≤ N 1 n ⋆ for all sufﬁciently large N. ∑ µ(n) f (Tp x) 0, ( ) N −−−−→N +∞ n=1 → This later result combined with Bourgain’s result [50] on the existence of dynamical system N 1 n > ⋆⋆ for which the Möbius-Liouville randomness ∑ µ(n) f (Tβ x) 0, β 1 ( ) N −−−−→N +∞ law hold (see also [17]) allows us to ask the n=1 → following question. for f a continuous function and x [0,1)? ∈ Question 3. For any ǫ > 0, can one construct Let us notice that if we consider the dou- a dynamical system with topological entropy ǫ for bling map x 2x. Then, it is easy to see 7→ which the Möbius-Liouville randomness law hold. that (⋆) holds for a dense periodic points. We furhter notice that the general case is related Acknowledgment. The author wishes to ex- to the problem of mutiplicative order function press his thanks to François Parreau for many stim- ℓ (n). Whence, according to the main result in ulating conversations on the subject and his sus- [9], for almost all integers q, for any p, we have tained interest and encouragement. He further wishes to express his thanks to Wen Huang for 1 N ∑ µ(n) exp 2πi2n pq 0. many interesting email exchanges on the subject N −−−−→N +∞ n=1 → related to this work. He would like also to thanks Igor Shparlinski and Ping Xi for their interest and For a nice account on the mutiplicative order valuable comments. problem we refer to [45].

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