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Comptes Rendus Mathématique
Volume 355, n° 8
pages 866-870 (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 multi-ergodiques pondérées

Aihua Fan a, b
a LAMFA, UMR 7352 CNRS, Université de Picardie, 33, rue Saint-Leu, 80039 Amiens, France
b School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, China

 Abstract

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.

 Résumé

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.

 Introduction and results

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.

 Theorem 1

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.

 Theorem 2

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 quasi-discrete 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]].

 Theorem 3

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.

 Theorem 4

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.

 Proof of 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 .

 Proof of Theorem 1

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.

 Acknowledgements

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.

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