Davenport series and almost-sure convergence

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Davenport series and almost-sure convergence

Julien Brémont

To cite this version:

Julien Brémont. Davenport series and almost-sure convergence. Quarterly Journal of Mathematics,

Oxford University Press (OUP), 2011, 62 (1), pp.825-843. �hal-00794135�

Davenport series and almost-sure convergence

Julien Br´ emont

Universit´ e Paris-Est, janvier 2010

Abstract

We consider Davenport-like series with coefficients inl² and discussL²-convergence as well as almost-everywhere convergence. We give an example where both fail to hold. We next improve former sufficient conditions under which these convergences are true.

1 Introduction

LetR\Zbe the Circle andL² be the restriction of the spaceL²(R\Z→R) to odd functions.

For a real parameterλ >1/2, we introduce the map : gλ(x) = X

m≥1

sin(2πmx) m^λ .

This function is defined everywhere onR\Z. It is continuous, except at 0 when 1/2< λ≤1, and belongs toL². For real sequences (an)∈l², we consider expansions based on the dilated functions system{gλ(nx)}_n≥1 of the following form :

Xangλ(nx), (1)

where we writePfor the summationP+∞

n=1. We are interested in the questions ofL²-convergence and Lebesgue almost-everywhere (a.e) convergence of such series.

This is a natural problem which can be formulated withgλ replaced by a generalg∈L²(R\Z).

A reason for focusing on odd functions is that sin series in general better converge than cos series.

Wheng(x) = sin(2πx), theL²-convergence ofPa_ng(nx) follows from the fact that the{g(nx)}n≥1

are orthonormal. A.e-convergence in this case is the difficult theorem of Carleson [4]. For a different g such that the {g(nx)}n≥1 are complete inL², the{g(nx)}n≥1 are not orthogonal, see Bourgin and Mendel [2], and the question of L²-convergence is already not clear. The case of g = g_λ was introduced by Wintner in [17]. A special motivation comes Arithmetics and the caseλ= 1, corresponding to the first Bernoulli polynomial or “sawtooth function”{x}:=x−[x]−1/2, where [x] is the integer part ofx∈R. Indeed :

{x} = −1 π

X

m≥1

sin(2πmx)

m .

Series of the formPa_n{nx} appear since long ago in the litterature, at the interface of Number Theory and Analysis. We recommend the very detailed presentation of such series by Jaffard in [12], where they are called Davenport series, due to Davenport’s initial systematic study [5, 6]. In

AMS2000subject classifications : 11A25, 26A30, 42A16, 42A61, 60F05

Key words and phrases : Davenport series, Carlitz’s lemma,L²-convergence, almost-everywhere convergence.

this article and for simplicity we callD_λ-series a series of the form (1), the case of Davenport series corresponding toλ= 1.

Beginning with a discussion in L², Wintner [17] established that the family {gλ(nx)}n≥1 is complete inL²for anyλ >1/2. We now consider theL²-convergence ofD_λ-series with (a_n)∈l². According to work by Wintner [17] and next Hedenmalm, Lindqvist and Seip [11], the{gλ(nx)}n≥1

form a Riesz basis whenλ >1. By a “Riesz basis” we mean a complete sequence (ξ_n) in L² such that for some constantC >0 :

C⁻¹X

a²_n≤ kX

anξnk²≤CX

a²_n, ∀(an)∈l².

Here and in the whole article we denote byk kthe usualL²-norm, with scalar producth,i. Lindqvist and Seip [15] provide the inequalities :

ζ(2λ) ζ(λ)²

Xa²_n≤ kX

angλ(nx)k²≤ ζ(λ)² ζ(2λ)

Xa²_n,

where ζ(s) = P

n≥1n^−s, for s > 1, is the Riemann Zeta function. Constants are optimal. This settles the question of L²-convergence when λ >1. In this case, the a.e-convergence ofDλ-series for (an)∈l² follows from Carleson’s theorem [4] on the a.e-convergence of Fourier series. Indeed :

N

X

n=1

angλ(nx) = X

m≥1

m^−λ

N

X

n=1

ansin(2πmnx). (2)

For eachm≥1,PN

n=1a_nsin(2πmnx) converges a.e, asN →+∞, by Carleson’s theorem. Next :

|

N

X

n=1

ansin(2πmnx)| ≤ sup

K≥1

|

K

X

n=1

ansin(2πmnx)|=:M(mx).

By classical work on the maximal operator,kMkL¹(R\Z)≤CP

a²_n (cf for instance Fefferman [7]).

Thus P

m≥1m^−λM(mx) is integrable and in particular a.e finite. One can now a.e apply the Lebesgue dominated convergence theorem in (2) to conclude. Of course this argument does not work when 1/2< λ≤1. Mention also that Carleson’s theorem uses properties of the Fourier basis.

There exists orthonormal bases of L² for whichL²-convergence does not imply a.e-convergence, see Rademacher [16].

For the rest of the article we suppose that 1/2 < λ≤1. As a consequence of an analysis by Wintner [17] of some Dirichlet series associated to Dλ-series, for any 1/2 < λ ≤ 1 there exists (an)∈l²such thatPangλ(nx) isL²-divergent. In particular for 1/2< λ≤1, the{gλ(nx)}_n≥1do not form a Riesz basis ofL². We now detail known sufficient conditions forL²and a.e-convergence.

We are essentially aware of results concerning Davenport series. Wintner [17] proved thatP an{nx}

converges inL² wheneveran =O(n^−κ), withκ >1/2. Extending this result, Jaffard [12] showed that for (an) ∈l² the sum P

an{nx} converges in a Sobolev space very close to L². About a.e- convergence, Davenport in his fundamental papers [5, 6] gave non-trivial arithmetical examples where a.e-convergence is true, such as :

+∞

X

n=1

λ(n) n {nx},

+∞

X

n=1

Λ(n) n {nx},

+∞

X

n=1

µ(n)

n² {n²x}, (3)

whereλ(n), Λ(n) and µ(n) are respectively Liouville’s function, Von Mangolt’s function and Mo- bius’ function. When the a_n are slowly varying, the a.e-convergence of Pa_n{nx} follows, via an Abel transform, from estimates on P

n<N{nx}, cf Lang [14]. Jaffard [12] deduced the a.e- convergence ofPa_n{nx}, whenever a_n =O(logn)^−α and a_n+1−a_n = O(n⁻¹(logn)^−(1+α)) for someα >2. In particular, Hecke series :

Hs(x) =

+∞

X

n=1

{nx}

n^s (4)

converge a.e for Re(s) > 0, a result already shown by Hardy and Littlewood [8]. For general sequences, Hartman proved in [9] the a.e-convergence of Pan{nx} when an = O(n^−κ), with κ > 2/3. Mention finally some results going further than a.e-convergence. Using P-summation techniques, de la Bretˆeche and Tenenbaum [3] proved that (4) is convergent whens = 1 outside a set of Hausdorff dimension zero that they describe. Also the second series in (3) is everywhere convergent.

We now detail the content of the article. We discuss theL²and a.e-convergence ofDλ-series of the form (1) for general (an)∈l². We first complete the L²-divergence result of Wintner [17] by an a.e-divergence result. We next improve former sufficient conditions forL² and a.e-convergence.

Theorem 1.1

Assume that1/2< λ≤1.

i) There exists(an)∈l² such that Pangλ(nx)is simultaneouslyL²-divergent and a.e-divergent.

ii) Suppose that for someε >0 :







Pa²_nn

(1+ε)(logn)−(2λ−1)

2(1−λ) log logn <∞, when 1/2< λ <1, Pa²_n(logn)³(log logn)^2+ε<∞, whenλ= 1.

ThenP

angλ(nx) converges inL² and a.e.

In particular, the latter conditions are verified if P

a²_nn^ε<∞, for some ε >0. For example, the following series converge inL²and a.e when s >1/2 :

+∞

X

n=1

λ(n) n^s {nx},

+∞

X

n=1

Λ(n)

n^s {nx}and

+∞

X

n=1

µ(n) n^2s {n²x}.

We note that whereas Wintner’s approach is abstract we build here an explicit example. The fact that (a_n) ∈ l² does not imply the a.e-convergence ofD_λ-series is not that surprising, since this condition is not the correct one forL²-convergence. One can make the second moment explode and develop a probabilistic argument based on the Central Limit Theorem. The true question, more difficult, is whetherL²-convergence implies a.e-convergence. A weak formulation is as follows : Question. If 1/2< λ≤1 andP

k≥1

P

n≥1n^−λ|akn|²

<+∞, doesP

angλ(nx) converge a.e ? The above condition is strictly stronger than (a_n)∈l², when 1/2< λ≤1. As detailed below, it ensures theL²-convergence ofPangλ(nx) and is necessary when thean have constant sign.

We next consider three classical situations, for instance Hadamard lacunarity, where we can proveL²-convergence, but a.e-convergence only under stronger conditions. We define the support supp(n) of an integernas its set of prime divisors and write|supp(n)|for the cardinal of this set.

Theorem 1.2

Suppose that1/2< λ≤1.

i) Let(nk)checknk+1/nk ≥c, wherec >1. ThenP

akgλ(nkx)converges inL²whenever(ak)∈l² and more precisely :

C₁ X

a²_k ≤ kX

a_kg_λ(n_kx)k²≤C₂ X a²_k,

where :

C1= (1−1/e)ζ(4λ) 2

2λ−1 2λ

ln(c^2λ−1) 1 + ln(c^2λ−1)

²

andC2= ζ(2λ) 2

c^λ+ 1 c^λ−1

.

If the stronger conditionP

a²_k(logk)²<∞ is verified, thenP

akgλ(nkx)is also a.e-convergent.

ii) If(a_n)∈ l² and {|supp(n)|, a_n 6= 0} is finite, then Pa_ng_λ(nx) converges in L². In fact, for N ≥1 there existsC(λ, N)>0such that for (an)∈l² withan= 0 if|supp(n)|> N, then :

C⁻¹(λ, N) X

a²_n≤ kX

angλ(nx)k²≤C(λ, N) X a²_n. If moreoverPa²_n(logn)²<∞, thenPangλ(nx)is also a.e-convergent.

iii) Let a_n = O(b_n), where (b_n)_n≥1 ∈ l²∩(R+)^N satisfies b_nm = b_nb_m whenever n and m are relatively prime. ThenPa_ng_λ(nx)converges inL².

A word on the method. The study of the convergence of Davenport series often starts with trying to writeP

an{nx}as a Fourier seriesP

cmsin 2πmx. It was indeed remarked by Davenport [5] that formally the (cm) are explicitly given in terms of the (an) and vice-versa. An alternative approach, developed here, when consideringL²-convergence is to orthonormalize the{gλ(nx)}n≥1. The Gram-Schmidt orthonormalisation procedure is explicit and a consequence of Carlitz’s lemma on the reduction of quadratic forms. We provide details for simplicity. This furnishes a rather simple characterization ofL²-convergence. Theorem 1.2 follows via more or less standard compu- tations. Concerning a.e-convergence, the orthonormalisation approach allows to adapt a technique of Rademacher [16] initially developed for the pointwise convergence of series built with general orthonormal systems.

A few notations. We writei∧j andi∨j respectively for the greatest common divisor and the smallest common multiple of integersiandj. The set of primes isP ={pn, n≥1}.

2 Orthonormalization

Recall that 1/2 < λ ≤1. We first study correlations. The following computation is already contained in Lindqvist and Seip [15].

Lemma 2.1

Let i, j≥1. Thenhg_λ(i.), g_λ(j.)i= ζ(2λ) 2

i∧j i∨j

^λ . Proof of the lemma :

Leti⁰ = i/i∧j, j⁰ = j/i∧j. Since Lebesgue measure on R\Z is invariant by x7−→ px for any integerp, we havehgλ(i.), gλ(j.)i=hgλ(i⁰.), gλ(j⁰.)i. Using the Fourier expansion ofgλ :

hgλ(i⁰.), gλ(j⁰.)i = X

k,l≥1

Z 1 0

sin(2πki⁰x) k^λ

sin(2πlj⁰x)

l^λ dx= 1 2

X

m≥1

1

(m²i⁰j⁰)^λ =ζ(2λ)

2 (i⁰j⁰)^−λ, since a relationki⁰ =lj⁰ reduces tok=j⁰mand l=i⁰m for some integerm. This concludes the

proof of the lemma.

Remark. — The correlations being positive, ifP

bngλ(nx) isL²-convergent with (bn)∈(R+)^Nand ifan =O(bn), thenP

angλ(nx) also converges inL².

We turn to the orthonormalization of the {g_λ(nx)}_n≥1. The following proposition is an appli- cation of Carlitz’s lemma, cf for instance Haukkanen, Wang and Sillanp [10].

Recall that the M¨obius functionµon the integers is defined byµ(1) = 1,µ(p_i1· · ·p_ik) = (−1)^k andµ(n) = 0 if n is not square-free. Iff and g are real maps defined on the integers related by f(n) =P

k|ng(k), then (M¨obius inversion formula) we haveg(n) =P

k|nµ(n/k)f(k).

Proposition 2.2 i) Let f_n,λ(x) =n^−λX

k|n

k^λµ(n/k)g_λ(kx),n≥1. Then{f_n,λ}_n≥1 is an orthogonal family with : kfn,λk²= ζ(2λ)

2 Y

p|n,p∈P

1−p^−2λ

∈ 1

2,ζ(2λ) 2

.

The{fn,λ}n≥1 form an orthogonal Riesz basis of L², with : 2ζ(2λ)⁻¹X

n≥1

hfn,λ, hi²≤ khk²≤2X

n≥1

hfn,λ, hi²,∀h∈L².

ii) An equality Pn

i=1aigλ(i.) = Pn

i=1bifi,λ holds if and only if bi = P^[n/i]

k=1 k^−λaki, 1 ≤ i ≤n.

These equalities are reversed intoai=P[n/i]

k=1 k^−λµ(k)bki, 1≤i≤n.

Proof of the proposition :

Letn≥1. We introducen-square matricesDandT, whereDis diagonal andTis upper-triangular.

SetT = (t_ij) witht_ij =j^−λ1_i|j and write D= diag(d_i), where the (d_i) are defined below. First : (^tT DT)ij = X

1≤k≤n

tkidktkj= (ij)^−λ X

k|i∧j

dk.

We chooseDso thatP

k|mdk= (ζ(2λ)/2)m^2λ, which by the M¨obius inversion formula corresponds to settingd_k= (ζ(2λ)/2)P

l|kµ(k/l)l^2λ. Lemma 2.1 gives (^tT DT)_ij =hg_λ(i.), g_λ(j.)i.

Next, the inverse ofT is given byT_ij⁻¹= 1_i|ji^λµ(j/i), since for anyi≤j :

n

X

k=1

1_i|ki^λµ(k/i)j^−λ1_k|j= 1_i|ji^λj^−λX

l|j/i

µ(l) = 1i=j,

using thatP

k|mµ(k) = 0, ifm≥2. Observe thatfi,λ=i^−λP

1≤k≤n(^tT⁻¹)ikgλ(k.), for 1≤i≤n.

The{fi,λ}_i≥1 are therefore orthogonal in L², with kfi,λk² =dii^−2λ. They also form a complete family, since it is the case for the{gλ(nx)}, cf Wintner [17]. Decomposingi=p^α_l¹

1 · · ·p^α_l^k

k in prime factors, we have :

kfi,λk²=ζ(2λ) 2

X

d|i

d^−2λµ(d) =ζ(2λ) 2

k

Y

j=1

X

d|p^αj_lj

d^−2λµ(d) = ζ(2λ) 2

Y

p|i, p∈P

(1−p^−2λ).

Finally :

n

X

i=1

aigλ(i.) =

n

X

i=1

ai n

X

k=1

(^tT)ikk^λfk,λ=

n

X

i=1

ai n

X

k=1

i^−λ1_k|ik^λfk,λ=

n

X

k=1

fk,λ [n/k]

X

i=1

i^−λaki.

The reversed formula is proved in a similar way.

We deduce the following characterization ofL²-convergence.

Corollary 2.3

i) The seriesPa_ng_λ(nx)converges inL²if and only if the numerical seriesP

k≥1k^−λa_kiconverge for alli≥1, together with the uniformity condition :

X

i≥1



 X

k>[n/i]

k^−λa_ki





2

→0, asn→ ∞.

ii) A sufficient condition forPangλ(nx)to beL²-convergent is :

X

i≥1



 X

k≥1

k^−λ|a_ki|





2

<+∞.

This condition is necessary when(a_n)∈(R+)^N. Proof of the corollary :

If Pa_ng_λ(nx) converges in L², by proposition 2.2 the component P[n/i]

k=1 k^−λa_ki with respect to each f_i,λ converges as n → +∞. The L²-limit then has to be P

i≥1f_i,λ(P

k≥1k^−λa_ki). The uniformity condition is a consequence from the fact that the norm offi,λbelongs to (1/2, ζ(2λ)/2).

The first assertion of the second item is an application of the Lebesgue Dominated Convergence Theorem, whereas the second one follows from the first item.

Remark. — Corollary 2.3 can also be obtained when considering directly the Fourier expansion ofP

angλ(nx) given bygλ. In the sequel, the orthonormalization point of view has the practical advantage to keep finite all partial sums.

3 A l

-example of a L

-divergent and a.e-divergent series

We prove theorem 1.1i), using that for 1/2< λ≤1 the seriesP

p∈Pp^−λ is divergent. For each integerK≥1, we choose a finite setPK ={pj,K}1≤j≤l_K of consecutive primes satisfying :

l_K

X

j=1

(pj,K)^−λ≥K. (5)

We fixmK ≥2 so that

m_K−1 mK

lK

≥1/2. Introduce sets :











F_1,K=n

p^u_1,K¹ · · ·p^u_l^lK

K,K ,1≤u₁,· · ·, u_lK ≤m_Ko F_1,K⁰ =n

p^u_1,K¹ · · ·p^u_l^lK

K,K ,1≤u₁,· · ·, u_lK ≤m_K−1o .

We have|F_1,K|= (m_K)^lK and|F_1,K⁰ |= (m_K−1)^lK. Letq_1,K= 1 and take next infinitely many primesq_2,K<· · ·< q_n,K<· · ·, larger thanp_lK,K and subject to the condition :

(p_1,K· · ·p_lK,K)^mK

1 + (m_K)^lK/2 K

^+∞

X

r=2

(q_r−1,K)^r−1 qr,K

≤ 1

K. (6)

Define the random variable :

X_1,K= 1 K|F1,K|^1/2

X

n∈F1,K

g_λ(nx).

It has zero mean and belongs toL². We write σ_K² =R1

0(X_1,K)²(x)dxfor its variance and choose an integerT_K≥K so that :

Z 1 0

(X_1,K)²(x)1_{|X1,K|>(TK)1/12σK} dx≤ 1

K. (7)

We define another collection of sets :







F_2,K=q_2,KF_1,K F_2,K⁰ =q2,KF_1,K⁰

· · ·







F_TK,K=q_TK,KF_1,K F_T⁰_K,K=qT_K,KF_1,K⁰ . Grouping sets, we define :

E_K =

TK

[

r=1

F_r,K andE_K⁰ =

TK

[

r=1

F_r,K⁰ .

We have|EK|=TK|F1,K|and|E_K⁰ |=TK|F_1,K⁰ |. In particular :

|E_K⁰ |

|EK| =|F_1,K⁰ |

|F1,K| =

m_K−1 mK

^lK

≥ 1

2. (8)

When considering the next integer (ieK+ 1) we start withp_1,K+1> q_TK,K(p_1,K· · ·p_lK,K)^mK. Observe that all the (Fr,K)_{K≥1,1≤r≤TK}, are pairwise disjoint and in particular the (EK)_K≥1, which furthermore are consecutive. We finally set :

an=







1

K|EK|^1/2 , whenn∈EK, for some K≥1,

0 , otherwise.

This completes the definition of the sequence (a_n). FormallyPa_ng_λ(nx) =P

K≥1Z_K, with : Z_K= 1

√T_K

TK

X

r=1

X_r,K(x) andX_r,K(x) = 1 K|F1,K|^1/2

X

n∈Fr,K

g_λ(nx). (9)

From the previous construction, observe that a partial sumPN

K=1Z_K(x) corresponds to a partial sum ofPa_ng_λ(nx). We now proceed to verifications.

i) The sequence (an) belongs tol². Indeed : X

n≥1

a²_n= X

K≥1

X

n∈EK

1

K²|EK| = X

K≥1

1 K² <∞.

ii) The seriesP

angλ(nx) isL²-divergent. Indeed, using (5) and (8) :

X

n≥1



 X

k≥1

k^−λakn





2

≥ X

K≥1

X

n∈E_K⁰



 X

k≥1

k^−λakn





2

≥ X

K≥1

X

n∈E_K⁰

X

k∈PK

k^−λakn

!²

≥ X

K≥1

X

n∈E_K⁰

1 K²|EK|

X

k∈PK

k^−λ

!²

≥ X

K≥1

|E_K⁰ | K²

K²|EK| ≥ X

K≥1

1

2 = +∞.

Since thea_n are positive, the conclusion comes from corollary 2.3.

iii) The seriesPa_ng_λ(nx) is a.e-divergent. This requires longer computations. For a fixedK≥1, all (X_r,K)_1≤r≤TK have the same law, due to the invariance of Lebesgue measure on R\Z under multiplication by an integer. They do not form a stationary process, but are nearly independent.

Under our hypotheses, it is routine to check that the law of (σ²_KTK)^−1/2PTK

r=1Xr,K is asymptoti- cally normal.

In a first step, we compute the variance σ_K² and verify that it grows rapidly to infinity, as suggested byii). Via lemma 2.1, we have :

σ²_K= Var(X1,K) = ζ(2λ) 2

1 K²(m_K)^lK

X

n,m∈F_1,K

n∧m n∨m

^λ

= ζ(2λ) 2

1 K²(mK)^lK

X

1≤a_j,b_j≤m_K,1≤j≤l_K l_K

Y

j=1

(pj,K)^{−λ|aj−bj|}

= ζ(2λ) 2

1 K²(m_K)^lK

l_K

Y

j=1





m_K

X

a,b=1

(pj,K)^−λ|a−b|





= ζ(2λ) 2

1 K²(mK)^lK

lK

Y

j=1

mK+ 2(pj,K)^{−λ(mK−1)}

mK−1

X

k=1

k(pj,K)^λ(k−1)

! .

Forx >1, we havePn−1

k=1kx^k−1 = ((n−1)xⁿ−nxⁿ⁻¹+ 1)/(x−1)² =nxⁿ⁻²(1 +o(1)), whenx andnare large. Inserting this in the previous calculations, we obtain, with a uniformo(1) :

σ²_K = ζ(2λ) 2

1 K²(mK)^lK

l_K

Y

j=1

mK+ 2mK(pj,K)^−λ(1 +o(1))

= ζ(2λ) 2

1

K²e^PlKj=1log(^1+2(pj,K)^−λ(1+o(1)))

= ζ(2λ) 2

1 K²e²

“PlK

j=1(pj,K)^−λ” (1+o(1))

≥e^K, (10)

for largeK, using (5).

We now establish the convergence : (σ_K²T_K)^−1/2

TK

X

r=1

X_r,K→ N(0,1), in law. (11)

We write E for the expectation under Lebesgue measure on R\Z. Set Yr,K = (σ_K²TK)^−1/2Xr,K

andSN =PN

r=1Yr,K. For 1≤r≤TK, introduce the finite partitions : Fr={[k/qr,K,(k+ 1)/qr,K), 0≤k < qr,K}.

EachY_r,K being (1/q_r,K)-periodic, for a bounded measurablef :

E(f(Y_r,K)) =E(f(Y_r,K)|Fr). (12) Fort∈Rand 2≤N≤TK, we have :

E(e^itSN) = E(e^itSN−1e^itYN,K)

= E(E(e^itSN−1|FN)e^itYN,K) +E((e^itSN−1−E(e^itSN−1|FN))e^itYN,K)

= A + B.

First of all, taking conditional expectation and using (12) :

A=E(E(e^itSN−1|FN)E(e^itYN,K|FN)) = E(E(e^itSN−1|FN)E(e^itYN,K))

= E(e^itSN−1)E(e^itYN,K). (13) Next :

|B| ≤E(|e^itSN−1−E(e^itSN−1|F_N)|). (14) The mapx7−→e^itxis|t|-Lipschitz. On each piece ofFN which contains no discontinuity ofS_N−1, when counting the oscillation we have :

|e^itSN−1−E(e^itSN−1|FN)| ≤ |t|

qN,K

(σ²_KTK)^−1/2 (K(m_K)^lK))^1/2

N−1

X

r=1

(mK)^lK(p1,K· · ·pl_K,K)^mKqr,K

≤ |t|

(mK)^lK/2

K (p1,K· · ·pl_K,K)^mK

(N−1)qN−1,K

q_N,K , (15)

sinceT_K ≥K and σ_K ≥1 for large K, by (10). Next, S_N−1 is continuous on the interior of each segment of the partition whose step⁻¹ is q1,K· · ·qN−1,K(p1,K· · ·pl_K,K)^mK. The total measure of the pieces ofFN which may contain a discontinuity ofSN−1 is bounded from above by :

q_1,K· · ·q_N−1,K(p_1,K· · ·p_lK,K)^mK 1

q_N,K ≤(p_1,K· · ·p_lK,K)^mK(qN−1,K)^N−1

q_N,K . (16)

From (14), (15) and (16), we deduce that :

|B| ≤ 2(1 +|t|)(p1,K· · ·pl_K,K)^mK

1 +(mK)^lK/2 K

(qN−1,K)^N−1

q_N,K . (17)

Using that for all 1≤N ≤TK, we have |E(e^itYN,K)| ≤1, when iterating the procedure with (13) and (14), we obtain via (6) :

|E(e^itSTK)−

TK

Y

r=1

E(e^itYr,K)| ≤ 2(1 +|t|)(p1,K· · ·p_lK,K)^mK

1 +(m_K)^lK/2 K

^TK

X

r=2

(q_r−1,K)^r−1 qr,K

≤ 2(1 +|t|)1

K. (18)

As a consequence of (18), in order to show (11) we only need to focus on :

T_K

Y

r=1

E(e^itYr,K) =E(e^itY1,K)^TK. (19) We now use the fact that for allt∈R:

|e^it−(1 +it−t²/2)| ≤min{|t|³/6,|t|²}, (20) which comes frome^it−(1 +it−t²/2) = i³/2Rt

0(t−s)²e^is ds=i²Rt

0(t−s)(e^is−1) ds. Via (20) and the property thatX_1,K has zero mean, we now deduce the following inequalities :

E e^itY1,K

−

1− t² 2T_K

≤ E

e^itY1,K−

1 +itY1,K−t² 2Y_1,K²

≤ E

e^itY1,K −

1 +itY_1,K−t² 2Y_1,K²

≤ E min{|tY1,K|³/6,|tY1,K|²} .

Withε= (T_K)^−5/12and using (7), as well as T_K ≥K andσ_K ≥1 for largeK:

E e^itY1,K

−

1− t² 2T_K

≤ |t|³

6 E |Y1,K|³1_|Y1,K|≤ε

+|t|²E |Y1,K|²1_|Y1,K|>ε

≤ |t|³

6(TK)^5/4 + |t|² σ²_KTK

E

|X1,K|²1_|X1,K|>(TK)1/12σ_K

≤ 1 TK

|t|³

6(TK)^1/4 + |t|² Kσ²_K

≤ 1 TK

|t|³

6K^1/4 +|t|² K

. (21)

SinceT_K →+∞, as K →+∞, we deduce from (18), (19) and (21) that E(e^itSTK)→e^−t2/2, as K→+∞, for allt∈R. This proves (11).

To conclude, for allL≥1 we chooseKL≥Lso that : P(|ST_KL| ≤1/L²)≤2

Z

|t|≤1/L²

dN(0,1)(t) =:δL. Clearly P

L≥1δL <∞, so by Borel-Cantelli’s lemma, for a.e x when L is large enough we have

|ST_KL| ≥1/L². For such ax, using (10) and whenLis large enough :

|ZK_L|=

1 pTK_L

T_KL

X

r=1

Xr,K_L(x)

=σK_L|ST_KL| ≥ e^KL/2

L² ≥ e^L/2 L² .

Since partial sumsPN

K=1ZK(x) are partial sums ofPangλ(nx), this preventsPangλ(nx) from converging atx. This completes the proof of itemi) of theorem 1.1.

4 Sufficient conditions for L

and a.e-convergence

We take a finite sequence (a_n) and writePa_ng_λ(nx) =Pb_nf_n,λ(x), where (b_n) is also finite.

By proposition 2.2 :

kX

a_ng_λ(nx)k²=kX

b_nf_n,λk²≤ ζ(2λ) 2

Xb²_n ≤ζ(2λ) 2

X

n≥1



 X

k≥1

k^−λ|a_kn|





2

.

Set ψλ(k) = k^1−λ(logk)² if 1/2 < λ < 1 and ψ1(k) = logk(log logk)^1+ε, for some ε > 0. For simplicity we write log(x) for max{1,log(x)}. Using Cauchy-Schwarz’s inequality :

X

n≥1



 X

k≥1

k^−λ|akn|





2

= X

k,k⁰≥1

(kk⁰)^−λX

n≥1

|ankank⁰| ≤ X

k,k⁰≥1

(kk⁰)^−λ



 X

n≥1

a²_nk





1/2

 X

n≥1

a²_nk0





1/2

≤





 X

k≥1

k^−λ



 X

n≥1

a²_nk





1/2





2

≤



 X

k≥1

k^−λ 1 ψλ(k)







 X

k≥1

k^−λψλ(k)X

n≥1

a²_nk





≤ C_εX

n≥1

a²_nX

k|n

k^−λψ_λ(k). (22)

We first consider the case 1/2< λ <1. Remark that 0<2λ−1<1. Using a classical upper-bound forP

k|nk^2λ−1, cf Kr¨atzel [13], we have for anyδ >0 : X

k|n

k^−λψ_λ(k) = X

k|n

k^1−2λ(logk)²≤(logn)²X

k|n

k^1−2λ≤(logn)²n^1−2λX

k|n

k^2λ−1

≤ (logn)²n^1−2λCδn^2λ−1e^{(1+δ)(logn)1}

−(2λ−1)

(1−(2λ−1)) log logn ≤C_δ⁰n^{(1+2δ)(logn)1}

−2λ 2(1−λ) log logn .

In the situation whenλ= 1, we have : X

k|n

k⁻¹ψ1(k)≤ψ1(n)X

k|n

k⁻¹=ψ1(n)n⁻¹X

k|n

k.

We use this time the inequalityP

k|nk≤Cnlog logn, see again [13]. As a result, for any ε >0 there is a constantCε>0 such that for any sequence (an) :







kPa_ng_λ(nx)k²≤C_εPa²_nn^(1+ε)(logn)

−(2λ−1)

2(1−λ) log logn , when 1/2< λ <1, kP

an{nx}k²≤CεP

a²_nlogn(log logn)^2+ε, whenλ= 1.

(23)

These properties imply theL²-convergence ofDλ-series under the assumptions of theorem 1.1.

We turn to the question of the a.e-convergence ofD_λ-series. The second item of theorem 1.1 is a consequence of inequalities (23) and of the following proposition. The latter is an adaptation of a method due to Rademacher [16] for the study of series based on a general orthonormal family.

Proposition 4.1

Let (an)_n≥1 and(ϕ(n))_n≥1 be such thatPa²_nϕ(n)(logn)²<∞ and for anyM ≤N :

N

X

n=M

angλ(nx)

2

≤

N

X

n=M

a²_nϕ(n).

ThenP

angλ(nx) converges a.e.

Proof of the proposition :

We can suppose that log is the logarithm in base 2. LetS(n)(x) =P

1≤k≤nakgλ(kx). Form < n, introduce the notations :

S(m, n)(x) = X

m≤k<n

a_kg_λ(nx) andσ_l(m, n) = X

m≤k<n

a²_kϕ(k)(logk)^l, forl∈ {0,1,2}.

Step 1. We show that (S(2ⁿ)(x)) converges for a.ex. Let 0< N < n. We have :

Z 1 0

X

N≤r<n

S(2^r,2ⁿ)²(x)dx ≤ X

N≤r<n

σ0(2^r,2ⁿ) = X

N≤r<n n−1

X

s=r

σ0(2^s,2^s+1)

≤

n−1

X

s=N

(s−N+ 1)σ0(2^s,2^s+1)

≤

n−1

X

s=N

sσ0(2^s,2^s+1)≤σ1(2^N,2ⁿ).

By Markov’s inequality,P

N≤r<nS(2^r,2ⁿ)²(x)≤σ1(2^N,2ⁿ)^2/3for allxin a Borel setEN,nwith : λ(EN,n)≥1−σ1(2^N,2ⁿ)^1/3≥1−σ1(2^N,∞)^1/3.

In particular, forx∈EN,nand allN ≤r < n, we haveS(2^r,2ⁿ)(x)≤σ1(2^N,2ⁿ)^1/3. Define a set E_N,n⁰ by the condition that for all N≤r≤r⁰ < n:

S(2^r,2^r0)(x)≤2σ1(2^N,∞)^1/3.

SinceE_N,n⊂E_N,n⁰ , we haveλ(E_N,n⁰ )≥1−σ₁(2^N,∞)^1/3.FixingN, theE_N,n⁰ are monotonic inn.

The setD_N defined by the condition that for allN ≤r ≤r⁰, S(2^r,2^r0)(x)≤2σ₁(2^N,∞)^1/3 has therefore a measure λ(DN) ≥ 1−σ1(2^N,∞)^1/3. Since λ(DN) → 1, as N → +∞, we deduce that λ(lim supDN) = 1. If x∈lim supDN, the sequence (S(2ⁿ)(x)) clearly satisfies the Cauchy criterion, so converges. This concludesstep 1.

Step 2. To complete the proof, we show that a.e sup₂r<n<2^r+1|S(2^r, n)(x)| →0, as r→+∞. Let 2^r< n <2^r+1 and decomposenin base 2 :

n= 2^r+

r

X

l=1

θl2^r−l, with θl∈ {0,1}.

Then :

S(2^r, n) =

r

X

l=1

S 2^r+

l−1

X

m=1

θm2^r−m,2^r+

l

X

m=1

θm2^r−m

! .

By convexity :

S(2^r, n)² ≤ r

r

X

l=1

S 2^r+

l−1

X

m=1

θm2^r−m,2^r+

l

X

m=1

θm2^r−m

!²

≤ r

r

X

l=1 2^r−l−1

X

h=0

S 2^r+h2^l,2^r+h2^l+ 2^l−12

=:T(r).

The quantityT(r) is independent on 2^r< n <2^r+1. Next :

Z 1 0

T(r)(x)dx = r

r

X

l=1 2^r−l−1

X

h=0

Z 1 0

S 2^r+h2^l,2^r+h2^l+ 2^l−12 (x)dx

≤ r

r

X

l=1 2^r−l−1

X

h=0

σ0(2^r+h2^l,2^r+h2^l+ 2^l−1)

≤ r

r

X

l=1

σ0(2^r,2^r+1) =r²σ0(2^r,2^r+1)≤σ2(2^r,2^r+1).

FixN and letr ≥ N. By Markov’s inequality, for xin a Borel setF_r(N) of Lebesgue measure λ(Fr(N))≥1−σ2(2^r,2^r+1)/σ2(2^N,∞)^2/3, we have :

sup

2^r<n<2^r+1

S(2^r, n)²(x)≤T(r)≤σ2(2^N,∞)^2/3. LetG_N =∩_r≥NF_r(N). Thenλ(G_N)≥1−P

r≥Nσ₂(2^r,2^r+1)/σ₂(2^N,∞)^2/3= 1−σ₂(2^N,∞)^1/3. Forx∈G_N :

∀r≥N, sup

2^r<n<2^r+1

|S(2^r, n)(x)| ≤σ2(2^N,∞)^1/3.

Asλ(G_N)→1, we getλ(lim supG_N) = 1. Ifx∈lim supG_N, then sup₂r<n<2^r+1|S(2^r, n)(x)|tends to 0, asr→ ∞. This concludes step 2 and the proof of the proposition.

5 Particular classes where L

-convergence is true

We consider the proof of theorem 1.2.

5.1 Proof of i)

Let (nk) be lacunary in the sense that nk+1/nk ≥ c > 1 and (ak) ∈ l². We first consider the upper bound. We can assume the sequence (ak) finite. By lemma 2.1, the L²-norm of (2/ζ(2λ))^1/2P

akgλ(nkx) is given by : X

k,l≥1

a_ka_l

n_k∧n_l nk∨nl

^λ

=X

a²_k+ 2X

k<l

a_ka_l(n_k∧n_l)^2λ (nknl)^λ . Using Cauchy-Schwarz’s inequality, the second term is bounded by :

X

k<l

|akal|(nk∧nl)^2λ (nknl)^λ ≤X

k<l

|akal| n^2λ_k

(nknl)^λ ≤ X

k<l

|akal|c^−λ(l−k)≤X

k≥1

|ak|X

l≥1

c^−λl|ak+l|

≤



 X

n≥1

a²_k





1/2



 X

k≥1



 X

l≥1

c^−λl|ak+l|





2





1/2

.

Next, still via Cauchy-Schwarz’s inequality :

X

k≥1



 X

l≥1

c^−λl|ak+l|





2

= X

l,l⁰≥1

c^−λ(l+l0)X

k≥1

|ak+la_k+l⁰|

≤ X

l,l⁰≥1

c^−λ(l+l0)



 X

k≥1

a²_k+l





1/2

 X

k≥1

a²_k+l0





1/2

≤





 X

l≥1

c^−λl



 X

k≥1

a²_k+l





1/2





2

≤ 1

(c^λ−1)² X

k≥1

a²_k.

As a consequence :

X

k<l

|a_ka_l|(nk∧nl)^2λ

(n_kn_l)^λ ≤ 1 (c^λ−1)

X

k≥1

a²_k.

Since 1 + 2/(c^λ−1) = (c^λ+ 1)/(c^λ−1), this completes the proof of the upper-bound.

For the lower bound, one can also suppose that (ak) is finite. We haveP

akgλ(nkx) =P bkfk,λ, where (b_k) is also finite. By proposition 2.2 :

kX

akgλ(nkx)k²=kX

bkfk,λk²≥1 2

Xb²_k.

Fixing 0< ε <1−(2λ)⁻¹, giving 2λ(1−ε)>1 :

Xa²_k=X

k≥1



 X

l≥1

l^−λµ(l)bln_k





2

≤ X

k≥1



 X

l≥1

l^{−2λ(1−ε)}µ(l)²







 X

l≥1

l^−2λεb²_lnk





≤ Y

i≥1

1 +p^{−2λ(1−ε)}_i



 X

l≥1

b²_l X

k,n_k|l

n_k l

^2λε





≤ Y

i≥1

1−p^{−4λ(1−ε)}_i 1−p^{−2λ(1−ε)}_i

!

 X

m≥0

c^−2λεm



 X

l≥1

b²_l

≤ ζ(2λ(1−ε))

ζ(4λ(1−ε))(1−c^−2λε)⁻¹X

l≥1

b²_l.

To complete the proof, we first use thatζ(4λ(1−ε))≥ζ(4λ) andζ(2λ(1−ε))≤1+1/(2λ(1−ε)−1).

Setε=ρ(1−1/(2λ)), with 0< ρ <1. We have :

1 + 1

2λ(1−ε)−1

1

1−c^−2λε ≤ 2λ 2λ−1

1

(1−ρ)(1−c^{−(2λ−1)ρ}).

Minimizing inρ, we takeρ= 1/(1 + lnc^2λ−1). We finally use the inequality 1−e^−x≥(1−1/e)x, for 0≤x≤1, giving (1−c^{−(2λ−1)ρ})≥(1−1/e)(1−ρ).

Concerning a.e-convergence, we can now apply proposition 4.1 with ϕ= 1.