On the spectral type of some class of rank one flows

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On the spectral type of some class of rank one flows

El Houcein El Abdalaoui

To cite this version:

El Houcein El Abdalaoui. On the spectral type of some class of rank one flows. 2012. �hal-00701070v2�

FLOWS

E. H. EL ABDALAOUI

Abstract. It is shown that a certain class of Riesz product type measure on Ris singular. This proves the singularity of the spectral types of some class of rank one flows. Our method is based on the extension of the Cen- tral Limit Theorem approach to the real line which gives a new extension of Salem-Zygmund Central Limit Theorem.

AMS Subject Classifications(2010): 37A15, 37A25, 37A30, 42A05, 42A55.

Key words and phrases: Rank one flows, spectral type, simple Lebesgue spectrum, singular spectrum, Salem-Zygmund Central Limit Theorem, Riesz products.

1. Introduction

The purpose of this paper is to study the spectral type of some class of rank one flows. Rank one flows have simple spectrum and using a random Ornstein procedure [22], A. Prikhod’ko in [26] produce a family of mixing rank one flows.

It follows that the mixing rank one flows may possibly contain a candidate for the flow version of the Banach’s well-known problem whether there exists a dynamical flow pΩ,A, µ,pTtqtPRq with simple Lebesgue spectrum ¹. In [25], A. Prikhod’ko introduced a class of rank one flows called exponential staircase rank one flows and state that in this class the answer to the flow version of Banach problem is affirmative. Unfortunately, as we shall establish, this is not the case since the spectrum of a large class of exponential staircase rank one flow is singular and this class contain a subclass of Prikhod’ko examples.

Our main tools are on one hand an extension toRof the CLT method (introduced in [1] for the torus) and on the other hand the generalized Bourgain methods [5]

obtained in [2] (in the context of the Riesz products onR).

This allows us to get a new extension of the Salem-Zygmund CLT Theorem [32]

to the trigonometric sums withreal frequencies.

Originally Salem-Zygmund CLT Theorem concerns the asymptotic stochastic behaviour of the lacunary trigonometric sums on the torus. Since Salem-Zygmund pioneering result, the central limit theorem for trigonometric sums has been in- tensively studied by many authors, Erd¨os [10], J.P. Kahane [13], J. Peyri`ere [24], Berkers [4], Murai [20], Takahashi [28], Fukuyama and Takahashi [11], and many others. The same method is used to study the asymptotic stochastic behaviour of Riesz-Raikov sums [23]. Nevertheless all these results concern only the trigonomet- ric sums on the torus.

Here we obtain the same result onR. The fundamental ingredient in our proof is based on the famous Hermite-Lindemann Lemma in the transcendental number theory [30].

Notice that the main argument used in the torus case [1] is based on the density of trigonometric polynomials. This argument cannot be applied here since the density of trigonometric polynomials in L¹pR, ωptqdtq (ω is a positive function in L¹pRq), is not verified unlessω satisfies some extra-condition. Nevertheless, using the density of the functions with compactly supported Fourier transforms, we are able to conclude.

The paper is organized as follows. In section 2, we review some standard facts from the spectral theory of dynamical flows. In section 3, we recall the basic construction of the rank one flows obtained by the cutting and stacking method and we state our main result. In section 4, we summarize and extend the relevant material on the Bourgain criterion concerning the singularity of the generalized Riesz products on R. In section 5, we develop the CLT method for trigonometric sums with real frequencies and we prove our main result concerning the singularity of a exponential staircase rank one flows.

1Ulam in his book [29, p.76] stated the Banach problem in the following form

Question 1.1 (Banach Problem). Does there exist a square integrable function fpxq and a measure preserving transformation Tpxq, ´8 ă x ă 8, such that the sequence of functions tfpTⁿpxqq;n“1,2,3,¨ ¨ ¨ uforms a complete orthogonal set in Hilbert space?

2. Basic facts from spectral theory of dynamical flows

A dynamical flow is a quadrupletpX,A, µ,pT_tqtPRqwherepX,A, µqis a Lebesgue probability space and pT_tqtPR is a measurable action of the group R by measure preserving transformations. (It means that

‚ eachTtis a bimeasurable invertible transformation of the probability space such that, for anyAPA,µpT_t^´1Aq “µpAq,

‚ for alls, tPR,T_s˝T_t“T_s`t,

‚ the mappt, xq ÞÑT_tpxqis measurable fromRˆX intoX.)

Let us recall some classical definitions. A dynamical flow is ergodic if every measurable set which is invariant under all the mapsTteither has measure zero or one. A number λis aneigenfrequency if there exists nonzero functionf PL²pXq such that, for alltPR,f˝Tt“e^iλtf. Such a functionf is called aneigenfunction.

An ergodic flowpX,A, µ,pTtqtPRqisweakly mixingif every eigenfunction is constant (a.e.). A flowpX,A, µ,pT_tqtPRqismixing if for allf, gPL²pXq,

ż

f ˝Ttpxqgpxqdµpxq ´´´´´Ñ

|t|Ñ`8

ż

fpxqdµpxq

ż

gpxqdµpxq.

Any dynamical flowpTtqtPRqinduces an action ofRby unitary operators acting on L²pXq according to the formula UTtpfq “ f ˝T_´t. When there will be no ambiguity on the choice of the flow, we will denote Ut“UTt.

The spectral properties of the flow are the property attached to the unitary representation associated to the flow. We recall below some classical facts; for details and references see [8] or [14].

Two dynamical flowspX1,A₁, µ1,pTtqtPRqandpX2,A₂, µ2,pStqtPRqaremetrically isomorphic if there exists a measurable mapφfrompX1,A₁, µ1qintopX2,A₂, µ2q, with the following properties:

‚ φis one-to-one,

‚ For allAPA₂,µ1pφ^´1pAqq “µ2pAq.

‚ St˝φ“φ˝Tt,@tPR.

If two dynamical flowspTtqtPRand pStqtPRare metrically isomorphic then the iso- morphism φ induces an isomorphism Vφ between the Hilbert spaces L²pX2q and L²pX1q which acts according to the formula Vφpfq “ f ˝φ. In this case, since VφUSt “UTtVφ, the adjoint groupspUTtq andpUStq are unitary equivalent. Thus if two dynamical flows are metrically isomorphic then the corresponding adjoint groups of unitary operators are unitary equivalent. It is well known that the con- verse statement is false [8].

By Bochner theorem, for any f P L²pXq, there exists a unique finite Borel measureσ_f onRsuch that

x σfptq “

ż

R

e^´itξ dσfpξq “ xUtf, fy “

ż

X

f˝Ttpxq ¨fpxqdµpxq.

σf is called thespectral measure of f. Iff is eigenfunction with eigenfrequencyλ then the spectral measure is the Dirac measure atλ.

The following fact derives directly from the definition of the spectral measure: let pa_kq1ďkďnbe complex numbers andpt_kq1ďkďnbe real numbers; considerf PL²pXq and denote F “ řn

k“1a_k ¨f ˝T_tk. Then the spectral measure σ_F is absolutely

continuous with respect to the spectral measureσ_f and

p1q dσ_F

dσfpξq “ ˇˇ ˇˇ ˇ

ÿn k“1

a_ke^itkξ ˇˇ ˇˇ ˇ

2

.

Here is another classical result concerning spectral measures : let pgnq be a sequence inL²pXq, converging tof PL²pXq; then the sequence of real measures pσgn´σfqconverges to zero in total variation norm.

Themaximal spectral type of pT_tqtPR is the equivalence class of Borel measures σ on R (under the equivalence relation µ₁ „ µ₂ if and only if µ₁ ăă µ₂ and µ₂ ăăµ₁), such that σ_f ăă σ for allf PL²pXqand if ν is another measure for whichσ_f ăăν for allf PL²pXqthenσăăν.

The maximal spectral type is realized as the spectral measure of one function:

there exists h1 P L²pXq such that σh1 is in the equivalence class defining the maximal spectral type ofpTtqtPR. By abuse of notation, we will call this measure the maximal spectral type measure.

The reduced maximal typeσ₀is the maximal spectral type ofpU_tqtPRonL²₀pXq^def“

$&

%f PL²pXq :

ż

f dµ“0 ,.

-. The spectrum ofpTtqtPR is said to be discrete (resp.

continuous, resp. singular, resp. absolutely continuous , resp. Lebesgue) if σ0 is discrete (resp. continuous, resp. singular with respect to Lebesgue measure, resp.

absolutely continuous with respect to Lebesgue measure).

The cyclic space ofhPL²pXqis

Zphq^def“ spantUth : tPRu.

There exists an orthogonal decomposition ofL²pXqinto cyclic spaces

p2q L²pXq “

à8 i“1

Zphiq, σh1 "σh2". . .

Each decomposition (2) is be called aspectral decomposition of L²pXq(while the sequence of measures is called a spectral sequence). A spectral decomposition is unique up to equivalent class of the spectral sequence. The spectral decomposition is determined by the maximal spectral type and themultiplicity functionM :RÑ t1,2, . . .u Y t`8u, which is defined σ_h1-a.e. byMpsq “ř8

i“11Yipsq, where Y₁“R andYi“supp_dσ^dσxi

x1 foriě2.

The flow has simple spectrum if 1 is the only essential value of M. The mul- tiplicity is homogeneous if there is only one essential value of M. The essential supremum ofM is called themaximal spectral multiplicity.

Von Neumann showed that the flowpT_tqtPRhas homogeneous Lebesgue spectrum if and only if the associated group of unitary operators pU_tqtPR satisfy the Weyl commutation relations for some one-parameter grouppV_tqtPRi.e.

UtVs“e^´istVsUt, s, tPR, wheree^´ist denotes the operator of multiplication bye^´ist.

It is easy to show that the Weyl commutation relations implies that the maxi- mal spectral type is invariant with respect to the translations. The proof of von Neumann homogeneous Lebesgue spectrum theorem can be found in [8].

3. Rank one flows by Cutting and Stacking method

Several approach of the notion of rank one flow have been proposed in the liter- ature. The notion ofapproximation of a flow by periodic transformations has been introduced by Katok and Stepin in [15] (see Chapter 15 of [8]). This was the first attempt of a definition of a rank one flow.

In [9], del Junco and Park adapted the classical Chacon construction [6] to produce similar construction for a flow. The flow obtain by this method is called the Chacon flow.

This cutting and stacking construction has been extended by Zeitz ([31]) in order to give a general definition of a rank one flow. In the present paper we follow this cutting and stacking (CS) approach and we recall it now. We assume that the reader is familiar with the CS construction of a rank one map acting on certain measure space which may be finite or σ-finite. A nice account may be founded in [12].

Let us fix a sequenceppnqnPNof integersě2 and a sequence of finite sequences of non-negative real numbers´

psn,jq^p_j^n´_“1¹¯

ną0.

LetB0be a rectangle of height 1 with horizontal baseB0. At stage one divideB0

intop0 equal partspA1,jq^pj“⁰1. LetpA1,jq^pj“⁰1 denotes the flow towers overpA1,jq^pj⁰“1. In order to construct the second flow tower, put over each towerA1,j a rectangle spacer of heights1,j(and base of same measure asA1,j) and form a stack of height h1“p0`řp⁰

j“1s1,jin the usual fashion. Call this second towerB1, withB1“A1,1. At the k^th stage, divide the base Bk´1 of the tower Bk´1 into pk´1 subsets pAk,jq^p_j“1^k´1 of equal measure. Let pAk,jq^p_j“1^k´1 be the towers overpAk,jq^p_j“1^k´1 respec- tively. Above each towerA_k,j, put a rectangle spacer of height s_k,j (and base of same measure asA_k,j). Then form a stack of heighth_k“p_k´1h_k´1`řpk´1

j“1 s_k,j in the usual fashion. The new base isB_k“A_k,1 and the new tower isB_k.

All the rectangles are equipped with Lebesgue two-dimensional measure that will be denoted byν. Proceeding this way we construct what we call a rank one flow pTtqtPRacting on a certain measure spacepX,B, νqwhich may be finite orσ´finite depending on the number of spacers added at each stage.

This rank one flow will be denoted by pT^tqtPRdef

“ ´ T_pp^t_n,ps

n`1,jq^pn_j“1qně0

¯

tPR

The invariant measureν will be finite if and only if

`8ÿ

k“0

řpk

j“1sk`1,j

pkhk ă `8.

In that case, the measure will be normalized in order to have a probability.

Remarks 3.1. The only thing we use from [31] is the definition of rank one flows.

Actually a careful reading of Zeitz paper [31] shows that the author assumes that for any rank one flow there exists always at least one time t0 such that Tt0 has rank one property. But, it turns out that this is not the case in general as proved by Ryzhikhov in [27]. Furthermore, if this property was satisfied then the weak closure theorem for flows would hold as a direct consequence of the King weak

closure theorem (CpT_tq Ă Cpt₀q ^J.King“ W CTpT_t0q Ă W CTpT_tq Ă CpT_tq, where Cpt0qis the centralizer ofTt0 andW CTpTt0qis the weak closure ofTt0).

3.1. Exponential staircase rank one flows of type I. The main issue of this note is to study the spectrum of a subclass of a rank one flows calledexponential staircase rank one flows of type Iwhich are defined as follows.

Let pmn, pnqnPN be a sequence of positive integers such that mn and pn goes to infinity as ngoes to infinity. Let εn be a sequence of rationals numbers which converge to 0. Put

ω_nppq “mn

ε²_n p_n´

exp`εn.p p_n

˘´1¯

for any pP t0,¨ ¨ ¨, p_n´1u, and define the sequence of the spacerspps_n`1,pqp“0,¨¨¨,pn´1qně0 by

hn`sn`1,p`1“ωnpp`1q ´ωnppq, p“0,¨ ¨ ¨, pn´1, nPN. In this definition we assume that

(1) m_něε_n.h_n, for anynPN. (2) logppnq

m_n ´´´´´Ñ

nÑ8 0 ifpně ^m_εnⁿ (3) logpp_nq

mn ´´´´´Ñ

nÑ8 0 and logpp_nq

pn ďε_n ifp_n ă^m_ǫnⁿ We will denote this class of rank one flow by

pT^tqtPRdef

“ ´

T_p^t_pn,ωnqně0¯

tPR.

It is easy to see that this class of flows contain a large class of examples introduced by Prikhodko [25]. Indeed, assume that h^β_n ě p_n ě h¹_n^`α, β ě 2 and α Ps0,¹₄r. Then, by assumption (1), we have

logpp_nq

mn ďβlogph_nq εnhn

, Takingβ“εn`

th^δ_nu`1˘

, 0ăδă1, we get logpp_nq

m_n ´´´´´Ñ_n

Ñ8 0.

In [2] it is proved that the spectral type of of any rank one flowpT_p^t_p

n,ps_n`1,jq^pn_j“1qně0qtPR

is given by some kind of Riesz-product measure onR. To be more precise, the au- thors in [2] proved the following theorem

Theorem 3.2 (Maximal spectral type of rank one flows). For any sP p0,1s, the spectral measureσ0,s is the weak limit of the sequence of probability measures

źn k“0

|P_kpθq|²K_spθqdθ, where

P_kpθq “ 1

?pk

˜_pk´1 ÿ

j“0

e^{iθpjhk`¯skpjqq}

¸

, s¯_kpjq “ ÿj i“1

s_k`1,i, s¯_kp0q “0.

and

K_spθq “ s 2π¨

˜sinp^sθ₂q

sθ 2

¸2

.

In addition the continuous part of spectral type of the rank one flow is equivalent to the continuous part of ÿ

kě1

2^´kσ_0,¹

k.

The theorem above gives a new generalization of Choksi-Nadkarni Theorem [7], [21]. We point out that in [3], the author generalized the Choksi-Nadkarni Theorem to the case of funny rank one group actions for which the group is compact and Abelian.

We end this section by stating our main result.

Theorem 3.3. LetpT^tqtPR“´ T_pp^t_n,ωnq

ně0

¯

tPRbe a exponential staircase rank one flow of type I associated to

ω_nppq “m_n

ε²_n p_nexp`εn.p pn

˘, p“0,¨ ¨ ¨, p_n´1.

Then the spectrum ofpT_tq^Ris singular.

4. On the Bourgain singularity criterium of generalized Riesz products on R

In this section, for the convenience of the reader we repeat the relevant material from [2] without proofs, thus making our exposition self-contained. Let us fix sP p0,1qand denote byµ_s the probability measure of densityK_s onR, that is,

dµspθq “Kspθqdθ

We denote byσthe spectral measure of ₁1_B0,sgiven as the weak limit of the following generalized Riesz products

dσ“W´ lim

NÑ`8

źN k“1

|Pk|²dµs, p3q

where

P_kpθq “ 1

?p_k

˜_pk´1 ÿ

j“0

e^{iθpjhk`¯skpjqq}

¸

, s¯_kpjq “ ÿj i“1

s_k`1,i, s¯_kp0q “0.

Let us recall the following Bourgain criterion established in [2].

Theorem 4.1 (Bourgain criterion). The following are equivalent (i) σis singular with respect to Lebesgue measure.

(ii) inf

$&

%

ż

R

źL ℓ“1

|P_nℓ| dµ_s : LPN, n₁ăn₂ă. . .ăn_L ,.

-“0.

As noted in [2] to prove the singularity of the spectrum of the rank one flow it is sufficient to prove that a weak limit point of the sequence`ˇˇ|P_m|²´1ˇˇ˘is bounded by below by a positive constant. More precisely, the authors in [2] established the following proposition.

Proposition 4.2. LetEbe an infinite set of positive integers. Suppose that there exists a constantcą0 such that, for any positive functionφPL²pR, µ_sq,

lim inf

mÝÑ`8

mPE

ż

R

φˇˇ|P_m|²´1ˇˇ dµ_sěc

ż

R

φ dµ_s. Thenσis singular.

The proof of the proposition 4.2 is based on the following lemma.

Lemma 4.3. Let E be an infinite set of positive integers. Let L be a positive integer and 0ďn1ăn2ă ¨ ¨ ¨ ănL be integers. DenoteQ“śL

ℓ“1|Pnℓ|. Then lim sup

mÝÑ`8

mPE

ż

Q|Pm| dµsď

ż

Q dµs´1 8

¨

˝lim inf

mÝÑ`8

mPE

ż

Qˇˇˇ|Pm|²´1ˇˇˇ dµs

˛

‚

2

. For sake of completeness we recall from [2] the proof of the proposition 4.2.

Proof of Proposition 4.2.

Let β “ inf

$&

%

ż

Q dµs : Q“ źL ℓ“1

|Pnℓ|, LPN,0ďn1ăn2ă ¨ ¨ ¨ ănL

,.

-. Then, for any suchQ, we have

ż

Q dµ_sěβ and lim inf

ż

Q|P_m|dµ_sěβ.

Thus by Lemma 4.3 and by taking the infimum over allQwe get βďβ´1

8pcβq² It follows that

β“0,

and the proposition follows from Theorem 4.1.

5. the CLT method for trigonometric sums and the singularity of the spectrum of exponential staircase rank one flows of type I The main goal of this section is to prove the following proposition

Proposition 5.1. LetpT^tqtPR“´ T_p^t_p

n,ωnqně0

¯

tPRbe a exponential staircase rank one flow of type I associated to

ωnppq “mn

ε²_n pnexp`εn.p pn

˘, p“0,¨ ¨ ¨, pn´1.

Then, there exists a constant c ą 0 such that, for any positive function f in L²pR, µsq, we have

lim inf

mÝÑ`8

ż

R

fptqˇˇ|Pmptq|²´1ˇˇ dµsptq ěc

ż

R

fptqdµsptq.

The proof of the proposition 5.1 is based on the study of the stochastic behaviour of the sequence |P_m|. For that, we follow the strategy introduced in [1] based on the method of the Central Limit Theorem for trigonometric sums.

This methods takes advantage of the following classical expansion exppixq “ p1`ixqexp´

´x²

2 `rpxq¯ ,

where |rpxq| ď |x|³, for all real number x², combined with some ideas developed in the proof of martingale central limit theorem due to McLeish [19]. Precisely, the main ingredient is the following theorem proved in [1].

Theorem 5.2. Let tXnj : 1 ďj ď kn, n ě1u be a triangular array of random variables andta real number. Let

Sn “

kn

ÿ

j“1

Xnj, Tn“

kn

ź

j“1

p1`itXnjq, and

U_n“exp

˜

´t² 2

kn

ÿ

j“1

X_nj² `

kn

ÿ

j“1

rptX_njq

¸ . Suppose that

(1) tTnuis uniformly integrable.

(2) EpTnq ´´´´´Ñ

nÑ8 1.

(3)

kn

ÿ

j“1

X_nj² ´´´´´Ñ

nÑ8 1 in probability.

(4) max

1ďjďkn|Xnj| ´´´´´Ñ

nÑ8 0 in probability.

ThenEpexppitSnqq ÝÑexpp´t² 2q.

We remind that the sequence tX_n, n ě 1u of random variables is said to be uniformly integrable if and only if

cÝÑ`8lim

ż

|Xn|ąc(

ˇˇX_nˇˇdP“0 uniformly inn.

and it is well-known that if sup

nPN

ˆE`ˇˇX<sub>n</sub>ˇˇ<sup>1`ε</sup>˘˙ ă `8, p4q

for someεpositive, then tXnuis uniformly integrable.

Using Theorem 5.2 we shall prove the following extension toRof Salem-Zygmund CLT theorem, which seems to be of independent interest.

Theorem 5.3. LetAbe a Borel subset of RwithµspAq ą0 and letpmn, pnqnPN

be a sequence of positive integers such thatmn andpn goes to infinity asngoes to infinity. Letεn be a sequence of rationals numbers which converge to 0 and

ω_npjq “ m_n

ε²_n p_nexp`ε_n.j p_n

˘ for any jP t0,¨ ¨ ¨, p_n´1u.

2this is a direct consequence of Taylor formula with integral remainder.

Assume that

(1) mněεn.hn, for anynPN. (2) logpp_nq

mn ´´´´´Ñ

nÑ8 0 ifp_ně ^m_εnⁿ (3) logppnq

mn ´´´´´Ñ

nÑ8 0 and logppnq

pn ďεn ifpn ă^m_ǫnⁿ. Then, the distribution of the sequence of random variables ?^?p²n

řpn´1

j“0 cospω_npjqtq converges to the Gauss distribution. That is, for any real numberx, we have

1 µ_spAqµs

#

tPA :

?2

?p_n

pnÿ´1 j“0

cospωnpjqtq ďx +

´´´´´Ñ

nÑ8

?1 2π

ż

´8

e^´12t2dt^def“ Nps´8, xsq. p5q

We start by proving the following proposition.

Proposition 5.4. There exists a subsequence of the sequence´ˇˇˇ|P_npθq|²´1ˇˇˇ¯

něN

which converges weakly in L²pR, µ_sq to some non-negative function φ bounded above by 2.

In the proof of Proposition 5.4 we shall need the following lemma proved in [2].

Lemma 5.5. The sequence of probability measures |Pnpθq|²Kspθqdθ converges weakly toKspθqdθ.

Proof of Proposition 5.4. Since for all n, we have }P_n}²L²pµsq “1. Therefore, the sequence´ˇˇˇ|P_n|²´1ˇˇˇ¯

nPNis bounded inL²` R, µ_s˘

, thus admits a weakly convergent subsequence. Let us denote byφone such weak limit function. Letf be a bounded continuous function on

ż

f¨ˇˇˇ|Pn|²´1ˇˇˇdµsď

ż

f¨ |Pn|²dµs`

ż

f dµs. By Lemma 5.5 we get

nÑ`8lim

ż

f ¨ˇˇˇ|P_n|²´1ˇˇˇdµ_sď2

ż

f dµ_s. Hence, for any bounded continuous functionf onR, we have

ż

f¨φ dµ_sď2

ż

f dµ_s.

which proves that the weak limitφis bounded above by 2.

Let us prove now that the functionφis bounded by below by a universal positive constant. For that we need to prove Proposition 5.1. Let nbe a positive integer and put

W_n^def“ !ÿ

jPI

ηjωnpjq : ηj P t´1,1u, IĂ t0,¨ ¨ ¨ , pn´1u) .

The elementw“ř

iPIη_jω_npjqis called a word.

We shall need the following two combinatorial lemmas. The first one is a classi- cal result in the transcendental number theory and it is due to Hermite-Lindemann.

Lemma 5.6(Hermite-Lindemann, 1882). Letαbe a non-zero algebraic num- ber. Then, the number exppαqis transcendental.

We state the second lemma as follows.

Lemma 5.7. For anynPN^˚.All the words ofW_n are distinct.

Proof. Letw, w¹ PW_n, write

w“ÿ

jPI

ηjωnpjq, w¹ “ ÿ

jPI¹

η¹_jω_npjq. Thenw“w¹ implies ÿ

jPI

η_jω_npjq ´ÿ

jPI¹

η_j¹ω_npjq “0

Hence ÿ

jPI

ηjexppεn

pn

jq ´ÿ

jPI¹

η_j¹exppεn

pn

jq “0

But Lemma 5.6 tell us thateεⁿ{p_n is a transcendental number. This clearly forces

I“I¹ and the proof of the lemma is complete.

Proof of Theorem 5.3. LetAbe a Borel set withµspAq ą0 and notice that for any positive integern, we have

ż

R

ˇˇ ˇ

?2

?p_n

pnÿ´1 j“0

cospωnpjqtqˇˇˇ²dµsptq ď1.

Therefore, applying the Helly theorem we may assume that the sequence

´ ?

?p2_n

pnÿ´1 j“0

cospωnpjqtq¯

ně0

converge in distribution. As is well-known, it is sufficient to show that for every real number x,

1 µspAq

ż

A

exp

#

´ix

?2

?pn pnÿ´1

j“0

cospω_npjqtq +

dµ_sptq ´´´´´Ñ

nÑ8 expp´x² 2 q. To this end we apply theorem 5.2 in the following context. The measure space is the given Borel setA of positive measure with respect to the probability measure µs on Requipped with the normalised measure µs

µ_spAq and the random variables are given by

Xnj “

?2

?pn

cospωnpjqtq, where 0ďjďpn´1, nPN.

It is easy to check that the variablestX_njusatisfy condition (4). Further, condition (3) follows from the fact that

ż

R

ˇˇ ˇ

pnÿ´1 j“0

X_nj² ´1ˇˇˇ²dµsptq ´´´´´Ñ

nÑ8 0.

It remains to verify conditions (1) and (2) of Theorem 5.2. For this purpose, we set

Θnpx, tq “

pn´1

ź

j“0

´1´ix

?2

?pn

cospωnpjqt¯

“1` ÿ

wPWn

ρ_w^pnqpxqcospwtq,

and

W_n “ď

r

W^prq

n , whereW^p_n^rqis the set of words of lengthr. Hence

ˇˇ

ˇΘnpx, tqˇˇˇď

#_pn´1 ź

j“0

´ 1`2x²

p_n

¯+¹2

. But, since 1`uďe^u, we get

ˇˇ

ˇΘnpx, tqˇˇˇďe^x2. p6q

This shows that the condition (1) is satisfied. It still remains to prove that the variablestXnjusatisfy condition (2). For that, it is sufficient to show that

ż

A p

ź

j“0

ˆ 1´ix

?2

?p_ncospω_npjqtq

˙

dµ_sptq ´´´´´Ñ_n

Ñ8 µ_spAq. p7q

Observe that

ż

A

Θnpx, tqdµsptq “µspAq ` ÿ

wPW_n

ρwpnqpxq

ż

A

cospwtqdµsptq and forw“řr

j“1ωnpqjq PW_n, we have

|ρwpnqpxq| ď 2^1´r|x|^r pn^r2

p8q , hence

wmaxPW_n|ρ_w^pnqpxq| ď|x| p

1

n2

´´´´´Ñ_n

Ñ8 0.

We claim that it is sufficient to prove the following

ż

R

φ

p

ź

´1´ix

?2

?pn

cospωnpjqtq¯

dt´´´´´Ñ

nÑ8

ż

R

p9q φdt,

for any functionφ with compactly supported Fourier transforms. Indeed, assume that (9) holds and letǫą0. Then, by the density of the functions with compactly supported Fourier transforms [16, p.126], one can find a functionφ_ǫwith compactly supported Fourier transforms such that

››

›χ_A.K_s´φ_ǫ›››

L¹pRqăǫ,

where χ_A is indicator function of A. Hence, according to (9) combined with p7q, fornsufficiently large, we have

ˇˇ ˇ

ż

A

Θnpx, tqdµ_sptq ´µ_spAqˇˇˇ“ˇˇˇ

ż

A

Θnpx, tqdµ_sptq ´

ż

R

Θnpx, tqφ_ǫptqdt` p10q

ż

R

Θnpx, tqφǫptqdt´

ż

R

φǫptqdt`

ż

R

φǫptqdt´µspAq| ăe^x2ǫ`2ǫ.

p11q

The proof of the claim is complete. It still remains to prove (9). For that, let us compute the cardinality of words of lengthrwhich can belong to the support ofφ.

By the well-known sampling theorem, we can assume that the support of φ is r´Ω,Ωs, Ωą0. First, it is easy to check that for all oddr,|w^prqn | ´´´´´Ñ_n

Ñ8 `8. It suffices to consider the words with even length. The case r“2 is easy, since it is obvious to obtain the same conclusion. Moreover, as we will see later, it is sufficient to consider the caser“2^k,kě2. We argue that the cardinality of words of length 2^k,kě2 which can belong to r´Ω,Ωsis less than Ω.p^k_n plogppnqq^k´1

mn ε^k´2n

. Indeed, for k“2. Write

w_n^p4q“η1ωnpk1q `η2ωnpk2q `η3ωnpk3q `η4.ωnpk4q, withη_i P t´1,1uandk_iP t0,¨ ¨ ¨, p_n´1u, i“1,¨ ¨ ¨,4,and put

e_nppq “exp`ε_n pn

.p˘

, pP t0,¨ ¨ ¨, p_n´1u. Ifř4

i“1η_i‰0 then there is nothing to prove sinceˇˇw^p4qn ˇˇ´´´´´Ñ_n

Ñ8 `8. Therefore, let us assume that ř4

i“1η_i “0. In this case, without loss of generality (WLOG), we will assume thatk₁ăk₂ăk₃ăk₄. Hence

w_n^p4q“η1ωnpk1q `η2ωnpk2q `η3ωnpk3q `η4ωnpk4q

“mn

ǫ²_n pnenpk1q´

η1`η2enpα1q `η3enpα2q `η4enpα3q¯

where, α_i “k<sub>i`1</sub>´k<sub>1</sub>, i “1,¨ ¨ ¨,3.At this stage, we may assume again WLOG thatη<sub>1</sub>`η₂“0 andη₃`η₄“0. It follows that

w^p_n^4q“ mn

ǫ²_n pnenpk1q´ η2`

enpα1q ´1˘

`η4enpα2q`

enpα¹₁q ´1˘¯

with α¹₁“α3´α2. Consequently, we have two cases to deal with.

‚ Case 1: η₂“η₄. Then ˇˇ

ˇw^p_n^4qˇˇˇěmn

ǫ²_n pnenpk1q´

enpα1q ´1¯ ěm_n

ǫ²_n p_nε_n

pn ´´´´´Ñ

nÑ8 `8.

‚ Case 2: η2“ ´η4.We thus get w_n^p4q“m_n

ǫ²_n p_n.e_npk₁qη₄´ e_npα₂q`

e_npα¹₁q ´1˘

´`

e_npα₁q ´1˘¯

Hence

ˇˇw^p4q_n ˇˇ“m_n

ǫ²_n p_ne_npk₁qˇˇˇe_npα₂q`

e_npα¹₁q ´1˘

´`

e_npα₁q ´1˘ˇˇˇ Therefore, we have three cases to deal with.

– Case 1: α¹₁ąα₁. In this case, ˇˇw_n^p4qˇˇ“ m_n

ǫ²_n p_ne_npk₁q´ e_npα₂q`

e_npα¹₁q ´1˘

´`

e_npα₁q ´1˘¯

ě mn

ǫ²_n pn`

enpα¹₁q ´enpα1q˘ ě m_n

ǫ²_n p_n`

e_npα¹₁´α₁q ´1˘

´´´´´Ñ

nÑ8 `8. – Case 2: α<sup>1</sup><sub>1</sub>ăα1. Write α1“α<sup>1</sup><sub>1</sub>`β. Thus ˇˇw_n^p4qˇˇ“ mn

ε²_n p_ne_npk₁qˇˇˇe_npα₂q`

e_npα¹₁q ´1˘

´`

e_npα¹₁`βq ´1˘ˇˇ ˇ ě m_n

ε²_n p_nˇˇˇe_npα¹₁`α<sub>2</sub>q ´e<sub>n</sub>pα<sub>2</sub>q ´e<sub>n</sub>pα<sup>1</sup><sub>1</sub>`βq `1ˇˇˇ ě m_n

ε²_n p_nˇˇˇ`

e_npα₂q ´1˘`

e_npα¹₁q ´1˘

´e_npα¹₁q`

e_npβq ´1˘ˇˇ ˇ ě mn

ε²_n p_nεnβα¹₁ p_n

´enpα¹₁q`

enpβq ´1˘

εnβα¹1

pn

´

`enpα2q ´1˘`

enpα¹₁q ´1˘

εnβα¹1

pn

¯

ě mn

ε_n βα¹₁´enpα¹₁q`

enpβq ´1˘

εnβα¹1

pn

´

`enpα2q ´1˘`

enpα¹₁q ´1˘

εnβα¹1

pn

¯

But for anyxP r0,logp2qr, we havexďe^x´1ď2x. Therefore

`enpα2q ´1˘`

enpα¹₁q ´1˘

εnβα¹1

pn

ď4εn. Hence

ˇˇw^p_n^4qˇˇ´´´´´Ñ

nÑ8 `8 – Case 3: α¹₁“α₁. In this case, we get

ˇˇw^p_n^4qˇˇě m_n ǫ²_n p_n´

e_npα₂q ´1¯´

e_npα₁q ´1¯ .

From this, we have ˇˇw_n^p4qˇˇě mn

ǫ²_n p_n.´εn

p_n

¯2

.α₂.α₁. ě m_n

pn

α2.α1

It follows that

ˇˇw_n^p4qˇˇďΩùñα2.α1ď pn

mn

.Ω.

But the cardinality ofpα1, α2qsuch thatα2.α1ďΩpn

m_n is less than Ωpn

m_n logppnq. This gives that the cardinality of words of length 4 which can belong to r´Ω,Ωsis less than Ωp²_n

m_n logpp_nq.

Repeated the same argument as above we deduce that the only words to take into account at the stagek“3 are the form

w^p8q_n “ mn

ε²_n pnenpαq´

penpα1q ´1qpenpα2q ´1q `η.enpα3qpenpα4q ´1qpenpα5q ´1q¯ , whereη“ ˘1. In the caseη“1, it is easy to see that

ˇˇw_n^p8qˇˇě m_nε_n

p²_n α₃αĂ₁Ăα₂, where αr_i“infpα_i, α_3`iq. Forη“ ´1, write

ˇˇw^p8q_n ˇˇ“m_n

ε²_n pnenpαqˇˇˇpenpα1q ´1qpenpα2q ´1q ´

penpα3q ´1qpenpα4q ´1qpenpα5q ´1q ´ penpα4q ´1qpenpα5q ´1qˇˇˇ. Using the following expansion

e_npxq “1`εn.x p<sub>n</sub> `op1q, p12q

we obtain, for a largen,

ˇˇw_n^p8qˇˇě m_nε_n

p²_n α₃αĂ₁αĂ₂.

We deduce that the cardinality of words of length 8 which can belong to r´Ω,Ωs is less than

Ω.pn. p²_n mnεn

ÿ

Ă α1,αĂ2

1 Ă

α1αĂ2 ďΩ. p³_n

mnεnplogpp_nqq².

In the same manner as before consider the words of lengthkin the following form w_n^p2kq“mn

ǫ²_n pnenpαq´

penpα1q ´1q ¨ ¨ ¨ penpαkq ´1q¯ . Therefore

ˇˇw^p_n^2kqˇˇěmn

ǫ²_n pn.´εn

p_n

¯k

.α1.α2¨ ¨ ¨αk. ě m_n

p^kn^´1

ε^k´2_n α₁.α₂¨ ¨ ¨α_k

which yields as above that the cardinality of words of length 2^k which can belong tor´Ω,Ωsis less than

Ω.pn. p^k´1_n mnε^k´2n

ÿ

α2,¨¨¨,αk

1

α₂¨ ¨ ¨α_k ďΩ.pn. p^k´1_n mnε^k´2n

plogppnqq^k´1. Now, ifris any arbitrary even number. Writerin base 2 as

r“2^ls` ¨ ¨ ¨ `2^l1 with lsą ¨ ¨ ¨ ąl1ě1.

and write

w^p_n^rq“w^p_n^2lsq` ¨ ¨ ¨ `w^p_n^2l1q, with

w^p2_n^ljq“η₁^pjqωnpk₁^pjqq ` ¨ ¨ ¨ `η^pjq

2^ljωnpk_l^p_j^jqq, j“1,¨ ¨ ¨, s, and

2^lj

ÿ

i“1

η^p_i^jq“0, j“1,¨ ¨ ¨, s.

Observe that the important case to consider is the case w^p2_n^ljq“ ˘m_n

ε²_n pn lj

ź

i“1

`epα^p_i^jqq ´1˘

, j“1,¨ ¨ ¨, s.

Using againp12qwe obtain for a largenthat ˇˇw^p_n^rqˇˇě mn

p^ln^s´1

ε^l_n^s´2

ls

ź

i“1

α^p_i^sq.

Hence, the cardinality of words of lengthrwhich can belong tor´Ω,Ωsis less than Ω. 1

m_n. p^tlogn ^2prqu

ε^tlogn ^2prqu´2

plogpp_nqq^tlog2prqu´1, and, in consequence, byp8q, we deduce that

ÿ

wPW_n

ˇˇ

ˇρ_w^pnqpxq ż

R

e^´iwtφptqdtˇˇˇď ÿ

wPWprq n ,r even

4ďrďpn

ˇˇ

ˇρ_w^pnqpxq ż

R

φptqe^´iwtdtˇˇˇ.

Therefore, under the assumption (2), we get ÿ

wPW_n

ˇˇ

ˇρwpnqpxq ż

R

e^´iwtφptqdtˇˇˇďΩ|x| m_n

ÿ

r even

4ďrďpn

plogppnqq^tlog2prqu´1 p

`_r

2´2tlog2prqu`2˘

n

. 1

`p_n.ε_n˘tlog2prqu´2

ďΩ.|x|.logppnq m_n

ÿ

r even

4ďrďpn

1 p

`_r

2´2tlog2prqu`2˘

n

.

The last inequality is due to the fact that for a largenwe may assume thatlogpp_nq mn

is strictly less than 1. In addition, sincep_n ě2, ÿ

r even

4ďrďpn

1 p

`_r

2´2tlog2prqu`2˘

n

is convergent.