L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Stéphane SEURET & Adrián UBIS

LocalL²-regularity of Riemann’s Fourier series Tome 67, n^o5 (2017), p. 2237-2264.

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LOCAL L

-REGULARITY OF RIEMANN’S FOURIER SERIES

by Stéphane SEURET & Adrián UBIS (*)

Abstract. — We are interested in the convergence and the local regularity of the lacunary Fourier series Fs(x) = P^+∞

n=1 e^2iπn2x

n^s . In the 1850’s, Riemann introduced the seriesF2 as a possible example of nowhere differentiable function, and the study of this function has drawn the interest of many mathematicians since then. We focus on the case when 1/2< s61, and we prove thatFs(x) converges whenx satisfies a Diophantine condition. We also study theL²- local regularity ofFs, proving that the localL²-norms ofFs around a pointxbehave differently around differentx, according again to Diophantine conditions onx.

Résumé. — Dans cet article, nous nous intéressons aux propriétés de conver- gence et de régularité locale des séries de Fourier lacunairesFs(x) =P+∞

n=1 e^2iπn2x

n^s . Dans les années 1850, Riemann avait proposé la sérieF2comme exemple possible de fonction continue nulle part dérivable. La non-dérivabilité deF2 et plus géné- ralement sa régularité locale ont depuis lors été étudiées par de nombreux mathé- maticiens, soulevant des questions d’analyse harmonique, d’analyse complexe et d’approximation diophantienne. Nous considérons le cas 1/2< s61, et trouvons un critère diophantien sur x ∈ Rpour la convergence de Fs(x). Nous étudions également la régularité locale deFs, en démontrant que les L²-exposants de Fs

dépendent de conditions diophantiennes surx. Les preuves utilisent des estimées locales sur la normeL²des sommes partielles deFs.

1. Introduction

Riemann introduced in 1857 the Fourier series R(x) =

+∞

X

n=1

sin(2πn²x) n²

Keywords: Fourier series, Diophantine approximation, local regularity, Hausdorff dimension.

2010Mathematics Subject Classification:42A20, 11K60, 28C15, 28A78.

(*) The first author’s research was partially supported by the ANR project MUTADIS, ANR-11-JS01-0009.

as a possible example of continuous but nowhere differentiable function.

Though it is not the case (R is differentiable at rationals p/q where p andqare both odd [5]), the study of this function has drawn the interest of many mathematicians, mainly because of its connections with several domains: complex analysis, harmonic analysis, Diophantine approximation, and dynamical systems [4, 5, 6, 7, 8, 9] and more recently [2, 3, 11].

In this article, we study the local regularity of the series

(1.1) Fs(x) =

+∞

X

n=1

e^2iπn2x n^s

whens∈(1/2,1]. In this case, the series converges inL²([0,1]); neverthe- less several questions considering its local behavior arise. First it does not converge everywhere, hence one needs to characterize its set of convergence points; this question was studied in [11], and we will first continue that work by finding slightly more precise conditions to decide the convergence.

Then, if one wants to characterize the local regularity of a (real) function, one classically studies the pointwise Hölder exponent defined for a locally bounded function f : R→R at a point xby using the functional spaces C^α(x):f ∈C^α(x) when there exist a constantC and a polynomialP with degree less thanbαcsuch that, locally aroundx(i.e. for smallH), one has

(f(·)−P(· −x))1^B(x,H)k∞

:= sup(|f(y)−P(y−x)|:y∈B(x, H))6CH^α, where B(x, H) = {y ∈ R: |y−x| 6H}. Unfortunately these spaces are not appropriate for our context since F_s is nowhere locally bounded (for instance, it diverges at every irreducible rationalp/qsuch thatq6= 2×odd).

Following Calderon and Zygmund in their study of local behaviors of solutions of elliptic PDE’s [1], it is natural to introduce in this case the pointwiseL²-exponent defined as follows.

Definition 1.1. — Let f : R→R be a function belonging to L²(R), α>0andx∈R. The functionf is said to belong toC₂^α(x)if there exist a constantCand a polynomialP with degree less thanbαcsuch that, locally aroundx(i.e. for smallH >0), one has

1 H

Z

B(x,H)

f(h)−P(h−x

|²dh

!^1/2

6CH^α.

Then, the pointwiseL²-exponent of f atxis αf(x) = sup

α∈R:f ∈C₂^α(x) .

This definition makes sense for the series F_s when s ∈ (1/2,1], and is based on a natural generalization of the spacesC^α(x) by replacing theL^∞ norm by the L² norm. The pointwise L²-exponent has been studied for instance in [10], and is always greater than−1/2 as soon asf ∈L².

Our goal is to perform the multifractal analysis of the seriesFs. In other words, we aim at computing the Hausdorff dimension, denoted by dim in the following, of the level sets of the pointwiseL²-exponents.

Definition 1.2. — Let f : R→R be a function belonging to L²(R).

TheL²-multifractal spectrum df :R⁺∪ {+∞} →R⁺∪ {−∞}of f is the mapping

df(α) := dimEf(α), where the level setEf(α)is

E_f(α) :={x∈R:α_f(x) =α}.

By convention one setsd_f(α) =−∞ifE_f(α) =∅.

Performing the multifractal analysis consists in computing itsL²-multi- fractal spectrum. This provides us with a very precise description of the distribution of the localL²-singularities off. In order to state our result, we need to introduce some notations.

Definition 1.3. — Let x be an irrational number, with convergents (pj/qj)j>1. Let us define

(1.2) x−pj

q_j =hj, |hj|=q_j^−rj

with26r_j<∞. Then the approximation rate ofxis defined by rodd(x) = lim{rj:qj 6= 2∗odd}.

This definition always makes sense because if qj is even, then qj+1 and q_j−1 must be odd (so cannot be equal to 2∗odd). Thus, we always have 2 6 rodd(x) 6 +∞. It is classical that one can compute the Hausdorff dimension of the set of points with the Hausdorff dimension of the points xwith a given approximation rater>2:

(1.3) for allr>2, dim{x∈R:rodd(x) =r}= 2 r.

Whens >1, the seriesFsconverges, and the multifractal spectrum ofFs

was computed by S. Jaffard in [9]. For instance, the multifractal spectrum of the classical Riemann’s series F2 (using the classical pointwise Hölder

exponents) reads

dF₂(α) =











4α−2 ifα∈[1/2,3/4], 0 ifα= 3/2,

−∞ otherwise.

Actually theL²-multifractal spectrum is the same as the Hölder spectrum forF2.

Here our aim is to somehow extend this result to the range 1/2< s61.

The convergence of the seriesFs is described by our first theorem.

Theorem 1.4. — Let s∈(1/2,1], and letx∈(0,1) with convergents (p_j/q_j)_j>1. We set for everyj>1

δj=

(1 ifs∈(1/2,1) log(q_j+1/q_j) ifs= 1, and

(1.4) Σs(x) = X

j:q_j6=2∗odd

δj

r qj+1

(qjqj+1)^s.

(1) Fs(x) converges whenever ^s−1+1/r₂^odd(x) > 0. In fact, it converges wheneverΣs(x)<+∞.

(2) F_s(x) does not converge if ^s−1+1/r₂^odd(x) < 0. In fact, it does not converge whenever

limj:q_j6=2∗odd δj

r qj+1

(qjqj+1)^s >0,

In the same way we could extend this results to rational pointsx=p/q, by proving thatF_s(x) converges forq6= 2∗odd and does not forq= 2∗odd.

Observe that the convergence of Σs(x) implies that rodd(x) 6 1−s¹ . Our result asserts that Fs(x) converges as soon as r_odd(x) < _1−s¹ , and also whenrodd(x) =_1−s¹ when Σs(x)<+∞.

Jaffard’s result is then extended in the following sense:

Theorem 1.5. — Lets∈(1/2,1].

(1) For everyxsuch thatΣs(x)<+∞, one has αF_s(x) = s−1 + 1/rodd(x)

2 .

(2) For everyα∈[0, s/2−1/4]

dF_s(α) = 4α+ 2−2s.

s/2−1/2 1

α d ( )^α

F s

s/2−1/4 0

Figure 1.1. L²-multifractal spectrum ofFs

The second part of Theorem 1.5 follows directly from the first one. In- deed, using part (1) of Theorem 1.5 and (1.3), one gets

dF_s(α) = dim

x: s−1 + 1/r_odd(x)

2 =α

= dim{x:rodd(x) = (2α+ 1−s)⁻¹}

= 2

(2α+ 1−s)⁻¹ = 4α+ 2−2s.

The paper is organized as follows. Section 2 contains some notations and preliminary results. In Section 3, we obtain other formulations forFsbased on Gauss sums, and we get first estimates on the increments of the partial sums of the series F_s. Using these results, we prove Theorem 1.4 in Sec- tion 4. Finally, in Section 5, we use the previous estimates to obtain upper and lower bounds for the localL²-means of the seriesF_s, and compute in Section 6 the localL²-regularity exponent of Fs at real numbersxwhose Diophantine properties are controlled, namely we prove Theorem 1.5.

Finally, let us mention that theoretically theL²-exponents of a function f ∈L²(R) take values in the range [−1/2,+∞], so they may have negative values. We believe that this is the case at pointsxsuch thatr_odd(x)> _1−s¹ , so that in the end the entire L²-multifractal spectrum of F_s would be dF_s(α) = 4α+ 2−2sfor allα∈[s/2−1/2, s/2−1/4].

Another remark is that for a givens∈(1/2,1), there is an optimalp >2 such that Fs belongs locally to L^p, so that the p-exponents (instead of the 2-exponents) may carry some interesting information about the local behavior ofFs.

2. Notations and first properties

In all the proofs, C will denote a constant that does not depend on the variables involved in the equations.

For two real numbersA, B>0, the notationABmeans thatA6CB for some constantC >0 independent of the variables in the problem.

In Section 2 of [3] (also in [2]), the key point to study the local behavior of the Fourier seriesF_swas to obtain an explicit formula forF_s(p/q+h)− Fs(p/q) in the range 1< s <2; this formula was just a twisted version of the one known for the Jacobi theta function. In our range 1/2< s61, such a formula cannot exist because of the convergence problems, but we will get some truncated versions of it in order to prove Theorems 1.4 and 1.5.

Let us introduce the partial sum Fs,N(x) =

N

X

n=1

e^2iπn2x n^s .

For any H6= 0 letµeH be the probability measure defined by

(2.1) µeH(g) =

Z

C(H)

g(h) dh 2H,

whereC(H) is the annulusC(H) = [−2H,−H]∪[H,2H].

Lemma 2.1. — Let f :R → Rbe a function in L²(R), and x∈ R. If αf(x)<1, then

(2.2) αf(x) = supn

β∈[0,1) :∃C >0,∃fx∈R

kf(x+·)−fxk_L2(

eµH)6C|H|^βo .

Proof. — Assume that α := α_f(x) < 1. Then, for every ε > 0, there exist C > 0 and a real numberfx ∈ R such that for every H > 0 small enough

1 H

Z

B(x,H)

f(h)−fx

2dh

!^1/2

6CH^α−ε. SinceC(H)⊂B(x,2H), one has

(2.3)

f(x+·)−fx

_L2(

eµH)= Z

C(H)

f(x+h)−fx

2 dh 2H

!^1/2

6C|2H|^α−ε |H|^α−ε. Conversely, if (2.3) holds for everyH >0, then the result follows from the fact thatB(x, H) ={x} ∪ x+S

k>1C(H/2^k)

.

So, we will use equation (2.2) as definition of the localL² regularity.

It is important to notice that the frequencies in different ranges are going to behave differently. Hence, it is better to look atNwithin dyadic intervals.

Moreover, it will be easier to deal with smooth pieces. This motivates the following definition.

Definition 2.2. — Let N > 1 and let ψ : R →R be a C^∞ function with support included in[1/2,2]. One introduces the series

(2.4) F_s,N^ψ (x) =

+∞

X

n=1

e^2iπn2x n^s ψn

N

,

and forR >0one also sets

w^ψ_R(t) =e^2iπRt2ψ(t) and

E_N^ψ(x) = 1 N

+∞

X

n=1

w_N^ψ2x

n N

,

For every functionψ, we will use the notation

(2.5) ψ_s(t) :=t^−sψ(t),

which is still C^∞ with support in (1/2,2) when ψ is. It is immediate to check that

(2.6) F_s,N^ψ (x) =N^1−sE_N^ψs(x).

3. Summation Formula for F_s,N and F_s,N^ψ

3.1. Poisson Summation

Letp, q be coprime integers, withq >0. In this section we obtain some formulas for Fs(p/q+h)−Fs(p/q) with h >0. This is not a restriction, since

(3.1) F_s

p q−h

=F_s −p

q +h

.

We are going to write a summation formula forE_N^ψ(p/q+h), withh >0.

Proposition 3.1. — We have

(3.2) E^ψ_N

p q +h

= 1

√q X

m∈Z

θ_m·w[^ψ_N2h

N m q

.

where fb(ξ) = Z

R

f(t)e^−2iπtξdt stands for the Fourier transform of f and (θ_m)m∈Z are some complex numbers whose modulus is bounded by√

2.

Proof. — We begin by splitting the series into arithmetic progressions E_N^ψ

p q +h

= 1 N

+∞

X

n=1

e^2iπn2pq ·w^ψ_N2h

n N

=

q−1

X

b=0

e^2iπb2pq

+∞

X

n≡bn=1:modq

1

N w^ψ_N2hn N

.

Since w_N^ψ2h is C^∞ and supported inside (1/2,2), we can apply Poisson Summation to get

+∞

X

n≡bn=1:modq

1 Nw^ψ_N2h

n N

=X

n∈Z

1 Nw^ψ_N2h

b+nq N

= X

m∈Z

1

qe^2iπbmq ·w[^ψ_N2h

N m q

.

This yields

E_N^ψ p

q +h

= 1 q

X

m∈Z

τm·w[^ψ_N2h

N m q

with

τm=

q−1

X

b=0

e^{2iπpb2 +mbq} .

This termτ_mis a Gauss sum. One has the following bounds:

• for everym∈Z,τm=θm

√qwithθm:=θm(p, q) satisfying 06|θm|6√

2.

• ifq= 2∗odd, thenθ0= 0.

• ifq6= 2∗odd, then 16|θ0|6√ 2.

Finally, we get the summation formula (3.2).

3.2. Behavior of the Fourier transform of w_R^ψ

To use formula (3.2), one needs to understand the behavior of the Fourier transform ofw^ψ_R

wc^ψ_R(ξ) = Z

R

ψ(t)e^{2iπ(Rt2−ξt)}dt.

On one hand, we have the trivial bound |wc_R^ψ(ξ)| 1 since ψ is C^∞, bounded and compactly supported. On the other hand, one has

Lemma 3.2. — LetR >0andξ∈R. Letψbe aC^∞function compactly supported inside[1/2,2]. Let us introduce the mappingg_R^ψ:R→C

g_R^ψ(ξ) =e^iπ/4e^{−iπξ2/(2R)}

√

2R ψ

ξ 2R

.

Then one has

(3.3) wc^ψ_R(ξ) =g_R^ψ(ξ) +Oψ

ρ_R,ξ

√

R + 1

(1 +R+|ξ|)^3/2

,

with

ρR,ξ=

(1 ifξ/2R∈[1/2,2]andR <1,

0 otherwise. .

Moreover, one has (3.4) wd^ψ_2R(ξ)−wc_R^ψ(ξ)

=g^ψ_2R(ξ)−g^ψ_R(ξ) +O_ψ

ρ_R,ξ√

R+ R

(1 +R+|ξ|)^5/2

,

Moreover, the constant implicit inOψ depends just on the L^∞-norm of a finite number of derivatives ofψ.

Proof. — For ξ/2R 6∈ [1/2,2] the upper bound (3.3) comes just from integrating by parts several times; forξ, R1 the bound (3.3) is trivial.

The same properties hold true for the upper bound in (3.4).

Let us assume ξ/2R ∈ [1/2,2], andR > 1. The lemma is just a conse- quence of the stationary phase theorem. Precisely, Proposition 3 in Chapter VIII of [12] (and the remarks thereafter) implies that for some suitable func- tionsf, g:R→R, and ifg is such thatg⁰(t0) = 0 at a unique point t0, if one sets

S(λ) = Z

R

f(t)e^iλg(t)dt−

s 2π

−iλg⁰⁰(t0)f(t₀)e^iλg(t0),

then|S(λ)| λ^−3/2and also|S⁰(λ)| λ^−5/2, where the implicit constants depend just on upper bounds for some derivatives off andg, and also on a lower bound forg⁰⁰.

In our case, we can apply it withf =ψ,λ=R, andg(t) = 2π(t²−tξ/λ) to get precisely (3.3). This is so because althoughgdepends onλ, since in our range forR we have|ξ/λ|64, the trivial upper bounds forg and its derivatives do not (and alsog⁰⁰(t) does not depend onλ).

With the same choices forf andg, by applying the Mean Value Theorem and our bound forS⁰(λ), we finally obtain formula (3.4).

3.3. Summation formula for the partial seriesFs,N.

A certain duality formula for the Riemann functionR(x) has been at the heart of the proofs of many of its convergence and differentiability prop- erties [6, 7, 9]. This formula in turn can be seen as coming from the sym- metries of the Jacobi theta function, and can be derived by using Fourier Analysis.

There is a related duality formula forF_s(x), but it does not work in our case due to convergence problems related to the range in whichs moves.

As a substitute we will use a truncated version of it that can be proved by similar techniques. This formula gives precise information about the behavior of the partial sum functionFs,2N −Fs,N near a rational number.

Proposition 3.3. — Letp, qbe two coprime integers,q >1. ForN >q and06h6q⁻¹, we have

Fs,2N

p q+h

−Fs,N

p q+h

= θ0

√q Z 2N

N

e^2iπht2 t^s dt

+Gs,2N(h)−Gs,N(h) +O(N^12−slogq), where

(3.5) G_s,N(h) = (2hq)^s−12e^iπ/4 X

16m62N hq

θm

m^se^−iπm

2 q2h.

Pay attention to the fact that G_s,N depends on pand q. We omit this dependence in the notation for clarity.

Proof. — We can write

F_s,2N(x)−F_s,N(x) =F_s,N^1(1,2)(x) +O(N^−s).

Hence, we would like to use the formulas proved in the preceding section, but those formulas apply only to compactly supportedC^∞ functions. We thus decompose the indicator function1(1,2) into a countable sum ofC^∞ functions, as follows. Let us considerη, aC^∞function with support [1/2,2]

such that

η(t) = 1−η(t/2) 16t62.

Then, the function

ψ(t) =X

k>2

η t

2^−k

has support in [0,1/2], equals 1 in (0,1/4] and isC^∞in (0,+∞). Therefore, we have

(3.6) 1(1,2)(t) =ψ(t−1) +ψ(2−t) +ψ(t)e

withψesomeC^∞ function with support included in [5/4,7/4]⊂[1,2].

In order to get a formula forF_s,N^1(1,2)(x), we are going to use (3.6) and the linearity inψof the formula (2.4). Indeed, writing for all k>2

(3.7) η^k(t) :=η((t−1)/2^−k) and ζ^k(t) :=η((2−t)/2^−k), one has the decomposition:

(3.8) F_s,N^1(1,2)(x) =F^ψe

s,N(x) +X

k>2

F_s,N^ηk(x) +X

k>2

F_s,N^ζk (x).

Our goal is to obtain several estimates to treat all these terms. The proof is decomposed into several lemmas, to deal with the different cases.

First, we obtain an estimate for F_s,N^φ , where φis anyC^∞ function sup- ported in [1/2,2]. In particular, this will apply to the termF^ψe

s,N(x).

Lemma 3.4. — Letφbe aC^∞function supported in[1/2,2]. Then, (3.9) F_s,N^φ

p q+h

= N^1−sθ0

√q Z

R

e^2iπN2ht2

t^s φ(t) dt+G^φ_s,N(h) +Oφ

q N^1/2+s

,

where

(3.10) G^φ_s,N(h) = (2hq)^s−12e^iπ/4X

m6=0

θm

m^se^{−iπ m}

2 2q2hφ

m 2N hq

.

Proof. — Recall the notation (2.5):φ_s(t) =t^−sφ(t).

First, by (2.6) one hasF_s,N^φ (x) =N^1−sE_N^φs(x). Further, by (3.2) one has E_N^φs

p q+h

= 1

√q X

m∈Z

θ_m·w[^φ_N^s2h

N m q

,

and then, applying Lemma 3.2 withξ=^{N m}_q andR=N²h, one gets E^φ_N^s

p q+h

=θ0

ˆ w^φ_N^s2h(0)

√q +e^iπ/4

√q

×X

m6=0



θ_mφ_s m

2N qh

e^{−iπ m}

2 2q2h

√

2N²h +O

(N m/q)^−3/2



.

When _2R^ξ = _{2N qh}^m > 2, φs

m 2N qh

= 0. Recalling that N > q, since φs(t) =φ(t)t^−s, the above equation can be rewritten

(3.11) F_s,N^φ p

q +h

= N^1−sθ₀

√q Z

R

e^2iπN2ht2

t^s φ(t) dt+G^φ_s,N(h) +N^1−s

√q X

m>1

O_φ

(N m/q)^−3/2 .

The last term is controlled by N^1−s

√q

1 (N/q)^3/2

= q

N^1/2+s,

which yields (3.9).

Second, one wants to obtain a comparable estimate for F_s,N^ηk (x), for all k>1. For large kthe mass of the function η^k is very small, so the same should happen toF_s,N^ηk (p/q+h) andG^η_s,N^k (h). Then the estimate we want will be true in a trivial way, namely because all the terms involved are small.

That is the content of the following lemma and its proof (an equivalent way to proceed would be to say that all terms coming from large k are negligible).

Lemma 3.5. — For anyk>1and0< h61/q, one has F_s,N^ηk

p q +h

= θ₀N^1−s

√q Z

R

e^2iπN2ht2

t^s η^k(t) dt+G^η_s,N^k (h) +Oη N^1−s2^−k .

Proof. — We are going to bound each sum and integral trivially, that is by introducing absolute values inside them. First, sinceη has support in [1/2,2], one has directly:

• by (2.4):

|F_s,N^ηk (x)|6

+∞

X

n=1

1 n^s

η

n N −1

2^−k

6

N+N2^−k+1

X

n=N+N2^−k−1

1

n^s N^1−s2^−k

• by (3.10):

|G^η_s,N^k (h)| (qh)^s−12 X

m:η^k(_{2N hq}^m )6=0

1 m^s

(qh)^s−122N hq2^−k (2N hq)^s p

qh2^−kN^1−s

• and

θ₀N^1−s

√q Z

R

e^2iπN2ht2

t^s η^k(t) dt

N^1−s q^1/2

Z 1+2^−k+1 1+2^−k−1

dt t^s

2^−kN^1−sq^−1/2, hence the result by (3.11), where we used thatqh61.

For small kthe function F_s,N^ηk should behave asF_s,N^φ , but the estimates needed in order to prove the estimate for it are more delicate than the ones used in the proof of Lemma 3.4. The proof can probably be skipped at first reading.

Lemma 3.6. — For every k>2and0< h61, one has F_s,N^ηk

p q +h

= θ0N^1−s

√q Z

R

e^2iπN2ht2

t^s η^k(t) dt+G^η_s,N^k (h) + Oη

√ q

N^s +N^1/2−sγ_k

,

where the sequence(γk)k>1 is positive and satisfiesP

k>1γk1.

Proof. — The proof starts as the one of Lemma 3.4. Recalling the no- tation (2.5) and the definition (3.7) ofη_k, one has (η^k)_s(t) = ^η((t−1)/2_ts ^−k). For this function, one has

w\_R^(ηk)s(ξ) = Z

R

(η^k)_s(t)e^{2iπ(Rt2−tξ)}dt= Z

R

η(₂^t−1−k)

t^s e^{2iπ(Rt2−tξ)}dt

= 2^−ke^2iπ(R−ξ) Z

R

η(u)

(1 +u2^−k)^se^{2iπ(R2−2ku2−2−k(ξ−2R)u)}du

= 2^−ke^2iπ(R−ξ) \ w^ηe^k

R2^−2k(2^−k(ξ−2R)),

wherefη^k(u) = _(1+u2^η(u)−k)^s. Hence,

(3.12) E_N^(ηk)s

p q +h

= 1

√q X

m∈Z

θm·w\^(η_N2^kh^)s

N m q

=2^−k

√q X

m∈Z

θm·e^{2iπ(N2h−N mq )}

× \

w^ηe^k

N²h2^−2k

2^−k

N m

q −2N²h

Here we apply again Lemma 3.2 and we obtain E_N^(ηk)s

p q +h

= 2^−k

√q X

m6=0

θ_me^{2iπ(N2h−N mq )}

×





e^iπ/4ηf^k



 2^−k

N m

q −2N²h 2N²h2^−2k



 e^−iπ

2−2k(^{N m}_q ^−2N2h)²

2N2h2−2k

√

2N²h2^−2k +O

ηe^k

ρR_k,ξ_k

√

2^−2kN²h+

1 + 2^−k

N m

q −2N²h

+N²h2^−2k

−3/2!!

.

withRk = 2^−2kN²hand ξk = 2^−k|N m/q−2N²h|. Finally, after simplifi- cation, one gets

F_s,N^ηk p

q+h

=N^1−sE_N^(ηk)s p

q +h

= (2qh)^s−12 e^−iπ/4

X

m∈Z

θm

m^se^−2iπm

2 q2hη^k

m 2N hq

(3.13)

= θ0N^1−s

√q Z

R

e^2iπN2ht2

t^s η^k(t) dt+G^η_s,N^k (h) +L^k_N, (3.14)

where by Lemmas 3.4 and 3.5 one has (3.15) L^k_N =O

ηe^k



 X

m∈J_N^k∩Z^∗

2^−kN^1−s/√

√ q

2^−2kN²h

+ X

m∈Z^∗

2^−kN^1−s/√ q

(1 + 2^−k|^{N m}_q −2N²h|+N²h2^−2k)^3/2

! ,

withJ_N^k =

(2 + 2^−k−1)N qh,(2 + 2^−k+1)N qh .

First, as specified in Lemma 3.2, the constants involved in the O

ηe^k de- pend on upper bounds for some derivatives offη^k, and then by the definition offη^k we can assume they are fixed and independent on bothk ands.

Let {x} stand for the distance from the real number xto the nearest integer.

The first sum inL^k_N is bounded above by:

• √

qN^−s+N^1/2−swhen 2^−k ∈

{2N hq}/4N hq,{2N hq}/N hq ,

• √

qN^−s otherwise.

In particular, xbeing fixed, the term N^1/2−s may appear only a finite number of times whenkranges inN.

In the second sum, there is at most one integer m for which |N m/q− 2N²h|< N/2q, and the corresponding term is bounded above by

2^−kN^1−sq^−1/2(1 +N/q2^−k+|N²h|^−2k)^−3/2

62^−kN^1−sq^−1/2(1 +N/q2^−2k)^−3/2

=N^1/2−s N^1/2q^−1/22^−k (1 +N/q2^−2k)^−3/2 6N^1/2−sγk,

where γk =q _u

k

(1+u_k)³ and uk = 2^−2kN/q. The sum over k of this upper bound is finite, and this sum can be bounded above independently onN andq.

The rest of the sum is bounded, up to a multiplicative constant, by Z +∞

u=0

√qN^−s(2^−kN/q)du

(1 +N²h2^−2k+ 2^−k|^{N u}_q −2N²h|)^3/2

√qN^−s (1 +N²h2^−2k)^1/2 , which is less than√

qN^−s. Hence the result.

Now we are ready to prove Proposition 3.3.

Recall that we have the decomposition (3.8). The first term F^ψe

s,N(x) is dealt with by Lemma 3.4. Now we deal with the sumP

k>1F_s,N^ηk (x). The third term P

k>1F_s,N^ζk (x) is absolutely similar to the second one (Lem- mas 3.5 and 3.6 must be adapted in a straightforward way; we let them to the reader).

Recall that N >q and 06h6q⁻¹. Let K be the unique integer such that 2^−K 6

√q

N <2^−(K+1). If we apply Lemma 3.5 toF_s,N^ηk for everyk > K, the error terms coming from that lemma give a contribution bounded, up

to a constant, by X

k>K

N^1−s2^−k=N^1−s2^−K N^1−s

√q N =

√q

N^s 6 1 N^s−1/2.

If we apply Lemma 3.6 toF_s,N^ηk for every k 6K, the error terms coming from that lemma give a contribution bounded, up to a constant, by

X

k6K

(

√q

N^s+N^1/2−sγk)logN

√q

N^s+N^1/2−slogq

√ N

N^s = logq N^s−1/2,

where we use that the mappingx7→

√x

logx is increasing for largex.

Gathering all the informations, and recalling thatP+∞

k=2η^k(t) =ψ(t−1), we have that

F_s,N^ψ(·−1) p

q+h

= N^1−sθ₀

√q Z

R

e^2iπN2ht2

t^s ψ(t−1) dt+G^ψ(·−1)_s,N (h) +Oη

logq N^s−1/2

.

As said above, the same inequalities remain true if we use the functions ζ^k(t) =η((2−t)/2^−k), so the last inequality also holds forψ(2− ·).

Finally, recalling the decomposition (3.6) and (3.8) expressing1(1,2)and F_s,N^1(1,2) in terms of smooth terms, we get

(3.16) F_s,N^1(1,2) p

q+h

= N^1−sθ₀

√q Z 2

1

e^2iπN2ht2

t^s dt+G¹_s,N^(1,2)(h) +Oη

logq N^s−1/2

,

and the result follows from the formula

G¹_s,N^(1,2)(h) =Gs,2N(h)−Gs,N(h) +O(N^−s+1/2).

4. Proof of the convergence Theorem 1.4 4.1. Convergence part: item (1) Letxbe such that Σs(x)<+∞.

Recall the Definition 1.3 of the convergents of x. We begin by bounding Fs,M(x)−Fs,N(x) for any

(4.1) 1/26qj/46N < M < qj+1/4.

We apply Proposition 3.3 withp/q=p_j/q_jandh=h_j, so thatx=p/q+h.

Due to (3.1), we can assume thathj >0. It is known that _2q ¹

jq_j+1 6hj =

|x−pj/qj|< _q ¹

jq_j+1. LetN⁰ ∈[N,2N]. Since 4N hjqj<4N/qj+1 <1, the sums (3.5) appearing inGs,N⁰(hj) andGs,N(hj) have no terms, hence are equal to zero. This yields

Fs,N⁰(x)−Fs,N(x) = θ0

√q_j Z N⁰

N

e^2iπhjt2

t^s dt+O(N^12−slogqj).

Using the trivial bound directly or after writing t^−se^2iπt2 = 1

4iπt^−s−1(e^2iπt2)⁰ and integrating by parts we get the bound

Z a⁰ a

t^−se^2iπt2dt

6a^−smin(s+ 1 a , a),

true for every 0< a6a⁰62aand everys >0. So,

Z N⁰ N

e^2iπhjt2 t^s dt

=|hj|^s/2−1/2

Z N⁰√

h_j N√

h_j

e^2iπu2 u^s du

|hj|^−1/2N^−smin(|Np

hj|⁻¹,|Np hj|).

One deduces (using thatqjhj is equivalent toq⁻¹_j+1) that (4.2) |F_s,N⁰(x)−F_s,N(x)|

|θ0|

√qj+1

N^s min( N

√qjqj+1

,

√qjqj+1

N ) +N^12−slogqj. Thus, writingF_s,M−F_s,N as a dyadic sum and using (4.2) for each term we have

|Fs,M(x)−Fs,N(x)| |θ0|δj

√qj+1

(√

qjqj+1)^s + logq_j q_j^s−1/2

for everyM, N as in (4.1). Finally, recalling thatθ₀ is equal to zero when qj 6= 2×odd, fixing an integerj0>3, for anyM > N > qj₀, one has

|Fs,M(x)−Fs,N(x)| X

j>j₀,q_j6=2∗odd

δj

√q_j+1 (√

q_jq_j+1)^s+X

j>j₀

logqj

q^s−1/2_j .

The second series goes to 0 when j₀ → ∞ and the first does when P

s(x)<∞.

4.2. Divergence part: item (2)

Let 0 < ε <1/2 a small constant. LetNj =εqj and Mj = 2ε√ qjqj+1. Proceeding exactly as in the previous proof we get

Fs,M_j(x)−Fs,N_j(x) = θ0

√q_j Z M_j

Nj

e^2iπhjt2

t^s dt+O q

1 2−s j logqj

.

Sincee^2iπhjt2 = 1 +O(ε) inside the integral, as soon asqj 6= 2∗odd, one has

|Fs,M_j(x)−Fs,N_j(x)|> |θ0|

√qj

Mj−Nj

2·M_j^s >|θ0|ε 2√

qj+1−√ qj

2^1+s·ε^s·(q_jq_j+1)^s/2

r q_j+1 (qjqj+1)^s,

which is infinitely often large by our assumption. Hence the divergence of the series.

5. Local L² bounds for the function Fs

Further intermediary results are needed to study the local regularity ofFs.

Proposition 5.1. — Leth >0,1/2< s61andq²h1. We have (5.1) Fs,N

p q+ 2h

−Fs,N

p q +h

= θ0

√q Z N

0

e^2iπ2ht2−e^2iπht2

t^s dt

+Gs,N(2h)−Gs,N(h) +O

|qh|^s−1/2 .

Proof. — First, one writes

(5.2) F_s,N(x) =F_s,N^1(0,1](x) = X

16m61+log₂N

F_s,N/2^1(1,2]_m(x).

As we did in the previous sections, we are going to estimate (5.1) using the