Central limit theorems for simultaneous Diophantine approximations

Central limit theorems for simultaneous Diophantine approximations Tome 4 (2017), p. 1-35.

<http://jep.cedram.org/item?id=JEP_2017__4__1_0>

© Les auteurs, 2017.

Certains droits réservés.

Cet article est mis à disposition selon les termes de la licence CREATIVECOMMONS ATTRIBUTION–PAS DE MODIFICATION3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/

L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.cedram.org/legal/).

Publié avec le soutien

du Centre National de la Recherche Scientifique

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

CENTRAL LIMIT THEOREMS FOR SIMULTANEOUS DIOPHANTINE APPROXIMATIONS

by Dmitry Dolgopyat, Bassam Fayad & Ilya Vinogradov

Abstract. — We study the distribution modulo1of the values taken on the integers ofrlinear forms indvariables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii n^−r/d. By the Khintchine-Groshev theorem on Diophantine approximations,r/dis the critical exponent for the infinite number of hits.

Résumé(Théorème central limite pour des approximations diophantiennes simultanées) Nous étudions la loi de probabilité modulo1des valeurs prises sur les entiers parrformes linéaires de dvariables à coefficients aléatoires. Nous montrons un théorème central limite,

« en moyenne » et « presque sûr », pour le nombre de points atteignant simultanément des cibles de rayon décroissant à une vitessen^−r/d. D’après le théorème de Khintchine-Groshev sur les approximations diophantiennes,r/dest le seuil critique à partir duquel le nombre des points tend vers l’infini.

Contents

1. Introduction. . . 2

2. Central Limit Theorems on the space of lattices. . . 4

3. Diagonal actions on the space of lattices and Diophantine approximations . . . . 7

4. An abstract Central Limit Theorem. . . 9

5. Preliminaries on diagonal actions and Siegel transforms . . . 19

6. Proof of the CLT for diagonal actions. . . 22

7. Related results. . . 26

8. Variances. . . 29

Appendix. Truncation and norms. . . 30

References. . . 34

Mathematical subject classification (2010). — 60F05, 37A17, 11K60.

Keywords. — Central limit theorem, weakly dependent random variables, diophantine approxima- tion, linear forms, space of lattices.

D.D. was partially supported by the NSF.

B.F. was partially supported by ANR-15-CE40-0001 and by the project BRNUH.

I.V. appreciates the support of Fondation Sciences Mathématiques de Paris during his stay in Paris.

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement no. 291147.

1. Introduction

1.1. Results. — An important problem in Diophantine approximation is the study of the speed of approach to0of a possibly inhomogeneous linear form of several variables evaluated at integers points. Such a linear form is given by`:T×T^d×Z^d→T

`_x,α(k) =x+

d

X

i=1

k_iα_i (mod 1).

More generally, forr >1, one can considerr linear forms`_xj,α^j(k) forj = 1, . . . , r corresponding to a := (α^j_i) ∈ T^d×r and x := (x¹, . . . , x^r) ∈ T^r, where each α^j, j= 1, . . . , r, is a vector inT^d.

Diophantine approximation theory classifies the matricesaand vectorsxaccording to how “resonant” they are; i.e., how well the vector (`_xj,α^j(k))^r_j=1 approximates 0:= (0, . . . ,0)∈R^r as k varies over a large ball inZ^d. One can then fix a sequence of targets converging to0, say intervals of radius rn centered at0 with rn →0, and investigate the integers for which the target is hit, namely the integers k such that

`^j_xj,αj(k)∈[−r_|k|, r_|k|]for everyj = 1, . . . , r. An important class of targets is given by radii following a power law,rn =cn^−γ for someγ, c >0 (see for example [18, 15, 30]

or [5] for a nice discussion related to the Diophantine properties of linear forms).

Fix a norm | · |onR^d, and letk · kdenote the Euclidean norm onR^r. Forc >0, define the following sets

B1(k, d, r, c) = [0, c|k|^−d/r]^r⊂R^r and

B2(k, d, r, c) ={a∈R^r:kak6c|k|^−d/r}.

Forι= 1 or2, we then introduce VN,ι(a,x, c) = #

06|k|< N: (`_xj,α^j(k))^r_j=1∈Bι(k, d, r, c) , UN,ι(a, c) =VN,ι(a,0, c).

(1.1)

A matrix a ∈ T^d×r is said to be badly approximable if for some c > 0, the se- quence UN,ι(a, c) is bounded. By contrast, matrices a for which U_N,ι^ε (a, c) is un- bounded for some ε >0, whereU_N,ι^ε (a, c) is defined asUN,ι, but with radiicn^−dr−ε instead of cn^−d/r are calledvery well approximable or VWA. The obvious direction of the Borel-Cantelli lemma implies that almost every a ∈ T^d×r is not very well approximable (cf. [7, Chap. VII]). The celebrated Khintchine-Groshev theorem on Diophantine approximation implies that badly approximable matrices are also of zero measure [17, 15, 12, 28, 6, 4]. Analogous definitions apply in the inhomogeneous case ofVN,ι(a,x, c), and similar results hold.

For targets given by a power law, the radii cn^−d/r are thus the smallest ones to yield an infinite number of hits almost surely. A natural question is then to investigate statistics of these hits, which we call resonances. In the present paper we address in this context the behavior of the resonances on average over a and x, or on average

overawhilexis fixed at0or fixed at random. LetVoldenote the Euclidean volume and consider the expected number of hits

VbN,ι= Vol(Bι(1, d, r, c)) lnN

whenxandaare uniformly distributed on corresponding tori. LetN (m, σ²)denote the normal distribution with meanmand varianceσ².

Theorem1.1. — Suppose (r, d) 6= (1,1), and let a be uniformly distributed on T^dr. Then,

U_N,ι(a, c)−Vb_N,ι

√ lnN

converges in distribution toN (0, σ_ι²) asN→ ∞, where σ₁²= 2c^rdζ(r+d−1)

ζ(r+d) Vol(B), σ²₂= π^r/2 Γ(^r₂+ 1)σ²₁, (1.2)

andB is the unit ball in| · |-norm.

Remark1.2. — The restriction(r, d)6= (1,1)above is necessary. In fact, it is shown in [25, 27] using continued fractions that in that case the Central Limit Theorem (CLT) still holds forUN, but the correct normalization should be√

lnNln lnN rather than

√ lnN.

Theorem1.3. — Let a andxbe uniformly distributed inT^dr. Then, VN,ι(a,x, c)−VbN,ι

q Vb_N,ι converges in distribution toN (0,1)asN → ∞.

The preceding theorems give CLTs in the cases of x fixed to be0 or x random.

The CLT also holds for almost every fixedx.

Theorem1.4. — Suppose (r, d) 6= (1,1). For almost every x, if a is uniformly dis- tributed inT^dr, then(VN,ι(a,x, c)−VbN,ι)/

q

VbN,ι converges in distribution to a nor- mal random variable with zero mean and variance one.

1.2. Plan of the paper. — Using a by now standard approach of Dani correspon- dence (cf. [9, 23, 24, 1, 2, 3, 21]) we deduce our results about Diophantine approxi- mations from appropriate limit theorems for homogeneous flows. Namely we need to prove CLTs for Siegel transforms of piecewise smooth functions; these limit theorems are formulated in Section 2. The reduction of the theorems of Section 1 to those of Section 2 is given in Section 3. The CLTs in the space of lattices are in turn deduced from an abstract Central Limit Theorem (Theorem 4.2) for weakly dependent random variables which is formulated and proven in Section 4. In order to verify the condi- tions of Theorem 4.2 for the problem at hand we need several results about regularity of Siegel transforms which are formulated in Section 5 and proven in the appendix.

In Section 6 we deduce our Central Limit Theorems for homogeneous flows from the abstract Theorem 4.2. Section 8 contains the proof of the formula (1.2) on the vari- ances. Section 7 discusses some applications of Theorem 4.2 beyond the subject of Diophantine approximation.

2. Central Limit Theorems on the space of lattices

2.1. Notation. — We letG= SL_d+r(R),Ge = SL_d+r(R)n R^d+r. The multiplication rule inGe takes form (A, a)(B, b) = (AB, a+Ab). We regard G as a subgroup of Ge consisting of elements of the form (A,0). We let L be the abelian subgroup of G consisting of matricesΛa, andLe be the abelian subgroup ofGe consisting of matrices (Λ_a,(0, y))where0is an origin inR^d,yis anr-dimensional vector,ais ad×rmatrix and

Λa=

Idd 0 a Id_r

.

Let M be the space of d+r dimensional unimodular lattices and Mfbe the space of d+r dimensional unimodular affine lattices. We identify M andMfrespectively withG/SLd+r(Z)andGe/SLd+r(Z)n Z^d+r.

We will need spacesC^s,r(R^p), C^s,r(M), andC^s,r(Mf)of functions which can be well approximated by smooth functions, given s,r > 0. Recall first that the space C^s(R^p)consists of functionsf:R^p→Rwhose derivatives up to ordersare bounded.

To define spacesC^s(M)andC^s(Mf), fix bases for Lie(G)and Lie(Ge); then,C^s(M) and C^s(Mf) consist of functions whose derivatives corresponding to monomials of order up tosin the basis elements are bounded (see Appendix for precise definitions).

Now we defineC^s,r-norm on a space equipped with aC^s-norm and anL¹-norm by kfkC^s,r = sup

0<ε<1

sup

f⁻6f6f⁺ kf⁺−f⁻k_L1<ε

ε^r(kf⁺kC^s+kf⁻kC^s).

(2.1)

In the examples considered above (R^d+r,M and Mf), the L¹-norm is taken with respect to the Haar measure. Some properties of these spaces are discussed in the Appendix.

Given a functionf onR^r+d we consider its Siegel transformsS(f) :M →Rand Sf(f) :Mf→Rdefined by

S(f)(L) = X

e∈L

f(e), Sf(f)(Lf) = X

e∈Lf

f(e).

We emphasize that Siegel transforms of smooth compactly supported functions are never bounded but their growth at infinity is well understood, see Subsection 5.3.

2.2. Results for the space of lattices. — In this section we present general Central Limit Theorems for Siegel transforms. The reduction of Theorems 1.1, 1.3, 1.4 to the results stated here will be performed in Section 3.

Letf ∈C^s,r(R^d+r)be a non-negative function supported on a compact set which does not contain 0. (The assumption that f vanishes at zero is needed to simplify

the formulas for the moments of its Siegel transform. See Proposition 5.1.) Denote f =RR

R^d+rf(x, y)dxdy.

Given positive numbers K and αwe say that a subset S ⊂ M is (K, α)-regular if S is a union of codimension 1 submanifolds and there is a one-parameter sub- grouphu⊂Lsuch that

µ(L :h_[−ε,ε]L ∩S6=∅)6Kε^α,

where µ denotes the Haar measure on M. We say that a function ρ: M → R is (K, α)-regular if supp(ρ) has a (K, α)-regular boundary and the restriction of ρ to supp(ρ)belongs toC^αwith

kρkC^α(supp(ρ))6K.

(K, α)-regular functions onMfare defined similarly.

Let A be subgroup of G consisting of diagonal matrices. We use the notation da for Haar measure on A. We say that ρ is K-centrally smoothable if there is a positive function φsupported in a unit neighborhood of the identity inAsuch that R

Aφ(a)da= 1and

ρφ(L) :=

Z

A

ρ(aL)φ(a)da

hasL^∞norm less thanK. We say that a functionρonMfisK-centrally smoothableif ρ^∗(L) = sup

x

ρ(L,x)

is a K-centrally smoothable function on M. As before, we write N (m, σ²) for the normal distribution with m and variance σ² and “=⇒” stands for convergence in distribution.

Forp∈Nandt∈R, we denote thep×pdiagonal matrix D_p(t) =





 2^t

. .. 2^t







and let

(2.2) g=

Dd(−1) 0 0 Dr(d/r)

. Theorem2.1. — Suppose that (r, d)6= (1,1).

(a) There is a constantσ such that ifL is uniformly distributed on M then PN−1

n=0 S(f)(gⁿL)−N f σ√

N =⇒ N (0,1)

asN → ∞.

(b) Fix constants C, u, α, ε with ε <1/2. Suppose that L is distributed according to a densityρN which is(CN^u, α)-regular andC-centrally smoothable. Then

PN−1

n=N^εS(f)(gⁿL)−N f σ√

N =⇒ N (0,1)

asN → ∞.

Theorem2.2

(a) LetLfbe uniformly distributed onMf. Then there is a constant eσsuch that PN−1

n=0 Sf(f)(gⁿLf)−N f eσ√

N =⇒ N (0,1)

asN → ∞.

(b) Fix constants C, u, α, ε with ε <1/2. Suppose that Lfis distributed according to a densityρN which is(CN^u, α)-regular andC-centrally smoothable. Then

PN−1

n=N^εSf(f)(gⁿLf)−N f eσ√

N =⇒ N (0,1)

asN → ∞.

LetDe be an unstable box, that is

De={(Λa,(0,x))Lf0}_{(a,x)∈R1×R2},

whereLf0is a fixed affine lattice andR₁andR₂are boxes inR^d×randR^rrespectively.

Consider a partitionΠofDe into L-boxes. Thus elements ofΠare of the form {(Λa,(0,x0))Lf0}_a∈R1

for some fixed x0.

Theorem2.3. — Suppose that(r, d)6= (1,1). Then for each unstable cubeDe and for almost everyL ∈De, ifLfis uniformly distributed in Π(L), then

PN−1

n=0 Sf(f)(gⁿLf)−N f eσ√

N =⇒ N (0,1)

asN → ∞.

The explicit calculation ofσandσewhenf is an indicator function (the case needed for Theorems 1.1-1.4) will be given in Section 8.

Remark2.4. — Central Limit Theorems for partially hyperbolic translations on ho- mogeneous spaces are proven in [10] (for bounded observables) and in [22] (for L⁴ observables); see also [31, 26] for important special cases. It seems possible to prove Theorem 2.1 for sufficiently large values of d+rby verifying the conditions of [22].

Instead, we prefer to present in the next section an abstract result which will later be applied to derive Theorems 2.1, 2.2, and 2.3. We chose this approach for three reasons.

First, this will make the paper self contained. Second, we replace theL⁴ assumption of [22] by a weaker L^2+δ assumption which is important for small d+r. Third, our approach allows to give a unified proof for Theorems 2.1, 2.2, and 2.3.

3. Diagonal actions on the space of lattices and Diophantine approximations

In this section we reduce Theorem 1.1, 1.3, and 1.4 to Theorems 2.1–2.3. To fix our notation we considerUN,1andVN,1, the analysis ofUN,2 andVN,2 being similar. We also drop the extra subscript and writeUN,1 andVN,1 as UN andVN, respectively, until the end of this section.

In Subsections 3.1 and 3.2 we explain how to reduce Theorem 1.1 to Theorem 2.1.

The reductions of Theorems 1.3 and 1.4 to Theorems 2.2 and 2.3 require only minor modifications which will be detailed in Section 3.3.

3.1. Dani correspondence. — In this subsection we use the Dani correspondence principle to reduce the problem to a CLT for the action of diagonal elements on the space of lattices of the formΛawhere ais random.

Letabe the matrix with rows α_i∈R^d,i= 1, . . . , r. Let Λa=

Idd 0 a Id_r

. Letφbe the indicator of the set

(3.1) E_c :=n

(x, y)∈R^d×R^r:|x| ∈[1,2],|x|^d/ry_j ∈[0, c] forj = 1, . . . , ro and consider its Siegel transformΦ =S(φ).

Nown= (n₁, . . . , n_d)with|n|6N contributes toU_N(a, c)(from (1.1)) precisely when there exists(m1, . . . , mr)∈Z^r such that

(3.2) X^d

i=1

n_iα_1,i+m₁, . . . ,

d

X

i=1

n_iα_r,i+m_r

∈B(n, d, r, c).

Clearly such a vector(m₁, . . . , m_r)is unique. It is elementary to see that (3.2) holds if and only if

(3.3) g^tΛ_a(n₁, . . . , n_d, m₁, . . . , m_r)∈E_c.

for some integert62^[log2N] whereg is the diagonal matrix defined in (2.2).

Below we will use the notion AN BN to mean that the ratio |AN|/BN is bounded. From (3.2)–(3.3) we obtain

Lemma3.1. — For each ε >0 U_N(a, c) =

[log₂N]

X

t=[(log₂N)^ε]

Φ◦g^t(Λ_aZ^d+r) +R_N, with kRNk_L1(a)(log₂N)^ε and L¹(a) denoting the L¹-norm with respect to the Lebesgue measure on the unit cube in R^d×r.

Proof. — From (3.2)–(3.3), it follows that R_N 6#

2^[log2N]6|n|< N,

or|n|62^(log2N)ε) : (`<sub>0,α</sub>1(n), . . . , `_0,αr(n))∈B(n, d, r, c)

.

Note that for a fixed n ∈ Z^d and j ∈ 1, . . . , r, the form{`j(n,0, αj)} is uniformly distributed on the circle. Hence

Leb

a∈T^rd: (`<sub>0,α</sub>1(n), . . . , `0,α^r(n))∈B(n, d, r, c) 1

|n|^d, and so

kRNk_L1(a) X

2[log2N]6|n|6N

1

|n|^d + X

|n|62(log2N)ε

1

|n|^d (log₂N)^ε. Hence, to prove Theorem 1.1 we can replaceUN(a, c)by

[log₂N]

X

t=[(log₂N)^ε]

Φ◦g^t(Λ_aZ^d+r).

3.2. Changing the measure. — Note that the action ofg^ton the space of latticesM is partially hyperbolic and its unstable manifolds are orbits of the action Λ_a with a∈M(d, r)ranging in the set ofd×rmatrices. This will allow us to reduce the proof of Theorems 1.1, 1.3 and 1.4 to CLTs for the diagonal action on the space of lattices.

A similar reduction is possible for theg^t-action on the space of affine lattices since in that case the unstable manifolds for the action of g^t are given by (Λ_a,(0,x)), with a∈M(d, r),x∈R^r.

Letη = 1/k^10d wherek = [log₂N]. For i= 1, . . . , r+d−1 letti be independent uniformly distributed in[−η, η]. Also introduce a random matrixb∈M(r, d) where all the entries ofbare independent uniformly distributed in[−1,1]. Let

D_t= diag

1 +t₁, . . . ,1 +t_d+r−1, Qr+d−1

`=1 (1 +t<sub>`</sub>)−1 and

Λ_b=

Idd b 0 Id_r

. LetΛ(a,e b,t) =DtΛbΛa.

It is clear that ifa is distributed uniformly in a unit cube, thenΛ(a,e b,t) is dis- tributed according to a (Ck^10d,1)-regular and C-centrally smoothable density for some constantC. Note that

g^tΛ(a,e b,t) =DtΛb_tg^tΛa, where|bt|6e^−t. Observe also that forh∈GandE⊂R^d+r, we have

S(1E)(hL) =S(1h⁻¹E)(L), and thus ifεis sufficiently small then fort>k^εwe have

S(1E_c)(g^tΛ(a,e b,t)Z^d+r)−S(1E_c)(g^tΛaZ^d+r)

6S(1_Ee)(g^tΛaZ^d+r), whereEedenotes aCk^−10d neighborhood of the boundary ofEc. Now the same argu- ment as in Lemma 3.1 gives

Lemma3.2. — For each ε >0,

[log₂N]

X

t=[(log₂N)^ε]

Φ◦g^t(ΛaZ^d+r) =

[log₂N]

X

t=[(log₂N)^ε]

Φ◦g^t(eΛ(a,b,t)Z^d+r) +ReN,

with eR_N

_L1(a,b,t)1.

Now Theorem 1.1 follows from Theorem 2.1 except for the formula forσwhich is derived in Section 8.

3.3. Inhomogeneous case. — The reduction of Theorems 1.3 and 1.4 to Theorems 2.2 and 2.3 requires only small changes compared to the preceding section. To wit, Lem- mas 3.1 and 3.2 take the following form.

Lemma3.3

(a) For eachε >0, V_N(a,x, c) =

[log₂N]

X

t=[(log₂N)^ε]

Sf(1Ec)(g^t(Λ_aZ^d+r+ (0,x))) +R_N(a,x, c), wherekR_Nk_L1(a)(log₂N)^ε.

(b) Let bandthave the same distribution as in Subsection 3.2andybe uniformly distributed in [−1,1]^d. Then for eachε >0

[log₂N]

X

t=[(log₂N)^ε]

Sf(1Ec)(g^t(Λ_aZ^d+r+ (0,x)))

=

[log₂N]

X

t=[(log₂N)^ε]

Sf(1E_c)(g^t(eΛ(a,b,t)Z^d+r+ (y,x))) +ReN(a,b,t,x,y, c), with

eRN

_L1(a,b,t,y)1.

Note that in part (a) the error has small L¹(a) norm for each fixed x. This fol- lows from the fact that for each x and k, ak+x is uniformly distributed on T^r. This is useful in the proof of Theorem 1.4 since we want to have a control for each (or at least, most)x. We also note that part (b) is only needed for Theorem 1.3 since in Theorem 2.3 we start with initial conditions supported on a positive codimension submanifold ofMf. (One of the steps in the proof of Theorem 2.3 consists of fattening the support of the initial measure, see Subsection 6.3, however Lemma 3.3(b) is not needed at the reduction stage).

4. An abstract Central Limit Theorem

In this section, we present an abstract Central Limit Theorem for weakly dependent random variables that is well adapted to variables coming from deterministic dynami- cal systems. In order to make our theorem applicable to mixing flows on non-compact manifolds as well as to non-uniformly hyperbolic systems we allow the variables to

be unbounded and take into account the existence of small exceptional “bad sets” on which the variables are not controlled. In Section 6, we will use the abstract CLT (The- orem 4.2) to prove Theorems 2.1, 2.2, 2.3. There, we will take the variablesξ<sub>`</sub> to be ξ`(L) = Φ(g^`L),Φ =S(f)(orSf(f)), for a positive functionf ∈C^s,r(R^d+r)sup- ported on a compact set which does not contain0. The functionsΦare not bounded but, since we excluded the cases of linear lattices with r=d= 1, they are inL^s for s >2. The latter fact follows from the results of [23] and [21] (which will be recalled in Section 5.3 and the Appendix).

4.1. Bounded random variables. — Before we state our CLT for variables in L^s, s >2, we give a simplified version of it in the case of bounded variables.

In what follows C, u >0, θ ∈ (0,1) are fixed constants. Let ξn be a sequence of random variables satisfying the following conditions. Write ξbn =ξn−E(ξn)for the corresponding centered random variable.

(H1)] There exists filtration{F`}`>0such that for every`, kthere exists a bounded F`+k-measurable random variableξ_`,`+k withEξ_`,`+k=Eξ_`such that

P(|ξ`−ξ`,`+k|>θ<sup>k</sup>)6C(`+ 1)^uθ^k.

(H2)] For`, k, there exists an eventG`,ksuch thatP(G^c<sub>`,k</sub>)6Cθ<sup>k</sup> and forω∈Gk,`

|E(bξ`+k|F`)(ω)|6C(`+ 1)<sup>u</sup>θ<sup>k</sup>. (H3)] There is a sequencebk such that forω∈Gk,` andk⁰>k

E(bξ_{`+k</sub>ξb<sub>`+k}⁰|F`)(ω)−b_k⁰_−k

6C(`+ 1)<sup>u</sup>e<sup>u(k0−k)</sup>θ<sup>k</sup>. Theorem4.1. — If ξ` is a bounded sequence satisfying(H1)–g (H3)g then

Pn−1

`=0 (ξ`−E(ξ`))

√n

converges as n→ ∞to the normal distribution with zero mean and variance σ²=b₀+ 2

∞

X

k=1

b_k.

Theorem 4.1 follows from a more general statement (Theorem 4.2) that we now formulate and prove.

4.2. The general statement. — In what followsC, u >0, θ∈(0,1), ands > 2 are fixed constants. Let ξn be a sequence of random variables satisfying the following conditions. Writeξb_n=ξ_n−E(ξ_n)for the corresponding centered random variable.

(H1) Given anyK, there is a sequencesξ_n^K of random variables such that (H1a) |ξ^K_n|6K almost surely;

(H1b) E |ξ_n^K−ξn|

=O(K^1−s),E((ξ_n^K−ξn)²) =O(K^2−s).

(H2) There exists a filtrationF`=F`,n defined for06` < n such that for every pair of nonnegative numbers`, kwith`+k < n, there exists a variableξ<sup>K</sup><sub>`,`+k</sub> that is F`+k-measurable,|ξ_`,`+k^K |6K almost surely,Eξ_`,`+k^K =Eξ^K_` , and

P(|ξ^K<sub>`</sub> −ξ<sup>K</sup><sub>`,`+k</sub>|>θ<sup>k</sup>)6CK<sup>u</sup>(`+ 1)^uθ^k.

(H3) For`, k, there exists an eventGk,`such thatP(G^c<sub>k,`</sub>)6Cθ<sup>k</sup> and forω∈Gk,`

|E(bξ^K<sub>`+k</sub>|F`)(ω)|6C(`+ 1)^uK^uθ^k.

(H4) There exists a numerical sequenceb_K,k fork>0 such that forω∈G_k,` and k⁰ >k,

E(bξ^K_{`+k</sub>ξb<sup>K</sup><sub>`+k}0|F`)(ω)−bK,k⁰−k

6C(`+ 1)^uK^ue^u(k0−k)θ^k.

In the following Theorem, part (a) is sufficient to prove results such as Theo- rem 2.1(a), where the distribution of the lattices is given by the Haar measure, while part (b) is tailored to adapt to the case of localized initial conditions such as in Theorem 2.1(b).

Theorem4.2

(a) Under conditions(H1)–(H4) ifω is distributed according toP then Pn−1

`=0 ξb`

√n =⇒ N (0, σ²), with

(4.1) σ²=σ0+ 2

∞

X

j=1

σj, σj= lim

K→∞bK,j.

(b) Suppose conditions (H1)–(H4)are satisfied. Fixε >0 such that ^1+ε_s +ε <1/2 and setKn=n^(1+ε)/s. Suppose thatω is distributed according to a measurePn which has a densityρ_n=dPn/dP satisfying

(D1) ρ_n6Cn^u;

(D2) for eachk there is an Fk-measurable density ρn,k such that P(|ρn−ρ_n,k|>θ^k)6Cn^uθ^k;

(D3) For eachn^ε6`6n,

En(|ξ`−ξ<sub>`</sub>^Kn|)6CK_n^1−s,

whereEn denotes the expectation with respect toPn, that is,En(η) =E(ηρn).

Then,

Pn−1

`=n<sup>ε</sup>ξb<sub>`</sub>

√n =⇒ N(0, σ²).

4.3. Limiting variance. — Here we show that the normalized variance converges.

Lemma4.3. — Under conditions (H1)–(H4) we have that

n→∞lim

E((Pn−1

`=0 ξb<sub>`</sub>)²) n =σ², with σas in (4.1).

Proof. — First we record a property of cross-terms in the sum that lets us pass fromξ_n to the truncated sequence. We have

D:=E(bξ_m+jξb_m)−E(bξ_m+j^K ξb_m^K) =O(K^2−s).

(4.2)

Indeed, we have

D=E(ξ_m+jξ_m)−E(ξ^K_m+jξ_m^K)

−E(ξ_m+j)E(ξ_m) +E(ξ^K_m+j)E(ξ_m^K)

=E(ξ_m+j^K (ξ_m−ξ^K_m)) +E(ξ_m^K(ξ_m+j−ξ^K_m+j)) +E((ξ_m−ξ_m^K)(ξ_m+j−ξ_m+j^K ))

−E(ξ^K_m+j)E(ξm−ξ_m^K)−E(ξ^K_m)E(ξm+j−ξ_m+j^K )−E(ξm−ξ^K_m)E(ξm+j−ξ^K_m+j).

Now applying the Cauchy-Schwarz inequality followed by (H1a) and (H1b), we arrive at the bound (4.2) sinces >2.

Next, applying (H1), (H3), and (H4) with`= 0 gives E(bξ_m+j^K ξb_m^K) =E(bξ_m+j^K ξb^K_m1G_m,0) +E(bξ_m+j^K ξb_m^K1G^c_m,0)

=b_K,j+O(K^ue^ujθ^m) +O(kbξ^K_m+jk_L^∞kξb_m^Kk_L^∞P(G^c_m,0))

=bK,j+O(K^ue^ujθ^m) +O(K²θ^m).

(4.3)

Combining (4.2) and (4.3) we get

bK,j=E(bξm+jξbm) +O(K^2−s) +O(K^ue^ujθ^m) +O(K²θ^m).

Take a small numbereε >0and assume thatj 6εK,e K <K <e 2K. Takingm=K/2 we see that

bK,j−b

K,je =O(K^2−s).

Therefore for eachj, the following limits exist, σ_j:= lim

K→∞b_K,j. and moreover

(4.4) b_K,j=σ_j+O(K^2−s).

We give now the key estimates on the cross-terms in the variance according to different regimes for`andj.

Sublemma4.4. — We have the following estimates.

(a) There existsA >0such that for all (`, `+j)∈[0, n]²

|E(bξ`ξb`+j)|6A.

(b) Let η= min

3(s−2) 20(1+u),¹₅

. Ifj>lnn² and`6n, then E(bξ<sub>`</sub>ξb_`+j) =O(θ^ηj).

(c) Ifj 6lnn² and`>lnn⁴, then

E(bξ`ξb`+j)−σj=O(n⁻²).

(d) There existsC >0 andν >0 such that forj>lnn²,

|σj|6Cθ^νj.

Before we prove the sublemma, we show how it implies Lemma 4.3. We have E((Pn−1

`=0 ξb`)²)

n = 1

n ⁿ⁻¹

X

`=0

E(bξ²_`) + 2 X

`∈[0,n−2]

j∈[1,n−`]

E(bξ_{`</sub>ξb<sub>`+j})

= 1 n

ⁿ⁻¹ X

`=lnn⁴

E(bξ_`²) + 2 X

`∈[lnn⁴,n−2]

j∈[1,lnn²]

E(bξ`ξb`+j)

+o(1)

=σ0+ 2

lnn²

X

j=0

σj+o(1)

=σ0+ 2

∞

X

j=0

σj+o(1),

where the second equality follows from parts (a) and (b) of Sublemma 4.4, the third equality follows from part (c), and the fourth equality from part (d).

Proof of Sublemma4.4. — Part (a) follows from (4.2) and (H1a).

To prove part (b) assume j >lnn² and let K =θ^−δj with δ= 3/20(1 +u). We claim that

|E(bξ_{`</sub><sup>K</sup>ξb<sub>`+j}^K )|6Cθ^j/5. This is seen by writing

E(bξ_{`</sub><sup>K</sup>ξb<sub>`+j}^K ) =E(bξ^K_`,`+j/2E(bξ<sub>`+j</sub><sup>K</sup> |F`+j/2))−E(bξ_{`+j</sub><sup>K</sup> (bξ<sup>K</sup><sub>`,`+j/2</sub>−ξb<sup>K</sup><sub>`} )) =I+II and then estimating

II 6CKh

θ^j/2+K^1+u(`+ 1)^uθ^j/2i

6CK^2+2uθ^j/2 using (H1a) and (H2), and

I6CK×K^u(`+ 1)^uθ^j/26CK^2+2uθ^j/2 using (H1a) and (H3).

Now (4.2) implies that

|E(bξ_{`</sub>ξb<sub>`+j})|6Cθ^j/5+Cθ^(s−2)/2uj =O(θ^ηj) as needed.

To prove part (c) we letK=n^−2/(2−s). Then (4.2) yields E(bξ`ξb`+j)−E(bξ_{`</sub><sup>K</sup>ξb<sub>`+j}^K ) =O(n⁻²).

Now (H4) and (4.4) imply

E(bξ^K_{`</sub> ξb<sup>K</sup><sub>`+j})−σ_j=O(n⁻²) +O(θ^lnn3) and (c) follows.

To prove part (d), fixj, let K=θ^−εj where ε1, and take`=j/ε. Then (H4) and (4.4) imply that

E(bξ^K_{`</sub> ξb<sub>`+j}^K )−σ_j =O(θ^νj), withν =ε(s−2)/2. From (4.2) it follows that

E(bξ`ξb`+j)−σj =O(θ^νj),

which together with (b) yields (d).

4.4. Proof of Theorem 4.2. — In our proof, we shall use a standard Bernstein method based on “big block-small block” technique. That is, we divide the inter- val[0, n]into big blocks of lengthn^εalternating with small blocks of lengthn^ε2. The big blocks will be almost “independent,” each from the preceding ones, due to the buffer zones that are the small blocks, while the contribution of these small blocks can be neglected.

From now on we fix ε >0 so that ^1+ε_s +ε < ¹₂ and let K =K_n =n^(1+ε)/s. Let

`_m=m[n^ε], m= 0, . . . ,[n^1−ε] :=m_n. Denote Zm=

`_m+1−1

X

`=`m+n^ε2

ξb_`^K, Zem=

`_m+1−1

X

`=`m+n^ε2

ξb`, Zeem=

`_m+n^ε2−1

X

`=`_m

ξb`, Zˇ=

n−1

X

`=`_mn

ξb`, so that

n−1

X

`=0

ξb`=

m_n−1

X

m=0

ee Zm+

m_n−1

X

m=0

Zem+ ˇZ.

We claim that (4.5)

Pmn−1

m=0 Zee_m+ ˇZ

√n and

Pmn−1

m=0 (Z_m−Ze_m)

√n

converge to0in probability; it will therefore suffice for the proof of Theorem 4.2 (a), to prove the following.

Lemma4.5

Pm_n m=0Zm

σ√

n =⇒ N (0,1).

We first prove the claim. To begin with, observe that following the proof of Lemma 4.3, we have

(4.6) E

X^mn

m=0

ee

Zm+ ˇZ²

=O(n^1−ε+ε2) =o(n).

As for the second sum in (4.5), we write Z_m−Ze_m=

`m+1−1

X

`=`_m+n^ε2

ξ_{`</sub><sup>K</sup>−ξ<sub>`}

− E(ξ_{`</sub><sup>K</sup>)−E(ξ<sub>`}) . Condition (H1b) gives

E|Zm−Zem|62

`m+1−1

X

`=`m+n^ε2

E(|ξ^K<sub>`</sub> −ξ`|)6Cn^εK^1−s. Therefore,

(4.7)

mn

X

m=1

E|Z_m−Ze_m|

√n =O K^1−s√ n

=O

n^{1+εs −12}

=O n^−ε

−→0.

We now prepare for the proof of Lemma 4.5. We start by defining an exceptional set G^c_m on which we will not be able to exploit the almost independence of Z_m+1 from (Z1, . . . , Zm). The exact reasons for the definition of each condition on the

“good set”Gm will become evident in the course of the proof.

Letb`<sub>m+1</sub>=`_m+1+n^ε2/2. With setsG_`,k as in hypotheses (H2)–(H4), we let G⁽¹⁾_m =

n^ε

T

k=n^ε2

G`_m+1,k∩G

b`_m+1,k

form6m_n. Next, define for`>`_m+1+n^ε2 Ge_`=n

ω:E

ξ_`^K−ξ^K

`,`+n^ε2/2

F_b`m+1

(ω)6θ^nε

2/10o

and set

G⁽²⁾_m =

n^ε

T

k=n^ε2

Ge_`m+1+k. Fork, k⁰∈[n^ε2, n^ε]withk⁰−k>n^ε2/2 define

Em,k,k⁰ =n ω⁰:

E(bξ^K_{`m+1+k</sub>0|F<sub>`m+1+k+n}ε2/2)(ω⁰) >θ^nε

2/10o

and let

Gm,k,k⁰ =n

ω:E 1E_m,k,k0(ω⁰) F_`bm+1

(ω)6θ^nε

2/10o

and

G⁽³⁾_m = T

k,k⁰∈[n^ε2,n^ε] k⁰−k>n^ε2/2

G_m,k,k⁰.

Finally define “the good set”

Gm=G⁽¹⁾_m ∩G⁽²⁾_m ∩G⁽³⁾_m. Observe that (H2)–(H4) show that

(4.8) P(G^c_m)6Cθ^nε

2/100

.

The main step in the proof of Lemma 4.5, and thus of our CLT, is the following

Sublemma4.6. — Forω∈Gm, (4.9) lnE e^iλZm+1√n

F_b`m+1

(ω) =−n^ε

nλ²σ²(1 +o(1)), whereo(1)is uniform inm= 1, . . . , mn.

Proof. — To prove (4.9), we expand the exponential E e^iλZm+1√n

F_b`m+1

(ω) = 1 +i λ

√nE(Z_m+1|F_b`m+1)(ω)−λ²

2nE(Z_m+1² |F_b`m+1)(ω) +O λ³

n^3/2E(Z_m+1³ |F_b`m+1)(ω)

= 1 +i λ

√nE(Zm+1|F_b`m+1)(ω)−λ²

2nE(Z_m+1² |F_b`m+1)(ω) +oλ²

2nE(Z_m+1² |F_b`m+1)(ω) , where the last step uses that|Zm+1|6Kn^ε=o(n^1/2).

Next, (H3) implies that onGm(which is contained inG

`b_m+1,kfor eachk∈[n^ε2, n^ε]) we have

E(Zm+1|F_b`m+1)(ω) =E ^nε

X

j=n^ε2

ξb^K_{`</sub> F<sub>`bm+1}

(ω) =o θ^0.5nε

2 .

To complete the proof of (4.9) it suffices to show that forω∈G_m,

(4.10)

E(Z_m+1² |F_b`m+1)(ω)−n^εσ²

=o(n^ε).

Note that

E(Z_m+1² |F_b`m+1)(ω) =

n^ε

X

k,k⁰=n^ε2

E(bξ_`^K

m+1+kξb^K_`

m+1+k⁰|F_b`m+1)(ω).

Let us estimate individual terms in this sum. Without loss of generality assume that k⁰ >k. LetRbe a large constant and consider two cases.

(a)k > R(k⁰−k). In this case (H4) and (4.4) give E(bξ_{`</sub><sup>K</sup><sub>m+1+k</sub>ξb<sub>`}^K_m+1+k0|F_b`m+1)(ω) +O eθ^k

=bK,k⁰−k =σk⁰−k+O K^2−s

+O θe^k . (b)k6R(k⁰−k)and hencek⁰−k > n^ε2/R. Then

E(bξ_`^K

m+1+kξb_`^K

m+1+k⁰|F_b`m+1)(ω) =E(bξ^K

`m+1+k,`m+1+k+n^ε2/2ξb_`^K

m+1+k⁰|F_`bm+1)(ω) +E

ξb_{`</sub><sup>K</sup><sub>m+1+k</sub>−ξb<sub>`}^K

m+1+k,`m+1+k+n^ε2/2

ξb^K_{`m+1+k</sub>0|F<sub>b`m+1} (ω)

=I+II.

The second term is O θ^nε

2/10

K

since ω ∈ Ge`_m+1+k. For the first term use that ω∈Gm,k,k⁰ to obtain

|I|= E(bξ^K

`m+1+k,`m+1+k+n^ε2/2E(bξ_`^K

m+1+k⁰|F`<sub>m+1</sub>+k+n<sup>ε2/2</sup>)|F<sub>b`m+1</sub>)(ω) 6K²E(1E_m,k,k0(ω⁰)|F_b`m+1)(ω) +Kθ^nε

2/10

6K²θ^nε

2/10

,

so bothIandII are negligible. Combining the estimates of cases (a) and (b) we obtain

(4.10) completing the proof of (4.9).

Proof of Lemma4.5. — It now remains to derive Lemma 4.5 from (4.9). For j 6m, set

Zbj=

`j+1

X

`=`j+n^ε2

ξb_`,n^K_ε2/2. Then

E ^m

X

j=0

bZ_j−Z_j

=O θ^nε

2/10

and so

E e^i√λn

Pm+1 j=0 Z_j

−E e^i√λn(

Pm

j=0Zb_j)+i√^λ nZ_m+1

=O θ^nε

2/10 , (4.11)

E e^i√λn

Pm j=0Z_j

−E e^i√λn

Pm j=0Zb_j

=O θ^nε

2/10 . (4.12)

Therefore, E e^i√λn

Pm+1

j=0 Z_j^(4.11)

= E

e^i√λn

Pm j=0Zb_j

E e^i√λnZm+1|F_`bm+1

+O θ^nε

2/10

(4.8)

= E

e^{i√λnPmj=0Zbj}1GmE e^i√λnZm+1|F_`bm+1

+O θ^nε

2/100

(4.9)

= e^−σ

2λ2nε 2n E e^i√λn

Pm j=0Zbj

+o(n^ε/n)

(4.12)

= e^−σ

2λ2nε 2n E e^i√λn

Pm j=0Z_j

+o(n^ε/n). Iterating this recurrence relationm_n times we obtain

E e^{i√λnPmnj=0Zm}

=e^−σ2λ2/2+o(1),

completing the proof of Lemma 4.5, and thus of part (a) of Theorem 4.2.

Part (b) of Theorem 4.2 can be established by a similar argument and we just briefly describe the necessary changes. To extend the proof of the Central Limit Theorem to the setting of part (b) we need to prove (4.5) and (4.9) with En instead ofE. For (4.5) we need to prove the analogues of (4.6) and (4.7).

We claim the following. First, (4.9) still holds withEn instead ofE. Second,

(4.13) En

X^mn

m=1

ee

Zm+ ˇZ²

=O(n^1−ε+ε2) =o(n).