Comptes Rendus Mathématique Volume 355, n° 8 pages 866870 (août 2017)
Doi : 10.1016/j.crma.2017.07.008
Received : 2 January 2017 ;
accepted : 10 July 2017    Fully oscillating sequences and weighted multiple ergodic limit Suites pleinement oscillantes et limite des moyennes multiergodiques pondérées  
Aihua Fan ^{a, }^{b}
^{ a} LAMFA, UMR 7352 CNRS, Université de Picardie, 33, rue SaintLeu, 80039 Amiens, France ^{ b} School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, China
We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact Abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems. The full text of this article is available in PDF format.Nous montrons que les suites pleinement oscillantes sont orthogonales aux réalisations d'une application affine d'entropie nulle sur un groupe abélien compact. Ceci est plus que ce que demande la conjecture de Sarnak à ces systèmes dynamiques. The full text of this article is available in PDF format.   A sequence of complex numbers is said to be oscillating of order d ( ) if for any real polynomial of degree less than or equal to d , we have (1)limN→∞1N∑n=0N−1cne2πiP(n)=0. It is said to be fully oscillating if it is oscillating of all orders. This notion of oscillation of higher order was introduced in [[7]]. The oscillation of order 1 was earlier considered in [[8]] in order to formulate some results related to Sarnak's conjecture. See [[19]], [[20]] for Sarnak's conjecture. See [[1]], [[2]], [[6]], [[9]], [[14]], [[17]], [[15]], [[23]] for some related recent works. The Möbius sequence is a typical example of fully oscillating sequences ([[5]], [[13]]). Recall that , if n is square free and has k distinct prime factors, and for other integers n . The random subnormal sequence is almost surely fully oscillating ([[7]]) and the sequence is fully oscillating for almost all ([[3]]). The oscillating sequences of orders d are characterized by their orthogonality to different classes of dynamical systems ([[21]]). Sarnak's conjecture states that for any topological dynamical system of zero entropy, for any continuous function and any point , we have limN→∞1N∑n=1Nμ(n)f(Tnx)=0. This conjecture remains open in its generality. The above equality is referred to as the orthogonality of Möbius sequence to the realization of the system , or as the disjointness of to the system . For functions , the sequence could be referred to as a multiple ergodic realization. Following Liu and Sarnak [[18]], we can prove the following orthogonality of fully oscillating sequences to the multiple ergodic realizations of affine linear maps on a compact Abelian group that are of zero entropy.
Let be an integer. Let G be a compact Abelian group. Assume that (i) is an affine linear map of zero entropy; (ii) is a fully oscillating sequence; (iii) are ℓ polynomials such that for all j. Then, for any continuous function and any point , we have (2)limN→∞1N∑n=0N−1cnF(Tq1(n)x,⋯,Tqℓ(n)x)=0. Recall that an affine map on a compact Abelian group G is defined by Tx=Ax+b where is an automorphism of G and . Theorem 1 generalizes the following theorem due to Liu and Sarnak, which holds for the Möbius sequence to fully oscillating sequences.
The Möbius sequence is linearly disjoint from any affine linear map on a compact Abelian group that is of zero entropy. The result of Theorem 1 was proved in [[7]], based on [[10]], [[11]], [[12]], for the class of topological systems of quasidiscrete spectrum in the sense of Hahn–Parry [[10]], including minimal affine linear maps on a connected compact Abelian group. The proof of Theorem 1 in this note will be based on ideas of Liu and Sarnak and on the fact that arithmetic subsequences of oscillating sequences are oscillating. One of the ideas of Liu and Sarnak is stated as follows. It is drawn from the proof of their first theorem in [[18]].
Let be an affine linear map of zero entropy on where and F is a finite Abelian group. Consider the automorphism W on the product group defined by . Then (i) for all ; (ii) there exist integers and such that where N is nilpotent in the sense . The following fact, which has its own interest, will also be useful in the proof of Theorem 1.
Let be an integer. A sequence is oscillating of order d if and only if the arithmetic subsequence is oscillating of order d for any integer . Before proving Theorem 1, we prove Theorem 4.
  If are oscillating of order d for , it is obvious that is oscillating of order d . Now assume that is oscillating of order d . First observe that from the definition, it is clear that any shifted sequence ( ) is oscillating of order d . So, it suffices to prove that is oscillating of order d . Since a can be decomposed into a product of primes, we have only to prove that is oscillating of order d for any prime . Let . For any , denote SN=∑0≤n<Nwne2πiP(n),SN,j=∑m:0≤pm+j<Nwpm+je2πiP(pm+j)(0≤j<p). We have the trivial decomposition(3)SN=SN,0+SN,1+⋯+SN,p−1. For any integer , denoteSNu=∑0≤n<Nwne2πi(P(n)+unp) where is a real polynomial. Write . We have the following decomposition(4)SNu=SN,0+ωuSN,1+ω2uSN,2⋯+ω(p−1)uSN,p−1, which is similar to ((3)) and which contains ((3)) as a particular case corresponding to . Taking sum over u , we get∑u=0p−1SNu=pSN,0+SN,1∑u=0p−1ωu+SN,2∑u=0p−1ω2u+⋯+SN,p−1∑u=0p−1ω(p−1)u. Since p is prime, any j with is invertible in the ring so that for all . ThusSN,0=1p∑u=1p−1SNu. Since the sequence is oscillating of order d , we have for all u as , so thatlimN→∞1N∑m=0[N/p]wpme2πiP(pm)=limN→∞SN,0N=0. This implies that is oscillating of order d .
  Let be the dual group of G . We have . Any continuous function can be uniformly approximated by trigonometric polynomials on , which are finite linear combinations of functions of the form where . So, it suffices to prove that (5)limN→∞1N∑n=0N−1cnϕ1(Tq1(n)x)⋯ϕℓ(Tqℓ(n)x)=0 holds for all and all (see [[7]] for details). Now, we mimic Liu and Sarnak [[18]]. First remark that the problem can be reduced to a torus. In fact, let . Recall that the action of A on is defined by for and . Let be the smallest A invariant closed subgroup of which contains Φ and 〈Φ〉⊥:={x∈G:γ(x)=1∀γ∈Gˆ} be the annihilator of , a closed subgroup of G . LetGΦ:=G/〈Φ〉⊥ be the quotient group. The A invariance of implies that∀x∈G,∀y∈〈Φ〉⊥,T(x+y)=Txmod〈Φ〉⊥. Thus T induces an affine map, which will be denoted by , on the quotient group . Being a factor of , the system has zero entropy. By Aoki's Theorem ([[4]], a statement in the proof on p. 13), is finitely generated. Remark that . Then , as dual group of , is isomorphic to for some and some finite Abelian group F . On the other hand, for any and any , we haveϕ(Tnx)=ϕ˜(TΦnx˜) where is the projection of x onto and is the character in induced by ϕ . Thus the proof of ((5)) is reduced to the dynamics . By Theorem 3, it suffices to treat the automorphism W appearing in Theorem 3. Recall that with . For any integer , write with and . The following expression was obtained in [[18]] (6)Wmν+rx=∑j=0min(m,κ)(mj)Njyr,j=∑j=0κ(mj)Njyr,j when , where . Notice that these with and are independent of m . For any polynomial q (a typical polynomial of ), we have where ( ) is independent of m too, and is a polynomial having the same degree as q . It follows from ((6)) that(7)Wq(mν+r)x=Wq′(m)ν+r′x=∑j=0κ(q′(m)j)Njyr′,j. This holds for m sufficiently large so that . Apply ((7)) to each ( ). Thenϕs(Wqs(mν+r))=∏j=0κϕs(Njyrs′,j)(qs′(m)j) Let be the argument of the complex number . Then we get∏s=1ℓϕs(Wqs(mν+r))=e2πiP(m) where is the real polynomialP(x)=∑s=1ℓ∑j=0κts,r,j(qs′(m)j). By Theorem 4, for any , we getlimM→∞1M∑m=1Mwmν+r∏s=1ℓϕs(Wqs(mν+r)x)=limM→∞1M∑m=1Mwmν+re2πiP(m)=0. FinallylimN→∞1N∑n=0N−1wn∏s=1ℓϕs(Wqs(n)x)=0. Addendum. The oscillation of order d is strongly related to the control of the th Gowers uniformity norm (see [[22]], [[16]]). We can use Gowers uniformity norms to study oscillating properties. This work is partially supported by National Natural Science Foundation of China (NSFS 11471132). The author, supported by Knuth and Alice Wallenberg Foundation, visited Lund University in the autumn 2016 where the work was done.    
 El Abdalaoui E.H., Lemańczyk M., de la Rue T. Automorphisms with quasidiscrete spectrum, multiplicative functions and average orthogonality along short intervals Int. Math. Res. Not. 2017 ; 2017 (14) : 43504368    El Abdalaoui E.H., Kasjan S., Lemańczyk M. 0–1 sequences of the Thue–Morse type and Sarnak's conjecture Proc. Amer. Math. Soc. 2016 ; 144 (1) : 161176    S. Akiyama, Y.P. Jiang, Higher order oscillation and uniform distribution, preprint.    Aoki N. Topological entropy of distal affine transformations on compact abelian groups J. Math. Soc. Jpn. 1971 ; 23 : 1117 [crossref]    Davenport H. On some infinite series involving arithmetical functions (II) Q. J. Math. 1937 ; 8 : 313320 [crossref]    Downarowicz T., Glasner E. Isomorphic extensions and applications Topol. Methods Nonlinear Anal. 2016 ; 48 (1) : 321338    A.H. Fan, Oscillating sequences of higher orders and topological systems of quasidiscrete spectrum, preprint.    Fan A.H., Jiang Y.P. Oscillating sequences, minimal mean attractability and minimal meanLyapunov stabilityErgod. Theory Dyn. Syst. doi:.
http://dx.doi.org/10.1017/etds.2016.121    S. Ferenzi, J. KulagaPrzymus, M. Lemańczyk, C. Mauduit, Substitutions and Möbius disjointness, preprint, 2015.    Hahn F., Parry W. Minimal dynamical systems with quasidiscrete spectrum J. Lond. Math. Soc. 1965 ; 40 : 309323 [crossref]    Hoare H., Parry W. Affine transformations with quasidiscrete spectrum (I) J. Lond. Math. Soc. 1966 ; 41 : 8896 [crossref]    Hoare H., Parry W. Affine transformations with quasidiscrete spectrum (II) J. Lond. Math. Soc. 1966 ; 41 : 529530 [crossref]    Hua L.G. Additive Theory of Prime Numbers Providence, RI, USA: American Mathematical Society (1966).    W. Huang, Z. Lian, S. Shao, X. Ye, Sequences from zero entropy noncommutative toral automorphisms and Sarnak conjecture, preprint.    W. Huang, Z.R. Wang, G.H. Zhang, Möbius disjointness for topological models of ergodic systems with discrete spectrum, preprint.    J. Konieczny, Gowers norms for the Thue–Morse and Rudin–Schapiro sequences, preprint.    KulagaPrzymus J., Lemańczyk M. The Möbius function and continuous extensions of rotations Monatshefte Math. 2015 ; 178 (4) : 553582 [crossref]    Liu J.Y., Sarnak P. The Möbius function and distal flows Duke Math. J. 2015 ; 164 (7) : 13531399 [crossref]    Sarnak P. Three Lectures on the Möbius Function, Randomness and Dynamics : (2009). MobiusFunctionsLectures(2).pdf    Sarnak P. Möbius randomness and dynamics Not. S. Afr. Math. Soc. 2012 ; 43 : 8997    R.X. Shi, Equivalent definitions of oscillating sequences of higher orders, preprint.    Tao T. Higher Order Fourier Analysis Providence, RI, USA: American Mathematical Society (2012).    Z.R. Wang, Möbius disjointness for analytic skew products, preprint.  
     
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