Generalized Riesz Products on the Bohr compactification of $\R$

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Generalized Riesz Products on the Bohr compactification of

El Houcein El Abdalaoui

To cite this version:

El Houcein El Abdalaoui. Generalized Riesz Products on the Bohr compactification of . 2012. �hal- 00703137v4�

GENERALIZED RIESZ PRODUCTS ON THE BOHR COMPACTIFICATION OF R^p˚q

E. H. EL ABDALAOUI

Abstract. We study a class of generalized Riesz products connected to the spectral type of some class of rank one flows onR. Applying a Central Limit Theorem of Kac, we exhibit a large class of singular gen- eralized Riesz products on the Bohr compactification ofR. Moreover, we discuss the problem of the flat polynomials in this setting.

˚Dedicated to Professors Jean-Paul Thouvenot and Bernard Host.

AMS Subject Classifications (2010) Primary: 42A05, 42A55; Secondary:

11L03, 42A61.

Key words and phrases: Generalized Riesz products, almost periodic func- tions, Bohr compactification, mean value, Besicovitch space, Kac Cen- tral Limit Theorem, Kakutani criterion .

1

1. Introduction

We stress that the main purpose of this paper is to extend and complete the study of the notion of generalized Riesz product associated to the rank one flows on R formulated in the same manner as Peyri´ere in [3]. The authors in [3] mentioned that Peyri´ere extended the notion of Riesz product to the real line using a class of kernel functions. Furthermore, it is noted by Peyri´ere in his pioneer paper [41] that an alternative extension of the classical Riesz products can be done using the Bohr compactification ofR.

Indeed, it is usual that the extension of some notions from the periodic setting to the almost periodic deals with the Bohr compactification bR of R (or more generally, the Bohr compactification of Local Abelian Groups).

The Bohr compactification plays in the almost periodic case the same role played by the torus T^def“ ttPC,|t| “1uin the periodic case as the domain of the fast scale variables. As opposed to the torus, the Bohr compactifi- cation is often a non-separable compact topological space and this lack of separability is a source of difficulties in trying to adapt the arguments from the periodic context to the almost periodic one. Peyri´ere [41] mentioned this difficulty and introduced the Riesz products on R associated to some class of kernels.

Notice that here we use the notion of separability used in Hirotada- Kakutani paper [26], That is, the topological group (or space) is called separable if it satisfies the second countability axiom of Hausdorff which means that it has a countable basis. In that paper, Hirotada and Kaku- tani established that the Bohr compactification of a given locally compact Abelian group has a countable basis if and only if the union of the spectrum of all almost periodic functions is countable.

Our analysis here is also motivated by the recent growing interest in the problem of the flat polynomials suggested by A.A. Prikhod’ko [42] in the context of R. It turns out that the main idea developed in [42] does not seem well adapted to the context of our extension of generalized Riesz prod- ucts to the Bohr compactification of R. This is due to the fact that the sequence of trigonometric polynomials constructed by A.A. Prikhod’ko is only locallyL¹-flat and not L¹-flat in the usual sense (see, for instance, Re- mark 6.11).

The paper is organized as follows. In section 2 we review some standard facts on the almost periodic functions including the ergodicity of the action ofRby translations on its Bohr compactification. In section 3 we define the notion of the generalized Riesz products on the Bohr compactification of R. In section 4 we summarize and extend the relevant material on the Kakutani criterion and the Bourgain criterion on the singularity of the generalized Riesz products introduced in section 3. In section 5 we state and prove the

Central Limit Theorem due to M. Kac. In section 6 we apply the Central Limit Theorem of Kac to prove our main result concerning the singularity of a large class of generalized Riesz products onbR. Finally, in the appendix, we consider the problem of the flat polynomials on the Bohr compactification ofRand we add a short note based on Hirodata-Kakutani paper [26] on the Bohr compactification compared with the Stone- ˇCech compactification.

2. The Bohr compactification of R

The Bohr compactification of R is based on the theory of almost peri- odic functions initiated by H. Bohr [7] in connection with the celebrated ζ-function of Riemann. In this section we are going to recall the basic in- gredients of this theory. For the basic facts about almost periodic functions and generalizations of this concept the reader is referred to the classical presentation of Bohr [7] and Besicovitch [6].

We point out that the theory of almost periodic functions can be extended to more general setting with applications in many context including the non- linear differential equations [15].

Definition 2.1. Let f : RÝÑ C be a bounded continuous function and εą0; we say that τ PR is anε-almost period for f if

sup

xPR|fpx`τq ´fpxq|<sup>def</sup>“ ||fp.`τq ´fp.q||8 ăε.

The mapping f is said to be almost periodic if for any ε ą 0 the set of ε-almost periods off is relatively dense, i.e., there isl“lpεq ą0 such that any interval with lengthl contains at least oneε-almost period.

The space of all almost periodic functions is denoted by APpRq. From the above definition we easily deduce thatAPpRqis a subspace of the space of bounded continuous functions on R. An important characterization of almost periodic functions is due to Bochner and it can be stated as follows Theorem 2.2 (Bochner’s characterization ofAPpRq). A bounded function f is almost periodic function if, and only if, the family of translatestfp.` tqu^tPRis relatively compact in the space bounded continuous functions onR endowed with the sup-norm topology.

The proof of Theorem 2.2 can be found in [7], [6] or [18]. Furthermore, we have the following fundamental theorem (see for instance [7] or [6]).

Theorem 2.3 (Bohr). A bounded continuous function f is almost peri- odic function if, and only if, f is uniformly approximated by finite linear combinations of functions in the set tcosptxq,sinptxqutPR.

The space of all continuous functions onbRis denoted byCpbRq. CpbRq is a commutative C^‹-algebra under pointwise multiplication and addition.

Below, we give an important topological characterization of the Bohr com- pactification of R due to Gelfand, Raikov and Chilov. They obtain this

characterization as an application of their theory of commutative Banach algebras.

Theorem 2.4 (Gelfand-Raikov-Chilov [21]). The group R, equipped with the usual addition operation, may be embedded as a dense subgroup of a compact Abelian group bR in such way as to make APpRq the family of all restrictions functions f_|R to R of functions f P CpbRq. The operator f ÞÝÑ f_|R is an isometric ‹-isomorphism of CpbRq onto APpRq. Moreover, the addition operation` : RˆRÝÑRextends uniquely to the continuous group operation of bR, ` :bRˆbRÝÑ bR.The group bRis called the Bohr compactification ofbR.

For simplicity of notation, for anyf inAPpRq, we use the same letterffor its canonical extension to bR. As a consequence of Theorem 2.4 combined with the Riesz representation Theorem we have

Theorem 2.5 ([18]). The dual of the spaceAPpRq is isometrically isomor- phic to the space MpbRq of all Radon measures on the Bohr compactifica- tion ofR. The isomorphismx^˚ÞÝѵ_x^˚ is given by the formula

x^˚pfq “

ż

bR

fptqdµ_x^˚ptq.

We recall in the following the definition of the characters. The characters play a important role in the Abelian group and, by Koopmann observation, in the spectral analysis of dynamical systems.

Definition 2.6. LetGbe an Abelian group andeits identity element, then a character of G is a complex valued function χ defined on G such that χpeq “1 and χpstq “χpsqχptq for all s, tPG.

Let us recall the following basic fact on compact Abelian groups due to Peter and Weyl.

Theorem 2.7(Peter-Weyl). LetGbe a compact Abelian group, withBpGq its Borel field andhits Haar measure. Then the set of continuous characters is fundamental both in CpGq and in L²pG,BpGq, hq.

The proof of Peter-Weyl Theorem can be found in [50] and for the thor- ough treatment we refer the reader to [50].

It is obvious that the continuous characters of bR are the functions e^iω : bR ÝÑ T. In addition the orthogonality of two distinct charac- ters can be checked directly. Indeed,

ż

bR

e^iωt.e^iω1tdhptq “ lim

TÝÑ`8

1 2T

ż

T

´T

e^ipω´ω1qtdt“0, whenever ω‰ω¹.

For f P APpRq we denote by

$

R

fptqdt the asymptotic mean value of f, given by

$

R

fptqdt“ lim

TÝÑ`8

1 2T

ż

T

´T

fptqdt.

As a consequence of the averaging properties of almost periodic functions we have the following

Lemma 1. For anyf PAPpRqwe have

ż

bR

f dhptq “

$

R

fptqdt,

where dhptq is the Haar measure in bR, normalized to be a probability measure, anddt is the usual Lebesgue measure inR. Moreover we have

ż

bR

fptqdhptq “ lim

TÝÑ`8

1 T|K|

ż

T.K

fptqdt,

where K is any bounded subset of R with |K| ‰ 0, |K| is the Lebesgue measure ofK.

Following [31], for any f PAPpRq, we introduce the notation fp`

tλu˘

“

$

R

fptqe^´iλtdt.

That is,` pf` tλu˘˘

λPRare the Fourier coefficients off relative to orthonormal family te^iλtu^λPR; the inner product is defined by

ăf, gą“ăf, gą“

ż

bR

fpκqgpκqdhpκq.

2.1. On Besicovitch space. Since the functionsf PAPpRqcorrespond to restrictions,f “fr_|R, of continuous functionsf on bR, a natural question is whether it is possible to define a class of functions f which correspond to

“restrictions”f “fr_|R, of functionsfrPL¹pbRq. This motivates the following definition.

Definition 2.8. Given p P r1,`8q the space BAP^ppRq, of Besicovitch’s generalized almost periodic functions on R, consists of those functions f P L^p_locpRq for which there exists a sequencef_nPAPpRq satisfying

nÝÑ`8lim lim sup

TÝÑ`8

1 2T

ż

T

´T

ˇˇ

ˇf_npxq ´fpxqˇˇˇ^pdx“0 p1q

We denote BAP^ppRq simply byB^ppRq.

The space of generalized almost periodic functionsB^ppRqwas introduced by Besicovitch, who also gave them a structural characterization. We refer to [6] for more details about functions in B^ppRq. We immediately have APpRq Ă B^ppRq Ă B¹pRq for any p ě 1 and it is easy to see that any f PB¹pRqhas the mean value property, that is, for any bounded measurable B ĂR, with|B| ‰0, we have

Mpfq “ lim

LÝÑ`8

1 L|B|

ż

B_L

f dx,

where BL “ tx PR : _L^x P Bu. The space corresponds to L^ppbRq in a way similar to the one in which the spaceAPpRqcorresponds toCpbRq. Indeed, notice first that the definition of B^ppRq immediately gives that the asymp- totic mean value

$

ˇˇfˇˇ^pdxof a function in B^ppRq is well defined; moreover, any approximating sequence fn PAPpRq satisfying p1q can be viewed as a Cauchy sequence in L^ppbRq and, hence, there exists frPL^ppbRq such that Ăfn converge tof inL^ppbRq. Sincefris easily seen to be independent of the approximating sequence, in this way we may associate with each f PB^ppRq a well determined function frP L^ppbRq which we may view as an “exten- sion” of f to bR. Notice that the mapf ÞÑfris a linear map and that the approximation procedure together with Lemma 1 show that

$

R

ˇˇ ˇfˇˇˇ^p “

ż

bR

ˇˇ

ˇ rfpκqˇˇˇ^pdκ,@f PB^ppRq. p2q

As a consequence, the kernel of the mapf ÞÝÑfris made by the functionsf such that the asymptotic mean value ofˇˇfˇˇ^pis 0. The corresponding quotient space is denoted by B^ppRq{ „. Forf PB^ppRq we set

|f|p def“

¨

˚˚

˝

$

R

ˇˇ ˇf

ˇˇ ˇ^pdx

˛

‹‹

‚

1 p

,

so thatˇˇˇˇ

p is a semi-norm onB^ppRq. It is well known thatB^ppRqis complete with respect to the semi-norm ˇˇˇˇ_p (see for instance [38]). The space B²pRq is endowed with the scalar product

ăf, gą“

$

R

fpxqgpxqdx“

ż

bR

frpκqrgpκqdκ, p3q

The second equality follows by p2q with p “2, implying that the scalar product is preserved under the map f ÞÑfr. Finally, we define

B⁸pRq^def“

# f P č

pě1

B^ppRq : sup

pě1

ˇˇfˇˇ

pă `8 +

,

Again,ˇˇˇˇ

8 is a semi-norm on B⁸pRq and the corresponding quotient space is denoted byB⁸pRq{ „. We finish this paragraph by stating the following important fact on the properties of the map f ÞÑ f. The proof of it is leftr to the reader.

Proposition 2.9. The map f ÞÑ fris an isometric isomorphism between the Banach spaces B^ppRq{ „and L^ppbRq for any pP r1,`8s

Furthermore, we point out that a suitable extension of Lebesgue and Fa- tou’s convergence results is obtained in [38].

Gelfand-Raikov-Chilov Theorem 2.4 allows us to define the Bohr com- pactification of R. But, in the Harmonic Analysis, it is well known that the Pontrygain Theorem gives an alternative definition. We briefly recall it here and we refer the reader to the large literature on the subject [50] and for a deeper discussion on the relation between the almost periodic functions and the Bohr compactification ofR to [32], [23].

LetGbe locally compact Abelian group and letGpbe its dual, i.e.,Gpis the set of the characters endowed with the topology inherited fromG. LetGpdbe Gp with the discrete topology. ThenGxx_d^def“ bGis the Bohr compactification of G. bGis a compact group such thatG is a dense subset ofbG.

We end this section by stating and proving the classical result on the ergodicity of the action ofRby translation onbR. We recall that the action ofRby translation is defined byτxpκq “κ`xwhere the extended addition is given by Theorem 2.4. Clearly, the family pτ<sub>x</sub>qxPR is a flow acting on bR since τ<sub>x</sub>pτ<sub>x</sub><sup>1</sup>pκqq “ τ<sub>x`x</sub>¹pκq and the Haar measure is invariant under translation.

Theorem 2.10. The action of R on bR is ergodic, that is, for any Borel set AĂbRwhich is invariant under the translation action we have hpAq P t0,1u, where hpAq denotes the normalized Haar measure of A. Moreover hpRq “0.

Proof. LetAĂbR be an invariant Borel set. We have hpAq “hpAXτxAq “

ż

bR

11Apβq¹1Apβ`xqdhpβq.

Now, translations are strongly continuous on L²pRq. Indeed, this is a stan- dard consequence of the density of CpbRq in L²pbRq, which follows from Theorem 2.4, and the invariance of the Haar measure. Therefore, the right- hand side is a continuous function of x, and so the identity still holds with xPbR. Hence we get, using Fubini theorem and the invariance of the Haar measure,

hpAq “

ż

bR

ż

bR

11_Apκq¹1_Apκ`ξqdhpκqdhpξq

“

ż

bR

11Apκqdhpκq

ż

bR

11Apξqdhpξq “hpAq²,

from which it follows thathpAq P t0,1u, as asserted. It remains to show that hpRq “ 0. First we observe that R is a Borel subset of bR, since it is the union of a countable family of compact sets, e.g., the images of the intervals r´k, ks, k PN. Since R is invariant under the translation action we have hpRq P t0,1u. But, for anyκPbRzR,κ`Ris also an invariant Borel set and RŞ

tκ`Ru “ H. By the invariance of the Haar measurehpκ`Rq “hpRq. Hence, we conclude that hpRq “0 and the proof is achieved.

3. Generalized Riesz Products on bR

Riesz products were discovered in 1918 by F. Riesz [46] to answer affir- matively a special question in the theory of Fourier series, namely, whether there exists a continuous measure whose Fourier coefficients do not vanish at infinity. Roughly speaking, the Riesz products are a kind of measures on the circle constructed inductively. The pioneer Riesz product construction gives a concrete example. Since then, the Riesz construction proved to be the source of powerful ideas that can be used to produce concrete coun- terexample of measures with a number of desired properties (controllability of the convergence of the Fourier coefficients being the goal of the original construction).

Later, A. Zygmund extended Riesz construction and introduced what it is nowadays called classical Riesz products [53, p.208].

In 1975, that Riesz products appear as a spectral type of some dynami- cal systems was shown by F. Ledrappier [36]. Ten years later, M. Queffelec [43], inspired by the work of Coquet-Kamae and Mandes-France [14], showed that the specific generalized Riesz products are the right tool to describe the spectrum of the class of dynamical systems arising from the substitution (see [44] and the references therein). In 1991, B. Host, J.-F. M´ela, F. Parreau in

[27] realized a large class of Riesz products as the maximal spectral type of the unitary operator associated with a non-singular dynamical system and a cocycle. Finally, in the more general setting, J. Bourgain established the connections between some class of generalized Riesz products on the circle and the maximal spectral type of a class of maps called rank one maps [10].

One year later, an alternative proof is given by Choksi-Nadkarni using the Host-M´ela-Parreau argument [12],[39] and at the same time a simple proof is obtained by Klemes-Reinhold [34] using the standard Fourier analysis ar- gument.

In [3], el Abdalaoui-Lema´nczyk-Lesigne and Ulcigrai proved that the gen- eralized Riesz products analogous to Peyri´ere-Riesz products can be realized as a spectral type of some class of rank one flows.

Here our aim is to extend the notion of generalized Riesz products to the Bohr compactification of R. It turns out that such generalization can be done directly. More precisely, in the study of the spectrum of some class of rank one flows, the following trigonometric polynomials on Rappears

P_kpθq “ 1

?p_k

ÿ

_p

k´1

j“0 e^iθpjhk`qsjq, p4q

wheresq0 “0 and qs_j “

j´1

ÿ

l“0

s_k,l, jě1,

withpp_kqkPNis a sequence of positive integers greater than 1 andps_k,jqj“0,¨¨¨,pk

is a sequence of positive real numbers with sk,0 “ 0 for any k PN; the se- quenceph_kq is defined inductively by

h0“1 andh<sub>k`</sub>1 “p<sub>k</sub>h<sub>k</sub>`

pk

ÿ

l“0

s_k,l. p5q

For simplicity, for any p, q P t0,¨ ¨ ¨, p_k ´1u we introduce the following sequence of real numbers

q sn,p,q “

maxpp,qq´1

ÿ

j“minpp,qq

sn,j, p6q

and for any real numbert, we put

eptq “e^it.

Theorem 3.1 (Generalized Riesz Products onbR). LetpP_nqnPNbe a fam- ily of trigonometric polynomials given by p4q. Then the weak limit of the sequence of probability measures on bR

ź

n k“0

ˇˇ ˇˇPkpθq

ˇˇ ˇˇ

2

dhpθq,

exists and is denoted by

ź

8 k“0

ˇˇ ˇˇP_kpθq

ˇˇ ˇˇ

2

dhpθq,

Proof. LetR_nptq “P0ptq ¨ ¨ ¨P_nptq, Q_nptq “ |R_nptq|²andσ_n“ |R_npκq|²dhpκq. By the definition ofP_n we have

ˇˇP_nptqˇˇ² “1`∆nptq with ∆nptq “ 1 pn

ÿ

p‰qe ˆ

i`

pp´qqh_n`qs_n,p,q˘ t

˙ and it is obvious that

$

R

Q_ndt“1.

Therefore, σ_n is a probability measure on bR. In addition, for any t P R, we have

Qn`1ptq “QnptqPnptq “Qnptq `Qnptq∆nptq Hence

z σ_n`1

`λ˘

“Qz_n`1

`tλuq “Qx<sub>n</sub>`

tλuq `Qx<sub>n</sub>˚∆xnpλq ěQx<sub>n</sub>`

tλuq “σx_n` λ˘ Consequently the limit r<sub>λ</sub> of the sequence pxσ<sub>n</sub>`

λ˘

qexists. Now, since bRis compact andpσnqis a sequence of probability measure onbRwe can extract a subsequencepσ_nkq which converge weakly to some probability measure on bR. We deduce that the limit of pσnq exists in the weak topology and this

finishes the proof.

The proof above is largely inspired by lemma 2.1 in [34]; it gives more, namely, the polynomials P_n can be chosen with positive coefficients and satisfying

$

_ˇ

ˇˇ ˇP_nptq

ˇˇ ˇˇ

2

dt“1 and

$ ź

n j“1

ˇˇ ˇˇP_jptq

ˇˇ ˇˇ

2

dt“1.

We mention also that we have

$ ź

k j“1

ˇˇ ˇˇPnjptq

ˇˇ ˇˇ

2

dt“1, p7q

for any given sequence of positive integersn1 ăn2 ă ¨ ¨ ¨ ăn_k, kPN^˚. We can now formulate our main result whose proof occupies all Section 6.

Theorem 3.2 (Main result). Letppmq^mPNbe a sequence of positive integers greater than 1 andpps_m,jq^pj“^m0^´1qmPNbe a sequence of positive real numbers.

Assume that there exists a sequence of positive integers m1 ă m2 ă ¨ ¨ ¨,

such that for any positive integerj, the numbersph_mj, s_mj,0,¨ ¨ ¨, s_mj,pmj´1q are rationally independent. Then the generalized Riesz product

µ“ źn k“0

ˇˇP_kpθqˇˇ²dhpθq,

where

Pkpθq “ 1

?p_k

ÿ

pk´1 j“0 e`

iθ`

jhk`qsk,0,j

˘˘.

is singular with respect to the Haar measure on bR.

4. On the Kakutani criterion and the Bourgain singularity criterion of the Riesz products on the Bohr

compactification of R

The famous dichotomy theorem of Kakutani has a rather long history. In his 1948 celebrated paper [30], Kakutani established a purity law for infinite product measures. Precisely, if P “ ^`8â

i“1

P_i and Q “ ^`8â

i“1

Q_i are a infinite product measures, where P_i,Q_i are probability measures such that P_i is absolutely continuous with respect to Q_i, for each positive integeri, then

P!Q or PKQaccording as ź

i

ż ˆdP_i dQi

˙

dQ_i converges or diverges.

There are a several proofs of Kakutani criterion in literature (see [9] and the references given there). For a proof based on the Hellinger integral we refer the reader to [17, p.60].

Kakutani’s Theorem was specialized to the Gaussian measures on Hilbert space with identical correlation operators in [22] and it was extended to Gaussian measures with non-identical correlation operators by Segal [51], Hajek [25], Feldman [20] and Rozanov [49]. Later, in 1979, Ritter in [47], [48] generalized Kakutani’s Theorem to a certain non-products measures with application to the classical Riesz products.

Here, applying the Bourgain methods, we obtain a new extension of Kaku- tani’s Theorem to the class of generalized Riesz products on the Bohr com- pactification ofR. Indeed, we show that the independence along subsequence suffices to prove the singularity.

Nevertheless, our strategy is similar to the strategy of [3] and it is based on the extension of Bourgain methods to the generalized Riesz products on the Bohr compactification of R combined with the Central Limit tools introduced in [2].

Moreover, having in mind applications beyond the context of this paper, we shall state and prove a Guenais sufficient condition on theL¹ flatness of the polynomials which implies the existence of generalized Riesz products onbRwith Haar component. We recall that the generalized Riesz products

µis given by

µ“

ź

`8 k“0

ˇˇP_kpθqˇˇ²dhpθq

where

P_kptq “ 1

?p_k

ÿ

pk´1 j“0 e`

iθpjh_k`qs_k,jq˘

and qs_k,j“

j´1

ÿ

l“0

s_k,l

Theorem 4.1 (bRversion of Bourgain criterion). The following are equiv- alent

(i) µ is singular with respect to Haar measure.

(ii) inf

$&

%

ż

bR

źL ℓ“1

ˇˇP_nℓˇˇdh : LPN, n1 ăn2 ă. . .ăn_L ,.

-“0.

The proof of bR version of Bourgain criterion is based on the following lemma.

Lemma 4.2. The following are equivalent (1)

ż

bR

źN k“0

ˇˇP_kˇˇdh´´´´Ñ

NÑ`8 0.

(2) inf

$&

%

ż

bR

źL ℓ“1

|P_nℓ|dh : LPN, n1 ăn2ă. . .ăn_L ,.

-“0.

Proof. The proof is a simple application of Cauchy-Schwarz inequality. Con- sider n1 ăn2 ă. . .ăn_L and N ěn_L. Denote N “ tn1 ăn2ă. . .ăn_Lu and N^c its complement in t1,¨ ¨ ¨ , Nu. Let aăb be two real numbers and define a probability measure on Rby

dλ_a,bptq “ ¹1_ra,bsptq

b´a dt, wheredt is the Lebesgue measure.

Then we have

ż

źN k“0

|P_k|dλ_a,b“

ż

ź

kPN

|P_k|¹² ˆ ź

kPN^c

|P_k|¹² źN k“0

|P_k|¹² dλ_a,b

ď

¨

˝

ż

ź

kPN

|P_k|dλ_a,b

˛

‚

1

2 ¨

˝

ż

ź

kPN^c

|P_k| ˆ źN k“0

|P_k|dλ_a,b

˛

‚

1 2

ď

¨

˝

ż

ź

kPN

|P_k|dλ_a,b

˛

‚

1

2 ¨

˝

ż

ź

kPN^c

|P_k|² dλ_a,b

˛

‚

1 4¨

˝

ż

źN k“0

|P_k|² dλ_a,b

˛

‚

1 4

By lettingb´a goes to infinity, we get

$

źN k“0

|P_k|dtď

¨

˝

$

ź

kPN

|P_k|dt

˛

‚

1 2¨

˝

$

ź

kPN^c

|P_k|² dt

˛

‚

1 4¨

˝

$

źN k“0

|P_k|² dt

˛

‚

1 4

“

¨

˝

$

ź

kPN

|P_k| dt

˛

‚

1 2

.

The last equality follows fromp7q.

Proof of Theorem 4.1. Assume that (i) holds. To prove that µ is singular , it suffices to show that for any ǫą0, there is a Borel set E withhpEq ăǫ and µpE^cq ăǫ. Let 0ăǫă1.

FixN0 such that for any N ąN0, we have ż

bR

źN k“0

|Pk|dhăǫ². The set E “!

ωPbR : śN

k“0|P_kpωq| ěǫ)

satisfies:

hpEq ď 1 ǫ

››

››

› źN k“0

P_k

››

››

›₁ ďǫ²{ǫ“ǫ,

and sinceE^c is open set, it follows from the Portmanteau Theorem that we have

µpE^cq ďlim inf

MÑ`8

ż

E^c

źM k“0

|P_k|² dh

ďlim inf

MÑ`8

ż

E^c

źN k“0

|Pk|² źM k“N`1

|Pk|² dh

ďǫ² lim

MÑ`8

ż

bR

źM k“N`1

|P_k|²dh“ǫ² ăǫ.

For the converse, given 0ăǫă1, there exists a continuous functionϕon bRsuch that:

0ďϕď1, µptϕ‰0uq ďǫ and hptϕ‰1uq ďǫ.

Letf_N “ źN k“1

|P_k|. By Cauchy-Schwarz inequality, we have ż

fN dh“ ż

tϕ‰1u

fN dh` ż

tϕ“1u

fN dh

ďhptϕ‰1uq^1{2 ˆż

bR

f_N² dh

˙¹_{²

`

˜ż

tϕ“1u

f_N² dh

¸¹_{²

hptϕ“1uq^1{2 ď?

ǫ` ˆż

bR

f_N² ϕ dh

˙¹_{² .

Since µis the weak limit off_N²dh, we have

NlimÑ8

ż

bR

f_N² ϕ dh“

ż

bR

ϕ dµďµptϕ‰0uq ďǫ.

Thus, lim sup ż

bR

fNdhď2?

ǫ. Sinceǫis arbitrary, we get lim

NÑ8

ż

bR

fN dh“

0,and this completes the proof.

From now on, letMbe a sequence of positive integers for whichph_m,ps_m,jq^pj^m“0^´1q are linearly independent over the rationals and let us fix some subsequence N “ tn1ăn2 ă. . .ănku ofM,kPNand mPMwith mąnk. Put

Qptq “

ź

k i“1

ˇˇ ˇˇP_niptq

ˇˇ ˇˇ.

We define the degree of any trigonometric polynomial f by degpfq “maxt|ξ|:fpptξuq ‰0u, p8q

and we denote d_m ^def“ degpP_mq. From equationsp5q and p6q, we have d_m“h_{m`</sub>1´h<sub>m</sub>´s<sub>m,pm</sub>ăh<sub>m`}1,

p9q

hmďhm`1{pmďhm`1{2.

p10q

Since n_j`1ďn<sub>j`</sub>1, telescoping we get q_k^def“ degpQ_kq “d_n1 `d<sub>n</sub>2 ` ¨ ¨ ¨ `d_nk

ď ph_n1`1´h<sub>n</sub>1q ` ph_n2`1´h<sub>n</sub>2q ` ¨ ¨ ¨ ` ph<sub>nk`</sub>1´h_nkq p11q

ăhn_k`1.

In the same spirit as above it is easy to see the following lemma. The proof of it in the case of the torus is given in [10] and [2].

Lemma 2. With the above notations we have

ż

bR

QˇˇP_mˇˇdhď

1 2

ˆ

ż

bR

Qdh`

ż

bR

QˇˇP_mˇˇ²dhpωq

˙

´1 8

ˆ

ż

bR

QˇˇˇˇP_mˇˇ²´1ˇˇdh˙² .

The following proposition is a simple extension of Proposition 2.4 in [2].

Proposition 4.3. We have

mlimÑ8

ż

bR

QˇˇPm

ˇˇ²dh“

ż

bR

Qdh

Proof. The sequence of probability measures ˇˇP_mpzqˇˇ²dh converges weakly

to the Haar measure.

From the proposition 4.3 and Lemma 2 we deduce the following Proposition 4.4. With the above notations we have

lim inf

ż

bR

QˇˇP_mˇˇdh ď

ż

bR

Qdh ´ 1 8

ˆ lim sup

ż

bR

QˇˇˇˇP_mˇˇ²´1ˇˇdh˙² .

Now, in the following lemma, we state a sufficient condition for the ex- istence of an absolutely continuous component with respect to the Haar measure for the given generalized Riesz product . In the case of Z action, the lemma is due to M´elanie Guenais [24], and the proof is similar.

Lemma 4.5. If

`8ÿ

k“1

gf ff e1´

ˆ

ż

bR

|P_k|dh

˙2

ă 8, then µ admits an abso- lutely continuous component.

Proof. We denote by} ¨ }^p the norm inL^pphq. For all functionsP and Qin L²phq, by Cauchy-Schwarz inequality we have

p12q }P}¹}Q}¹´ }P Q}¹“ ´ ż

p|P| ´ }P}¹q p|Q| ´ }Q}¹q dh

ď }|P| ´ }P}¹}² }|Q| ´ }Q}¹}², and by assumption,

p13q

`8ÿ

k“1

b

1´ }P_k}²¹ă 8,

hence

`8ÿ

k“1

1´ }P_k}²¹ ă 8and the infinite productź

k

}P_k}¹ is convergent:

p14q

`8ź

k“1

}P_k}¹ ą0.

Letn0 ďnbe a positive integers and take P “P_n andQ“

nź´1 k“n0

P_k, then }|P| ´ }P}¹}² “a

1´ }P}²¹ and }|Q| ´ }Q}¹}² ď1 ; hence by p12q we have

}P Q}¹ ě }P}¹}Q}¹´ b

1´ }P}²¹. Using the fact that }P_k}¹ď1, we obtain by induction

››

››

› źn k“n0

P_k

››

››

›1

ě źn k“n0

}P_k}¹´ ÿn k“n0

b

1´ }P_k}²¹ě źn k“n0

}P_k}¹´

`8ÿ

k“n0

b

1´ }P_k}²¹.

From p13qcombined with p14q we deduce that for large enoughn0 nÑ`8lim

źn k“n0

}P_k}¹´

`8ÿ

k“n0

b

1´ }P_k}²1 ą0, hence the sequence `śn

k“n0P_k˘

does not go to zero in L¹-norm. It follows from Bourgain criterion that the generalized Riesz product ś_`8

k“n0|P_k|² is not purely singular. Asśn0´1

k“1 |P_k|²has only countably many zeros, we con- clude that µ admits also an absolutely continuous component with respect

to the Haar measure.

5. On the Kac Central Limit Theorem

The Kac Central Limit Theorem in the context of the Bohr compactifi- cation of Ris stated and proved in [28]. For sake of completeness we prove it here using the standard probability arguments.

Definition 5.1. The real numbersω1, ω2,¨ ¨ ¨ , ωrare called rationally inde- pendent if they are linearly independent over Z , i.e. for all n1,¨ ¨ ¨, n_rPZ,

n1ω1` ¨ ¨ ¨ `n_rω_r“0ùñn1“ ¨ ¨ ¨ “n_r “0.

Theorem 5.2 (M. Kac [28]). Letω1, ω2,¨ ¨ ¨ , ω_n,¨ ¨ ¨ be rationally indepen- dent. Then, the functions cospω1tq,cospω2tq,¨ ¨ ¨,cospω_ntq,¨ ¨ ¨ are stochasti- cally independent with respect to the Haar measure of the Bohr compacti- fication of R.

Proof. It is sufficient to show that for any positive integerkand for a given positive integers l1, l2,¨ ¨ ¨, l_k, we have

$

R

ˆ

cospω1tq

˙l1ˆ

cospω2tq

˙l2

¨ ¨ ¨ ˆ

cospω_ktq

˙lk

dt

“

$

R

ˆ

cospω1tq

˙l1

dt

$

R

ˆ

cospω2tq

˙l_k

dt¨ ¨ ¨

$

R

ˆ

cospω_ktq

˙l_k

dt

Write

cospωjtq “ 1 2

ˆ

e^iωjt`e^´ωjt

˙

, j“1,2,¨ ¨ ¨ , tPR, and recall that

$

R

e^iαtdt“ lim

TÝÑ`8

1 2T

ż

T

´T

e^iαtdt“

#1, if α “0 0, if not. Hence

ˆ

cospω1tq

˙l1

¨ ¨ ¨ ˆ

cospω_ktq

˙lk

“

ź

k j“1

1 2^lj

ź

k j“1

ˆ

e^iωjt`e^´iωjt

˙lj

“

ź

k j“1

1 2^lj

ź

k j“1

ˆ

ÿ

_l

j

rj“0

ˆlj

r_j

˙

e^{ip2rj´ljqωjt}

˙ .

Whence

ź

k j“1

ˆ

cospω_jtq

˙lj

“

ź

k j“1

1 2^lj

ˆ

ÿ

_l

1,l2,¨¨¨,l_k r1,r2,¨¨¨,r_k“0

ˆl1

r1

˙ˆl2

r2

˙

¨ ¨ ¨ ˆl_k

r_k

˙

e^{ipřkj“1p2rj´ljqωjqt}

˙ .

Because of linear independence,

ÿ

k

j“1p2rj´l_jqω_j

can be zero only if 2ri “l_i, for any i“1,¨ ¨ ¨ , k,and thus it follows that

$

R

źk j“1

ˆ

cospω_jtq

˙lj

dt“

$’

&

’% źk j“1

1 2^lj

ˆl_j

lj

2

˙

, if all l_i are even

0, otherwise.

We conclude that

$

R

ź

k j“1

ˆ

cospω_jtq

˙lj

dt“

ź

k j“1

$

R

ˆ

cospω_jtq

˙lj

dt

and this finish the proof of the theorem.

In the following we recall the classical well known multidimensional Cen- tral Limit Theorem in probability theory [16, p.81] stated in the following forms

Theorem 5.3 (Multidimensional CLT Theorem). LetpZ_n,kq¹ďkďkn, ně1 be a triangle array of random variables vectors in C^dand put

Zn,k“ pZ_n,k¹ , Z_n,k² , . . . , Z_n,k^d q. Suppose that

(1) The random variables Z_n,k^j are square integrable.

(2) For each nPN, theZ_n,k,1ďkďk_n, are independents.

(3) For any εą0, lim

nÑ8 kn

ÿ

k“1

P´

}Z_n,k} ěε? k_n¯

“0.

(4) lim

nÑ8

?1 k_n

kn

ÿ

k“1

EpZ_n,kq “0.

(5) For each j, lP t1,¨ ¨ ¨ , du, (a) lim

nÑ8

1 k_n

kn

ÿ

k“1

CovpZ_n,k^j , Z_n,k^l q “γ_j,l. (b) lim

nÑ8

1 k_n

kn

ÿ

k“1

EpZ_n,k^j .Z_n,k^l q “0.

Then Γ is a hermitian non-negative definite matrix and the sequence of

random vectors ˜

?1 k_n

kn

ÿ

k“1

Z_n,k

¸

ně1

converges in distribution to the complex Gaussian measure N_Cp0,Γq onC^d. We also need the following important and classical fact from Probability Theory connected to the notion of the uniform integrability.

Definition 5.4. The sequence tXn, n ě1u of random variables is said to be uniformly integrable if and only if

cÝÑ`8lim

ż

|Xn|ąc(

ˇˇX_nˇˇdP“0 uniformly inn.

It is well-known that if sup

nPN

ˆE`ˇˇX<sub>n</sub>ˇˇ<sup>1`ε</sup>˘˙

ă `8, p15q

for someεpositive, then tXnuare uniformly integrable.

Let us mention that the convergence in distribution or probability does not in general imply that the moments converge (even if they exist). The useful condition to ensure the convergence of the moments is the uniform integrability. Indeed, we have

Theorem 5.5. If the sequence of random variables tXnu converges in dis- tribution to some random variable X and for somepą0, sup

nPN

`Ep|X_n|^pq˘

“ M ă `8, then for each răp,

nÝÑ`8lim EˆˇˇX_nˇˇ^r˙

“EˆˇˇXˇˇ^r˙

For the proof of Theorem 5.5 we refer the reader to [5, p.32-33] or [13, p.100].

Now let us state and prove the Kac Central limit Theorem.

Theorem 5.6 (Kac CLT [28]). Let pλ_nqnPN be a sequence of rationally independent real numbers. Then the functions cospλ_ntq `isinpλ_ntq, n “ 1,¨ ¨ ¨, are stochastically independent under the Haar measure of the Bohr compactification ofRand converge in distribution to the complex Gaussian measure N_Cp0,1q on C.

Proof. By Theorem 5.2 the functions cospλ_ntq `isinpλ_ntq, n “ 1,¨ ¨ ¨, are stochastically independent under the Haar measure ofbRand it is straight- forward to verify that the hypotheses of Central Limit Theorem 5.3 are satisfied. We conclude that the sequence

˜ 1

?n ÿn k“1

e^iλkt

¸

ně1

converges in distribution to the complex Gaussian measure N_Cp0,1q on C.

6. Proof of the main result (Theorem 3.2)

Using the analogous lemma of F´ejer’s Lemma [53, p.49] combined with the CLT methods introduced in [2], we shall give a direct proof of the singularity of a large class of generalized Riesz products onbR. Therefore, our strategy is slightly different from the strategy of the proofs given by many authors in the case of the torus [10], [33], [34], [2]. Indeed, the crucial argument in their proofs is to estimate the following quantity

ż

QˇˇˇˇP_mˇˇ²´1ˇˇdh.

p16q

Precisely, they showed that the weak limit point of the sequence`ˇˇ|P_m|²´1ˇˇ˘

is bounded below by a positive constant and it is well-known that this implies the singularity of the generalized Riesz products (see for instance [2] or [33]).

Let us start our proof by proving the following lemma analogous to F´ejer’s Lemma [53, p.49]

Lemma 3. With the above notations we have lim sup

mÝÑ`8

mPM

ż

bR

QˇˇP_mˇˇdh“ ˆ

ż

bR

Qdh

˙ˆ

lim sup

mÝÑ`8

mPM

ż

bR

|P_m|dh

˙ .

Proof. By our assumption the sequence `

h_m,ps_m,pm´1q˘

mPM is rationally independent. Hence, by Kac Theorem 5.2, for m ąh_nk`1, the function Q and Pm are stochastically independent. This allows us to write

ż

bR

QˇˇP_mˇˇdh“

ż

bR

Qdh

ż

bR

|P_m|dh,

which proves the lemma.

Proof of Theorem 3.2. Applying Lemma 3, we proceed to construct induc- tively the sequence n1 ă n2 ă n_k ă ¨ ¨ ¨ , such that, for any k ě 1, we

have

ż

bR k`1

ź

j“1

ˇˇP_njˇˇdhď

?51π 100

ż

bR

ź

k j“1

ˇˇP_njˇˇdh.

p17q

Indeed, by our assumption combined with Kac CLT 5.2, it follows thatpP_mq converges in distribution to the complex Gaussian measure N_Cp0,1q on C.

But, according top15q,pP_mq is uniformly integrable. Hence, from Theorem 5.5, we get

mÝÑ`8lim

ż

bR

|P_m|dh“

ż

C

ˇˇzˇˇdN_Cp0,1qpzq “

?π 2 . p18q

We remind that the density of the standard complex normal distribution N_Cp0,1q is given by

fpzq “ 1 πe^´|z|2.

Now assume that we have already construct n1 ăn2 ăn3 ă ¨ ¨ ¨ ă n_k and applyp18q withε“

?π

100 combined with Lemma 3 to get mąn_k such that

ż

bR

Q|P_m|dhď 51? π 100

ż

bR

Qdh.

Putn<sub>k`</sub>1“m. Therefore the inequalityp17qholds and by lettingkÝÑ `8 we conclude that

ż

bR

ź

k j“1

ˇˇP_njˇˇdh´´´´Ñ

kÑ`8 0.

which yields by Bourgain criterion that µ is singular with respect to Haar

measure and completes the proof.

Remark 6.1. The fundamental argument in the proof above is based on Lemma 3 and therefore strongly depended on the assumption that along subsequence the positive real numbers`

h_m,ps_m,pm´1q˘

are linearly indepen- dent over the rationals. We argue that in the general case, one may use the methods of Bourgain [10], Klemes-Reinhold [33], Klemes [33] and el Ab- dalaoui [2] to establish the singularity of a large class of generalized Riesz products on bR. In particular, the case when ppmq is bounded. In the forthcoming paper, we will show how to extend a classical results from the torus and real line setting to the generalized Riesz products on the Bohr compactification ofR.

Appendix.I. On The Flatness problem on bR.

We are concerned here with the flat polynomials issue in bR. First, we recall briefly the relevant fact on the flatness problem in the torus T.

The problem of flatness go back to Littlewood in his 1968 famous paper [37]. In that paper, Littlewood introduce two class of complex polynomials G_n and F_n where n is a positive integer. The class G_n is a class of those polynomials Ppzq “ řn

k“0a_kz^k that are unimodular, that is, |a_k| “ 1, for k“ t0,¨ ¨ ¨ , nu. F_n is the subclass ofG_nwith real coefficients, i.e.,a_k“ ˘1, fork“ t0,¨ ¨ ¨ , nu. The polynomialsP inF_nare nowadays called Littlewood polynomials. By Parseval’s formula

1 2π

ż

²π 0

ˇˇ ˇˇPpe^itq

ˇˇ ˇˇ

2

dt“n`1, for all P PG_n. Therefore, for allP PG_n,

min

|z|“1

ˇˇPpzqˇˇă?

n`1ămax

|z|“1

ˇˇPpzqˇˇ

In [37] Littlewood raised the problem of the existence of a sequence pP_nq of unimodular polynomials such that

max

tPR

ˇˇP_npe^itqˇˇ

?n`1 ´´´´Ñ_n

Ñ8 1.

Such sequence of unimodular polynomials are called ultraflat. Precisely, the usual definition of ultraflatness is given as follows