LOCAL $L^2$-REGULARITY OF RIEMANN'S FOURIER SERIES

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LOCAL L

-REGULARITY OF RIEMANN’S FOURIER SERIES

Stéphane Seuret, Adrian Ubis

To cite this version:

Stéphane Seuret, Adrian Ubis. LOCAL L²-REGULARITY OF RIEMANN’S FOURIER SERIES.

Annales de l’Institut Fourier, Association des Annales de l’Institut Fourier, 2017. �hal-01612284�

RIEMANN’S FOURIER SERIES

ST ´EPHANE SEURET^∗AND ADRI ´AN UBIS

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

n=1 e^2iπn2x

n^s . In the 1850’s, Riemann in- troduced the seriesF2as 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< s≤1, and we prove thatFs(x) converges whenxsatisfies a Diophantine condition. We also study theL²- local regularity ofFs, proving that the localL²-norm ofFsaround a pointxbehave differently around differentx, according again to Diophantine conditions onx.

1. Introduction Riemann introduced in 1857 the Fourier series

R(x) =

+∞

X

n=1

sin(2πn²x) n²

as a possible example of continuous but nowhere differentiable function. Though it is not the case (R is differentiable at rationals p/qwhere pand q are both odd [5]), the study of this function has, mainly because of its connections with several domains: complex analysis, harmonic analysis, Diophantine approximation, and dynamical systems [6, 7, 5, 4, 8, 10] and more recently [2, 3, 12].

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

(1.1) F_s(x) =

+∞

X

n=1

e^2iπn2x n^s

when s∈(1/2,1). In this case, several questions arise before considering its local behavior. First it does not converge everywhere, hence one needs to characterize its set of convergence points; this question was studied in [12], and we will first find a slightly more precise characterization. Then, if one wants to characterize the local regularity of a (real) function, one classically studies the pointwise H¨older exponent defined for a locally bounded functionf :R→Rat a point x by using the functional spaces C^α(x): f ∈ C^α(x) when there exist a constant C and a

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

Key words and phrases. Fourier series, Diophantine approximation, local regularity, Hausdorff dimension.

∗Research partially supported by the ANR project MUTADIS, ANR-11-JS01-0009.

1

polynomialP with degree less thanbαcsuch that, locally aroundx(i.e. for small H), one has

(f(·)−P(· −x))11_B(x,H)k_∞:= sup(|f(y)−P(y−x)|:y∈B(x, H))≤CH^α, where B(x, H) = {y ∈ R : |y −x| ≤ H}. Unfortunately these spaces are not appropriate for our context since Fs is nowhere locally bounded (for instance, it diverges at every irreducible rationalp/q such thatq 6= 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 ≥ 0 and x∈R. The functionf is said to belong toC₂^α(x) if there exist a constant Cand a polynomialP with degree less thanbαcsuch that, locally aroundx(i.e. for small H >0), one has

1 H

Z

B(x,H)

f(h)−P(h−x

|²dh

!1/2

≤CH^α.

Then, the pointwise L²-exponent of f atx is αf(x) = sup

α∈R:f ∈C₂^α(x) .

This definition makes sense for the series F_s when s∈ (1/2,1), and are based on a natural generalizations of the spacesC^α(x) be replacing theL^∞ norm by the L² norm. The pointwise L²-exponent has been studied for instance in [11], and is always greater than -1/2 as soon as f ∈L².

Our goal is to perform the multifractal analysis of the seriesF_s. 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). The L²- multifractal spectrum d_f :R⁺∪ {+∞} →R⁺∪ {−∞}of f is the mapping

df(α) := dimEf(α), where the iso-H¨older set Ef(α) is

E_f(α) :={x∈R:α_f(x) =α}.

By convention one setsdf(α) =−∞ ifEf(α) =∅.

Performing the multifractal analysis consists in computing its L²-multifractal spectrum. This provides us with a very precise description of the distribution of the local L²-singularities of f. In order to state our result, we need to introduce some notations.

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

(1.2) x−p_j

qj

=h_j, |h_j|=q^−r_j ^j

with 2≤rj <∞. Then the approximation rate of xis defined by r_odd(x) = lim{r_j :qj 6= 2∗odd}.

This definition always makes sense because ifqjis even, thenqj+1andqj−1must be odd (so cannot be equal to 2∗odd). Thus, we always have 2≤r_odd(x)≤+∞.

It is classical that one can compute the Hausdorff dimension of the set of points with the Hausdorff dimension of the points x with a given approximation rate r≥2:

(1.3) for all r≥2, dim{x∈R:r_odd(x) =r}= 2 r.

When s >1, the series F_s converges, and the multifractal spectrum ofF_s was computed by S. Jaffard in [10]. For instance, for the classical Riemann’s seriesF2, one has

d_F2(α) =







4α−2 ifα∈[1/2,3/4], 0 ifα= 3/2,

−∞ otherwise.

Here our aim is to somehow extend this result to the range 1/2< s≤1. The convergence of the seriesF_s is described by our first theorem.

Theorem 1.4. Let s ∈ (1/2,1], and let x ∈ (0,1) with convergents (p_j/q_j)j≥1. We set for every j≥1

δ_j =

( 1 if s∈(1/2,1)

log(q_j+1/q_j) if s= 1, and

(1.4) Σ_s(x) = X

j:qj6=2∗odd

δ_j

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

(i) F_s(x) converges whenever ^s−1+1/r₂^odd(x) >0. In fact, it converges when- ever Σs(x)<+∞.

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

limj:qj6=2∗odd δj

r qj+1

(qjqj+1)^s >0,

In the same way we could extend this results to rational points x = p/q, by proving thatF_s(x) converges forq6= 2∗odd and does not forq6= 2∗odd. Observe that the convergence of Σ_s(x) implies that r_odd(x)≤ _1−s¹ . Our result asserts that Fs(x) converges as soon as rodd(x) < _1−s¹ , and also when rodd(x) = _1−s¹ when Σ_s(x)<+∞.

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

Theorem 1.5. Let s∈(1/2,1].

s/2−1/2 1

α

d ( )_{F s} ^α

s/2−1/4 0

Figure 1. L²-multifractal spectrum of Fs

(i) For everyx such that Σs(x)<+∞, one hasα_Fs(x) = s−1 + 1/r_odd(x)

2 .

(ii) For every α∈[0, s/2−1/4]

d_Fs(α) = 4α+ 2−2s.

The second part of Theorem 1.5 follows directly from the first one. Indeed, using part (i) of Theorem 1.5 and (1.3), one gets

d_Fs(α) = dim

x: s−1 + 1/r_odd(x)

2 =α

= dim{x:r_odd(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 pre- liminary 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 Fs. Using these results, we prove Theorem 1.4 in Section 4. Finally, in Section 5, we use the previous estimates to obtain upper and lower bounds for the local L²- means of the seriesF_s, and compute in Section 6 the localL²-regularity exponent of Fs at real numbers x whose Diophantine properties are controlled, namely we prove Theorem 1.5.

Finally, let us mention that theoretically the L²-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 points x such that rodd(x) > _1−s¹ , so that in the end the entire L²-multifractal spectrum ofFs would be dFs(α) = 4α+ 2−2sfor all α∈[s/2−1/2, s/2−1/4].

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

2. Notations and first properties

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

For two real numbers A, B ≥0, the notation A B means thatA ≤CB 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 series F_s was to obtain an explicit formula for F_s(p/q+h)−F_s(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< s≤1, 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 anyH 6= 0 letµeH be the probability measure defined by

(2.1) µe_H(g) =

Z

C(H)

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

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

(2.2)

α_f(x) = sup n

β ∈[0,1) :∃C >0,∃fx ∈R kf(x+·)−fxk_L2(µe_H)≤C|H|^βo . Proof. Assume thatα:=α_f(x)<1. Then, for everyε >0, there existC >0 and a real numberfx ∈Rsuch that for every H >0 small enough

1 H

Z

B(x,H)

f(h)−fx

2dh

!1/2

≤CH^α−ε.

Since C(H)⊂B(x,2H), one has (2.3)

f(x+·)−f_x

L²(eµH)= Z

C(H)

f(x+h)−f_x

2 dh 2H

!1/2

≤C|2H|^α−ε|H|^α−ε.

Conversely, if (2.3) holds for every H > 0, then the result follows from the fact thatB(x, H) =x+S

k≥1C(H/2^k).

So, we will use equation (2.2) as definition of the local L² regularity.

It is important to notice that the frequencies in different ranges are going to behave differently. Hence, it is better to look atN within dyadic intervals. More- over, 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 ψ

n N

,

and for R >0 one also sets

w_R^ψ(t) =e^2iπRt2ψ(t) and

E_N^ψ(x) = 1 N

+∞

X

n=1

w_N^ψ2x

n N

,

For the function ψs(t) =t^−sψ(t) (which is stillC^∞ with support in (1/2,2)) it is immediate to check that

(2.5) 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. Let p, q be coprime integers, with q > 0. In this section we obtain some formulas forF(p/q+h)−F(p/q) withh >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 3.2. We have

(3.2) E_N^ψ

p q +h

= 1

√q X

m∈Z

θm·w[^ψ_N2h

N m q

.

where fb(ξ) = Z

R

f(t)e^−2iπtξdt stands for the Fourier transform off 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^2iπn2

p

q ·w^ψ_N2h

n N

=

q−1

X

b=0

e^2iπb2

p q

+∞

X

n≡bn=1:modq

1 N w_N^ψ2h

n N

.

Now we apply Poisson Summation to the inner sum to get

+∞

X

n≡bn=1:modq

1 Nw_N^ψ2h

n N

=

+∞

X

n=1

1 Nw_N^ψ2h

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[_N^ψ2h

N m q

with

τ_m =

q−1

X

b=0

e^2iπpb2+mbq .

This term τm is a Gauss sum. One has the following bounds:

• for every m∈Z,τ_m =θ_m√

q withθ_m:=θ_m(p/q) satisfying 0≤ |θ_m| ≤√

2.

• ifq = 2∗odd, thenθ0= 0.

• ifq 6= 2∗odd, then 1≤θ₀≤√ 2.

Finally, we get the summation formula (3.2).

3.3. Behavior of the Fourier transform of w_R^ψ. To use formula (3.2), one needs to understand the behavior of the Fourier transform of w_R^ψ

wc_R^ψ(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 by 1 and compactly supported. On the other hand, one has

Lemma 3.4. Let R >0and ξ∈R. Letψ be a C^∞ function compactly supported inside [1/2,2]. Let us introduce the mapping g_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^ψ_R(ξ) +O_ψ ρ_R,ξ

√

R + 1

(1 +R+|ξ|)^3/2

,

with ρR,ξ =

( 1 if ξ/2R∈[1/2,2]and R <1,

0 otherwise. . Moreover, one has

(3.4) wd_2R^ψ (ξ)−wc^ψ_R(ξ) =g_2R^ψ (ξ)−g_R^ψ(ξ) +O_ψ

ρ_R,ξ√

R+ R

(1 +R+|ξ|)^5/2

,

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

Proof. Forξ/2R6∈[1/2,2] the upper bound (3.3) comes just from integrating by parts several times; for ξ, R 1 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], and R > 1. The lemma is just a consequence of the stationary phase theorem. Precisely, Proposition 3 in Chapter VIII of [13]

(and the remarks thereafter) implies that for some suitable functionsf, g:R→R, and if g 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 of f and g, and also on a lower bound forg⁰⁰.

In our case, we can apply it withf =ψ,λ=R, andg(t) = 2π(t²−ξ/λ) to get precisely (3.3).

With the same choices forf andg, by applying the Mean Value Theorem and our bound for S⁰(λ), we finally obtain formula (3.4).

3.5. Summation formula for the partial seriesF_s,N. This important formula will be useful to study the convergence of F_s(x).

Proposition 3.6. Let p, q be two coprime integers. For N ≥q and 0≤h≤q⁻¹, we have

F_s,2N p

q +h

−F_s,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) Gs,N(h) = (2hq)^s−12e^iπ/4

b2N hqc

X

m=1

θm

m^se^−iπ

m2 q2h.

Pay attention to the fact that Gs,N depends on p and q. We omit this depen- dence in the notation for clarity.

Proof. We can write

F_s,2N(x)−F_s,N(x) =F_s,N^11[1,2](x).

Hence, we would like to use the formulas proved in the preceding section, but those formulas apply only to compactly supported C^∞ functions. We thus decompose the indicator function11_[1,2] into a countable sum ofC^∞functions, as follows. Let us consider η, a C^∞ function with support [1/2,2] such that

η(t) = 1−η(t/2) 1≤t≤2.

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 [1/4,1/2]. Therefore, we have

(3.6) 11_[1,2](t) =ψ(t−1) +ψ(2−t) +ψ(t)e withψesomeC^∞ function with support included in [1,2].

In order to get a formula forF_s,N^11[1,2](x), we are going to use (3.6) and the linearity inψ of the formula (2.4).

We will first get a formula forF_s,N^φ , where φ is any C^∞ function supported in [1/2,2]. In particular, this will work with φ=ψ.e

Lemma 3.7. Let φbe a C^∞ function supported in[1/2,2]. Then, (3.7) 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) +O_φ q N^1/2+s

,

where

(3.8) G^φ_s,N(h) = (2hq)^s−12e^iπ/4 X

m6=0

θ_m m^se^−iπ

m2 2q2hφ

m 2N hq

.

Proof. First, by (2.5) 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

θm·w[_N^φs2h

N m q

,

and then, applying Lemma 3.4 with ξ= ^{N m}_q and R=N²h, one gets E_N^φs

p q +h

= θ₀wˆ^φ_N^s2h(0)

√q

+e^iπ/4

√q X

m6=0



θmφs

m 2N qh

e^−iπ

m2 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

F_s,N^φ p

q +h

= N^1−sθ₀

√q Z

R

e^2iπN2ht2

t^s φ(t)dt+G^φ_N(h) (3.9)

+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 (3.7).

Now, one wants to obtain a comparable formula forη^k(t) :=η((t−1)/2^−k) for all k≥1. We begin with a bound which is good just for largek.

Lemma 3.8. For any k≥1 and 0< h≤1/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) +O_η

N^1−s2^−k . Proof. First, whenkbecomes large, sinceηhas support in [1/2,2], one has directly:

- by (2.4):

|F_s,N^ηk (x)| ≤

+∞

X

n=1

1 n^s

η

n N −1

2^−k

≤

N+N2^−k+1

X

n=N+N2^−k−1

1

n^s N^1−s2^−k - by (3.8):

|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−sq^−1/2

Z ₁₊₂^−k+1

1+2^−k−1

dt

t^s 2^−kN^1−sq^−1/2, hence the result by (3.9), where we used thatqh≤1.

One can obtain another bound that is good for any k.

Lemma 3.9. For every k≥2 and 0< h≤1, 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) +O_η √

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 3.7. Using the fact that for any (η^k)_s(t) = ^η((t−1)/2_ts ^−k) and that

w\_R^(ηk)s(ξ) = Z

R

(η^k)_s(t)e^{2iπ(Rt2−tξ)}dt= Z

R

η(^t−1₂−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^η_R2^fk−2k(2^−k(ξ−2R)),

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

p q +h

= 1

√q X

m∈Z

θ_m· \ w_N^(ηk2h^)s

N m q

= 2^−k

√q X

m∈Z

θ_m·e^{2iπ(N2h−N mq )} \ w^η_N^fk2h2^−2k

2^−k

N m

q −2N²h (3.10)

Here we apply again Lemma 3.4 and we obtain E_N^(ηk)s

p q +h

= 2^−k

√q X

m6=0

θ_me^{2iπ(N2h−N mq )}

×





e^iπ/4fη^k



 2^−k

N m

q −2N²h 2N²h2^−2k



 e^−iπ

2−2k(^{N m}q −2N2h)²

2N2h2−2k

√

2N²h2^−2k +O

fη^k

ρ_Rk,ξk

√

2^−2kN²h+

1 + 2^−k

N m

q −2N²h

+N²h2^−2k

−3/2!!

. with Rk = 2^−2kN²h and ξk = 2^−k|N m/q−2N²h|. Finally, after simplification,

one gets F_s,N^ηk

p q +h

= N^1−sE^(η_N^k)s p

q +h

= (2qh)^s−12 e^−iπ/4

X

m∈Z

θ_m m^se^−2iπ

m2 q2hη^k

m 2N hq

(3.11)

(3.12)

= θ0N^1−s

√q Z

R

e^2iπN2ht2

t^s η^k(t)dt+G^η_s,N^k (h) +L^k_N, where by Lemmas 3.7 and 3.8 one has

L^k_N =O

ηf^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 3.4, the constants involved in the O

fη^k depend on upper bounds for some derivatives of fη^k, and then by the definition of fη^k we can assume they are fixed and independent on bothk ands.

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

The first sum inL^k_N is bounded above by:

• √

qN^−s+N^1/2−s when 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 whenk ranges inN.

In the second sum, there is at most one integerm 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

≤ 2^−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

≤ N^1/2−sγ_k, whereγ_k=q _u

k

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

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 √ qN^−s,

hence the result.

Now we are ready to prove Proposition 3.6.

Recall that N ≥ q and 0 ≤ h ≤ q⁻¹. Let K be the unique integer such that 2^−K ≤

√q

N <2^−(K+1). We need to bound by above the sum of the errors L^k_N:

• when k≥K: we use Lemma 3.8 to get X

k≥K

|L^k_N| N^1−s2^K N^1−s

√q N =

√q

N^s ≤ 1 N^s−1/2.

• the remaining terms are simply bounded using Lemma 3.9 by

K

X

k=2

|L^k_N| K

√q

N^s+N^1/2−slogN

√q

N^s +N^1/2−s logq

√ N

N^s = logq N^s−1/2, where we use that the mapping x7→

√x

logx is increasing for largex.

Gathering all the informations, and recalling that P+∞

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) +O_η

logq N^s−1/2

. The same inequalities remain true if we use the functionsfη^k=η((2−t)/2^−k), so the last inequality also holds for ψ(2− ·)

Finally, recalling the decomposition (3.6) expressing 11_[1,2] in terms of smooth functions, we get

(3.13) F_s,N^11[1,2]

p q +h

= N^1−sθ0

√q Z 2

1

e^2iπN2ht2

t^s dt+G¹_s,N^1[1,2](h) +O_η

logq N^s−1/2

,

and the result follows.

4. Proof of the convergence theorem 1.4

4.1. Convergence part: item (i). Letx be such that (1.4) holds true.

Recall the definition (1.2) of the convergents of x. We begin by bounding Fs,M(x)−Fs,N(x) for any

qj/4≤N < M < qj+1/4.

We apply Proposition 3.6 with p/q = p_j/q_j and h = h_j, so that x = p/q+h.

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

jqj+1 ≤ hj =

|x−pj/qj|< _q ¹

jqj+1.

First, since 4N hjqj <4N/qj+1 <1, the sums (3.5) appearing inGs,2N(hj) and G_s,N(h_j) have no terms, hence are equal to zero. This yields

F_s,2N(x)−F_s,N(x) = θ0

√q_j Z _2N

N

e^2iπhjt2

t^s dt+O(N^12−slogq_j).

It is immediate to check that R2a

a t^−se^2iπt2dtmin(a^−s−1, a^−s+1), thus

Z 2N N

e^2iπhjt2 t^s dt

|h_j|^s/2−1/2

Z 2N√

hj

N√

hj

e^2iπu2 u^s du

|h_j|^−1/2N^−smin(|Np

h_j|⁻¹,|Np h_j|).

One deduces (using thatq_jh_j is equivalent to q_j+1⁻¹ ) that

|F_s,2N(x)−Fs,N(x)| |θ₀|

√qj+1

N^s min( N

√qjqj+1

,

√qjqj+1

N ) +N^12−slogqj. Thus, by writingF_s,M(x)−F_s,N(x) as a dyadic sum we have

|F_s,M(x)−Fs,N(x)| |θ₀|δ_j

√q_j+1 (√

q_jq_j+1)^s + logqj

q_j^s−1/2 .

Recalling thatθ₀ is equal to zero when q_j 6= 2×odd, fixing an integerj₀ ≥1, for any M > N > qj0, one has

|F_s,M(x)−F_s,N(x)| X

j≥j0,qj6=2∗odd

δ_j

√qj+1

(√

qjqj+1)^s +X

j≥j0

logq_j q^s−1/2_j

+X

j≥j0

1 q^s−1/2_j+1

.

The second and third series always converge when j₀ → ∞, and the first does when Σ_s(x)<∞.

4.2. Divergence part: item (ii). Let 0 < ε < 1/2 a small constant. Let N_j =εq_j and M_j = 2ε√

q_jq_j+1. Proceeding exactly as in the previous proof we get

Fs,Mj(x)−Fs,Nj(x) = θ0

√q_j Z Mj

Nj

e^2iπhjt2

t^s dt+O

q

1 2−s j logqj

.

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

|F_s,Mj(x)−Fs,Nj(x)| ≥ |θ₀|

√qj

Mj−Nj

2·M_j^s ≥ |θ₀|ε 2√

qj+1−√ qj

2^1+s·ε^s·(qjqj+1)^s/2

r qj+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. Let h >0, 1/2< s <3/2 and q²h1. We have

F_s,N p

q + 2h

−F_s,N p

q +h

= θ₀

√q Z N

0

e^2iπ2ht2 −e^2iπht2 t^s dt (5.1)

+G_s,N(2h)−G_s,N(h)

+O

|qh|^s−1/2 . Proof. First, one writes

(5.2) F_s,N(x) =F_s,N^11[0,1](x) = X

m≥1

F_s,N/2^11[1,2]m(x).

Observe that when N is divisible by 2, there may be some terms appearing twice in the preceding sum, so there is not exactly equality. Nevertheless, in this case, only a few terms are added and they do not change our estimates. This is left to the reader.

We are going to estimate (5.1) but with F_s,N^11[1,2] and G¹_s,N^1[1,2] instead ofF_s,N and Gs,N, with an error term suitably bounded by above. Then, using this result with N substituted by N/2^m, and then summing overm= 1, ...,blog₂Ncwill give the result (form >blog₂Nc, the sumF_s,N/2^11[1,2]m is empty).

We start from equation (3.10) applied withhand 2h, and then we apply Lemma 3.4, but this time equation (3.4) instead of (3.3). Let us introduce for all integers kthe quantity

E_N^k := F_s,N^ηk p

q + 2h

−F_s,N^ηk p

q +h (5.3)

+θ₀N^1−s

√q Z

R

e^2iπN22ht2−e^2iπN2ht2

t^s η^k(t)dt +G^η_s,N^k (2h)−G^η_s,N^k (h),

with η^k defined as in Proposition 3.6. By the exact same computations as in Lemma 3.9, one obtains the upper bound

|E_N^k| β_N^k X

m∈J_k∩Z^∗

2^−kN^1−s/√ q

(2^−2kN²h)^−1/2 + X

m∈Z^∗

(2^−kN^1−s/√

q)N²h2^−2k

(1 +N²h2^−2k+ 2^−k|^{N m}_q −2N²h|)^5/2, withJ_N^k =

(2 + 2^−k−1)N qh,(2 + 2^−k+1)N qh and β_N^k =

( 1 if 2^−2kN²h≤1, 0 otherwise.

Then, as at the end of the proof of Lemma 3.9, since hq⁻², we can bound the sums by

|E_N^k| β_N^k2^−2kN^2−s

√

√h

q +(N^1/2−s)p

(N/q)2^−2kN²h2^−2k (1 +N²h2^−2k+ (N/q)2^−2k)^5/2 +(√

qN^−s)N²h2^−2k (1 +N²h2^−2k)^3/2 and then adding up ink≥1 we get

∞

X

k=1

|E_N^k| Ee_N =

p1/hq

N^s min(1, N qh) +

√N

N^s min(1, N qh) +

√q

N^smin(1, N²h).

The same holds true for the functions fη^k = η((2−t)/2^−k), and for ψe since it is similar to η¹, so by (3.6) we finally obtain that

F_s,N^11[1,2]

p q + 2h

−F_s,N^11[1,2]

p q +h

= θ₀

√q Z N

0

e^2iπ2ht2−e^2iπht2 t^s dt (5.4)

+G¹_s,N^1[1,2](2h)−G¹_s,N^1[1,2](h)

+O

Ee_N .

The same holds true with N/2^m instead of N. To get the result, using (5.2), it is now enough to sum the last inequality over m= 1, ...,blog₂Nc. Let us treat the first term. One has

blog₂Nc

X

m=1

p1/hq

N^s min(1, N qh) =

blog₂Nc

X

m=1

p1/hq

(N2^m)^smin(1, N2^mqh)

=

p1/hq N^s

blog₂Nc

X

m=1

min(2^ms, N2^m(1−s)qh)

≤

p1/hq N^s

+∞

X

m=1

min(2^ms, N2^m(1−s)qh)

p1/hq N^s 2^{M s},

whereM is the integer part of the solution of the equation 2^{M s} =N2^m(1−s)qh, i.e.

2^M hN qh. Hence the first sum is bounded above by

√

1/qh

(1/qh)^s. The other terms are

treated similarly, and finally (5.1) is true with an error term bounded by above by O

p1/qh (1/qh)^s +

√q (h^−1/2)^−s

!

which isO((qh)^s−1/2) onhq⁻².

We also need to control theL² norm of the main term.

Lemma 5.2. Let 0< s≤1 and fix 0< H <1. Let f_s,N(·) =

Z N 0

e^{2iπt2(δ+2·)}−e^{2iπt2(δ+·)}

t^s dt.

Then for any N >0 , kf_s,N(·)k_L2(eµH)min H^(s−1)/2, H|δ|^(s−3)/2 .

Proof. •Let us treat first the case|δ|< H/4. Using a change of variable, one has f_s,N(h) =H^(s−1)/2

Z N√ H 0

e^2iπt2δ+2hH −e^2iπt2δ+hH

t^s dt.

We are interested in the range H < h < 2H, and in this case the ratios ^δ+2h_H ,

δ+h

H are bounded, so that the integral is bounded by a constant independent ofN. One deduces thatkf_s,N(·)k_L2(µeH)H^(s−1)/2.

• Assume then that |δ|> 4H. Assume that δ > 0 (the same holds true with negative δ’s). Using a change of variable, one has

f_s,N(h) =|δ|^(s−1)/2 Z N√

|δ|

0

e^{2iπt2(1+2hδ )}−e^{2iπt2(1+hδ)}

t^s dt.

The integral between 0 and 1 is clearlyO(h/|δ|). For the other part, one has (after integration by parts)

Z N√ δ 1

e^{2iπt2(1+2hδ )}−e^{2iπt2(1+hδ)}

t^s dt = O(h/|δ|),

so that |f_s,N(h)| H|δ|^(s−3)/2 for any H < h < 2H. Hence kf_s,N(·)k_L2(eµH) H|δ|^(s−3)/2.

• It remains us to deal with the caseH/4< δ≤4H. One observes that f_s,N(h) =D(δ+ 2h)−D(δ+h) +O(H), where D(v) =

Z N 1

t^−se^2iπvt2dt.

It is enough to get the bound Z H

0

|D(v)|²dvH^s,

which follows from the fact that |D(v)| |v|^(s−1)/2 when s < 1 and |D(v)|

1 + log(1/|v|) whens= 1.

Finally, the oscillating behavior of Gs,N(h) gives us the following.

Proposition 5.3. Let 0< H ≤q⁻² and |δ| ≤√

H/q. Let g_s,N(·) =F_s,N

p

q +δ+ 2·

−F_s,N p

q +δ+·

− θ₀

√qf_s,N(·).

One haskg_s,Nk_L2(eµH)H^s−1/22 .

Proof. We considerµH = (eµH)|R⁺. By (3.1), it is enough to treat the caseδ+h >0 and δ+ 2h >0. Proposition 5.1, applied successively with hn := 2⁻ⁿ(δ+h) and eh_n:= 2⁻ⁿ(δ+h/2), and summing overn≥0, we get that

g_s,N(h) =G_s,N(δ+ 2h)−G_s,N(δ+h) +O

(q(|δ|+|h))^s−1/2 . (5.5)

Thus, sinceq(|δ|+|h|)√

H, it is enough to show that (5.6) kG_s,N(δ+·)k_L2(µH) H^s−1/22 .

Assume first that |δ| ≥3H. By expanding the square and changing the order of summation, and using thatδ+ 2H ≤2|δ|, we have for some cn,m ≥0

kG_s,N(δ+·)k²_L2(µ

H) (q|δ|)^2s−1

2b2N|δ|qc

X

n,m=1

|θ_m| m^s

|θ_n| n^s

Z δ+2H δ+H+cn,m

e^2iπ

n2−m2 q2h dh

H

(q|δ|)^2s−1

2b2N|δ|qc

X

n,m=1

|θ_m| m^s

|θ_n| n^s

Z δ+2H δ+H

e^2iπ

n2−m2 q2h dh

H . Since for |M| ≥1 and 0< ε1

Z 1+ε 1

e^2iπMt dt 1

|M|, the previous sum is bounded above by

(q|δ|)^2s−1



 X

m≥1

1 m^2s +|δ|

Hq²|δ|X

m≥1

1 m^1+s

X

j≥1

1 j^1+s



,

withj=|n−m|. The term between brackets is bounded by a universal constant (since q²δ²/H ≤1), hence (5.6) holds true. It is immediate that the same holds true with (µeH)_|R⁻.

Further, assume that |δ|<3H. SettingH_k= 2^−kH, one has kG_s,N(δ+·)k²_L2(eµH)≤

Z 5H 0

|G_s,N(h)|²dh

H ≤ X

k≥−2

2^−kkG_s,N(·)k²_L2(µ_Hk). Now, observing that [Hk, Hk−1] ⊂ 3Hk+ ([−2H_k,−H_k]∪[Hk,2Hk]), one can apply (5.6) with H=H_k and δ= 3H_k to get

kG_s,N(·)k²_L2(µ_Hk) ≤ kG_s,N(δ_k+·)k_L2(µ_Hk)≤H

s−1/2 2

k =H^s−1/22 2^−ks−1/22 .