Jean-Pierre Conze

Stéphane Le Borgne

On the CLT for rotations and BV functions

Let x → x + α mod 1 be a rotation on the circle and let ϕ be a step function. We denote by ϕ n (x) the corresponding ergodic sums n-1 j=0 ϕ(x + jα). For a class of irrational rotations (containing the class with bounded partial quotients) and under a Diophantine condition on the discontinuity points of ϕ, we show that ϕ n / ϕ n 2 is asymptotically Gaussian for n in a set of density 1. The proof is based on decorrelation inequalities for the ergodic sums taken at times q k , where (q k ) is the sequence of denominators of α. Another important point is the control of the variance ϕ n 2 2 for n belonging to a large set of integers. When α is a quadratic irrational, the size of this set can be precisely estimated.

Introduction

For a dynamical system (X, µ, T ) and an observable ϕ on X, a general question is the asymptotic behaviour in distribution of the ergodic sums L-1 0 ϕ•T k after normalisation.

For a large class of observables and chaotic systems, many results of convergence toward a Gaussian distribution are known.

When the dynamical system has zero entropy, in particular for a rotation, the situation is dierent. Nevertheless one can ask if, at least, there are observables satisfying a non degenerate Central Limit Theorem. In this direction there are positive answers: R. Burton and M. Denker [START_REF] Burton | On the central limit theorem for dynamical systems[END_REF] in 1987, then T. de la Rue, S. Ladouceur, G. Peskir and M. Weber [START_REF] De La Rue | On the central limit theorem for aperiodic dynamical systems and applications[END_REF], M. Lacey [START_REF] Lacey | On central limit theorems, modulus of continuity and Diophantine type for irrational rotations[END_REF] proved for rotations the existence of functions whose ergodic sums satisfy a CLT after self-normalization. In general for a measure preserving aperiodic system, further results by D. Volný and P. Liardet [START_REF] Liardet | Sums of continuous and dierentiable functions in dynamical systems[END_REF], J.-P. Thouvenot and B. Weiss [START_REF] Thouvenot | Limit laws for ergodic processes[END_REF] showed that any distribution can appear as a limiting distribution of the ergodic sums of some functions after normalisation.

A dierent question is to ask if, for smooth systems, there is a CLT for explicit functions in a certain class of regularity. Here we consider step functions on X = R/Z and their ergodic sums ϕ n (x) := n-1 0 ϕ(x + jα) over an irrational rotation x → x + α mod 1.

By the Denjoy-Koksma inequality, if ϕ is a centered function with bounded variation, the sequence (ϕ n ) is uniformly bounded along the sub-sequence of denominators of α. But, besides, a stochastic behaviour at a certain scale can occur along other sub-sequences

(n k ). We propose a quantitative analysis of this phenomenon.

Let us mention the following related papers. For ψ := 1 [0, 1 2 [ -1 [ 1 2 ,0[ , F. Huveneers [15] studied the existence of a sequence (n k ) n∈N such that (ψ n k ) after normalization is asymptotically normally distributed. In [START_REF] Conze | Diusive behaviour of ergodic sums over rotations[END_REF] it was shown that, when α has unbounded partial quotients, along some subsequences the ergodic sums of ϕ in a class of step functions can be approximated by a Brownian motion.

Here we will use as in [START_REF] Huveneers | Subdiusive behavior generated by irrational rotations, Ergodic Theory Dynam[END_REF] a method based on decorrelation inequalities which applies in particular when the sequence of partial quotients of α is bounded (α is said to be of bounded type or bpq) or under a slightly more general Diophantine assumption. It relies on an abstract central limit theorem valid under some suitable decorrelation conditions.

If ϕ is a step function, we give conditions which ensure that for n in a set of integers of density 1, the distribution of ϕ n / ϕ n 2 is asymptotically Gaussian (Theorem 2.4). Beside the remarkable recent temporal limit theorems for rotations of bounded type (see [START_REF] Beck | Randomness of the square root of 2 and the giant leap, part 1, 2[END_REF], [START_REF] Beck | Probabilistic Diophantine approximation, Randomness in lattice point counting[END_REF],

[10], [START_REF] Bromberg | A temporal central limit theorem for real-valued cocycles over rotations[END_REF]), this shows that a spatial asymptotic normal distribution can also be observed for n in a large set of integers.

An important point is the control of the variance ϕ n 2 2 . In Section 1, we study the set of integers for which the variance ϕ n 2 2 of the ergodic sums is big (expected to be of order ln n for n belonging to a set of density 1, in the case α bpq). The most precise information is obtained in the special case where α is a quadratic irrational in Subsection 1.4.

The central limit theorem is presented in Section 2. It is based on the decorrelation between the ergodic sums at times q k (the denominators of α) and on an abstract central limit theorem. To apply the results to a step function, a Diophantine condition is needed on the discontinuities of ϕ which holds generically.

The proofs of the CLT and the decorrelation are given in Sections 3 and 4. In Appendix 1, we prove a proposition used for quadratic numbers in the study of the variance.

The results of this paper have been announced in [START_REF] Conze | On the CLT for rotations and BV functions: note aux C[END_REF]. The authors thank the referees for their helpful remarks.

Variance of the ergodic sums

Notation The uniform measure on T 1 identied with X = [0, 1[ is denoted by µ. A function ϕ on T 1 is viewed as a 1-periodic function of a real variable. We denote by V (ϕ) the variation of the restriction of ϕ to [0, 1] and write BV for with bounded variation.

The class of real centered BV functions on T 1 is denoted by C. It contains the 1-periodic step functions with a nite number of discontinuities. The Fourier coecients of a function ϕ are denoted by φ(r). For ϕ ∈ C, we can write: φ(r) = γ r (ϕ) r , r = 0, with K(ϕ) := sup

r =0 |γ r (ϕ)| ≤ V (ϕ) 2π < +∞. (1)
Let α = [0; a 1 , a 2 , . . .] be an irrational number in ]0, 1[, with partial quotients a n = a n (α), numerators p n and denominators q n , n ≥ 1.

The ergodic sums n-1 j=0 ϕ(x + jα) of a 1-periodic function ϕ for the rotation by α are denoted by ϕ n (x). Their Fourier expansion is ϕ n (x) = In this subsection, we recall some classical results on diophantine approximation. For this material we refer to [START_REF] Khinchin | Continued Fractions, English translation[END_REF] or [START_REF] Lang | Introduction to Diophantine Approximations[END_REF], as well as J. Beck's book [START_REF] Beck | Probabilistic Diophantine approximation, Randomness in lattice point counting[END_REF].

For u ∈ R, {u} denotes its fractional part and u := inf n∈Z |u -n| = min({u}, 1 -{u}) its distance to Z. Recall that 2 x ≤ | sin πx| ≤ π x , ∀x ∈ R.

For n ≥ 1, writing α = p n q n + θ n q n

, we have 1 a n+1 + 2 ≤ q n q n+1 + q n ≤ q n q n α = q n |θ n | ≤ q n q n+1 = q n a n+1 q n + q n-1 ≤ 1 a n+1 ,

θ n = (-1) n q n α , α = p n q n + (-1) n q n α q n , 1 2 q -1 n+1 ≤ θ n ≤ q -1 n+1 ,

q n+1 /q n+k ≤ C ρ k , ∀n, k ≥ 1, with C = 5 + √ 5 2 , ρ = √ 5 -1 2 < 1. (3)
Let us show the last inequality: for n ≥ 1 xed, putting r 0 = q n , r 1 = q n+1 , r k+1 = r k + r k-1 , for k ≥ 1, we have q n+k ≥ r k , ∀k ≥ 0, by induction and (4) follows easily.

For n ≥ 1, we denote by m(n) the integer such that n ∈ [q m(n) , q m(n)+1 [.

If α has bounded partial quotients (i.e., sup a n < ∞), then m(n) is of order ln n.

Ostrowski's expansion ( [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF], [START_REF] Beck | Probabilistic Diophantine approximation, Randomness in lattice point counting[END_REF])

Every integer n ≥ 1 can be represented as follows (α-Ostrowski's expansion):

if n < q m+1 , n = m k=0 b k q k , with 0 ≤ b 0 ≤ a 1 -1, 0 ≤ b k ≤ a k+1 for 1 ≤ k ≤ m. (5)
Indeed, if n ∈ [q 0 , q 1 = a 1 [, then ( 5) is satised, and if n ∈ [q m , q m+1 [ with m ≥ 1, we write n = b m q m + r, with 1 ≤ b m ≤ a m+1 , 0 ≤ r < q m . By iteration, we get either r = 0 at some point and the algorithm stops, or n ∈ [q 0 , q 1 [. In either cases we obtain [START_REF] Conze | On the CLT for rotations and BV functions: note aux C[END_REF].

In this way, we can code every n < q m+1 by a word b 0 ...b m , with b 0 ∈ {0, 1, ..., a 1 -1} and b j ∈ {0, 1, ..., a j+1 }, j = 1, ..., m.

In this representation, b m(n) = 0 and b j = 0 for m(n) < j ≤ m when m > m(n). In the latter case, there are m -m(n) zero's at the right end. For a given m and n < q m+1 , this Ostrowski's expansion is proper (without zeros at the end) if m = m(n).

For m ≥ 0, we call admissible of length m + 1 a nite word b 0 ...b m such that b 0 ∈ {0, 1, ..., a 1 -1}, b j ∈ {0, 1, ..., a j+1 }, for j = 1, ..., m and such that, for two consecutive letters

b j-1 , b j , if b j = a j+1 then b j-1 = 0. Remark that if b 0 ...b m is admissible, m ≥ 1, then b 0 ...b m-1 is admissible.
Let us show by induction that the Ostrowski's expansion of an integer n is admissible. Let n be in [q m , q m+1 [. We start the construction of the expansion of n as above. Now the following steps of the algorithm yield the Ostrowski'

s expansion of n -b m q m . Since n -b m q m ∈ [0, q m [, the Ostrowski's expansion of n -b m q m is admissible. It remains to check that, if b m = a m+1 , then b m-1 = 0. But if b m-1 = 0, we would have n ≥ a m+1 q m + q m-1 = q m+1 , a contradiction.
Conversely, if b 0 ...b m is admissible, one shows by induction that b 0 +b 1 q 1 +...+b m q m < q m+1 . This holds if m = 0, since b 1 < q 1 = a 1 . Assume that this is true for the length m. Let b 0 ...b m b m+1 be admissible of length m + 1.

If b m+1 = a m+2 , then b m = 0 and b 0 + b 1 q 1 + ... + b m q m = b 0 + b 1 q 1 + ... + b m-1 q m-1 < q m , so that b 0 + b 1 q 1 + ... + b m+1 q m+1 < q m + a m+2 q m+1 = q m+2 . If b m+1 ≤ a m+2 -1, then b 0 + b 1 q 1 + ... + b m+1 q m+1 < q m+1 + (a m+2 -1) q m+1 < q m+2 .
Therefore, if we associate to an admissible word the integer n = b 0 + b 1 q 1 + ... + b m q m , there is a bijection between the Ostrowski's expansions of integers n < q m+1 and the set of admissible words of length m + 1. The number of admissible words of length m is q m -1.

For n given by ( 5), putting n 0 = b 0 , n k = k t=0 b t q t , for k ≤ m(n), we have

ϕ n (x) = m(n) k=0 n k -1 j=n k-1 ϕ(x + jα) = m(n) k=0 b k q k -1 j=0 ϕ(x + n k-1 α + jα) = m(n) k=0 b k -1 i=0 ϕ q k (x + (n k-1 + iq k )α) = m(n) k=0 f k (x), (6)
with f k (x) := b k -1 i=0 ϕ q k (x + (n k-1 + iq k )α) = ϕ b k q k (x + n k-1 α), (7)
By convention, we put b k -1

i=0 ϕ q k (x + (n k-1 + iq k )α) = 0, if b k = 0. If ϕ is a BV centered function, then it holds (Denjoy-Koksma inequality): ϕ q ∞ = sup x | q-1 i=0 ϕ(x + iα)| ≤ V (ϕ), if q is a denominator of α. ( 8
)
One can also show that if ϕ satises (1) then ϕ qn 2 ≤ 2π K(ϕ). By (8), we have for f k dened by [START_REF] Conze | On recurrence and ergodicity for geodesic ows on non-compact periodic polygonal surfaces[END_REF]:

f k ∞ ≤ b k V (ϕ) ≤ a k+1 V (ϕ).

Bounds for the variance.

Let ϕ ∈ C and n ∈ [q -1 , q [. The variance is bounded from below as follows:

ϕ n 2 2 = 2 k>1 | ϕ(k)| 2 (sin πnkα) 2 (sin πkα) 2 ≥ 2 j=1 | ϕ(q j )| 2 (sin πnq j α) 2 (sin πq j α) 2 ≥ c 0 j=1 | ϕ(q j )| 2 nq j α 2 q j α 2 , with c 0 = 8 π 2 . Therefore, by (2) we have, for 0 < δ < 1 2 , ϕ n 2 2 ≥ c 0 j=1 |γ q j (ϕ)| 2 a 2 j+1 nq j α 2 ≥ c 0 δ 2 j=1 |γ q j (ϕ)| 2 a 2 j+1 1 nq j α ≥δ . (9)
An upper bound for the variance and a lower bound for the mean of the variance are

shown in [START_REF] Conze | Diusive behaviour of ergodic sums over rotations[END_REF]: there are constants C, c > 0 such that

ϕ n 2 2 ≤ C K(ϕ) 2 m(n) j=0 a 2 j+1 , ( 10
)
1 n n-1 k=0 ϕ k 2 2 ≥ c m(n)-1 j=0 |γ q j (ϕ)| 2 a 2 j+1 . (11)
Inequality [START_REF] De La Rue | On the central limit theorem for aperiodic dynamical systems and applications[END_REF] gives a semi explicit lower bound for the variance. Note that by [START_REF] Conze | Recurrence, ergodicity and invariant measures for cocycles over a rotation[END_REF], the variance is small if n is a denominator q i of α. In this case, as expected, one nds that the lower bound given by ( 9) is small. Indeed, by (4), we have q i q j α ≤ C 1 ρ |i-j| , with ρ < 1, for a constant C 1 , so that, for a given δ > 0, the number of j's less than such that q i q j α ≥ δ is bounded independently from . Now our rst goal will be to bound from below the variance ϕ n 2 by a big value for n in a large set.

Bounds for the variance for n in a large set of integers According to [START_REF] De La Rue | On the central limit theorem for aperiodic dynamical systems and applications[END_REF], a lower bound for ϕ n 2 depends on two separate conditions:

First we need the following condition on the Fourier coecients of ϕ:

∃M, η, θ > 0 such that Card {j ≤ N : a j+1 |γ q j (ϕ)| ≥ η} ≥ θ N, ∀N ≥ M. (12)
This condition clearly holds when ϕ is the function ϕ 0 (x) = {x} -1 2 , since in this case |γ q j (ϕ 0 )| = 1 2π , ∀j. Its validity, related to Diophantine conditions on the points of discontinuities, will be discussed for some step functions in Subsection 2.2.

For j < , we will estimate how many times {nq j α} ∈

I δ := [0, δ] ∪ [1 -δ, 1] for n ≤ q
and deduce from this estimation that j=1 1 I δ ({n q j α}) = j=1 1 nq j α ≤δ is small for a large set of values of n.

Lemma 1.1. For every δ ∈]0, 1 2 [ and every interval of integers I = [N 1 , N 1 + L[, we have

N 1 +L-1 n=N 1 1 I δ ({n q j α}) ≤ 20 (δ + q -1 j+1 ) L, ∀j such that q j+1 ≤ 2L. (13)
Proof. For a xed j and 0 ≤ N 1 < N 1 + L, let us describe the behaviour of the sequence ( n q j α , n = N 1 , ..., N 1 + L -1).

Recall that (modulo 1) we have q j α = θ j , with θ j = (-1) j q j α (see [START_REF] Bromberg | A temporal central limit theorem for real-valued cocycles over rotations[END_REF]). We treat the case j even (hence θ j > 0). The case j odd is analogous.

We are going to count how many times, for j even, we have {n θ j } < δ or 1 -δ < {n θ j }.

We start with n 1 := N 1 . Putting w(j, 1) := {n 1 θ j }, we have {n θ j } = w(j, 1) + (n -n 1 )θ j , for n = n 1 , n 1 + 1, ..., n 2 -1, where n 2 is such that w(j, 1) + (n 2 -1 -n 1 ) θ j < 1 < w(j, 1) + (n 2 -n 1 ) θ j .

Putting w(j, 2) := {n 2 θ j }, we have w(j, 2) = w(j, 1) + (n 2 -n 1 ) θ j -1 < θ j . Starting now from n 2 , we have {n θ j } = w(j, 2) + (n -n 2 )θ j for n = n 2 , n 2 + 1, ..., n 3 -1, where n 3 is such that w(j, 2)

+ (n 3 -1 -n 2 ) θ j < 1 < w(j, 2) + (n 3 -n 2 ) θ j .
We iterate up to R(j), where n R(j)-1 < N 1 + L ≤ n R(j) . This construction yields a sequence n 1 < n 2 < ... < n R(j) such that {nθ j } = w(j, i)

+ (n -n i )θ j , ∀n ∈ [n i , n i+1 [, and w(j, i) + (n i+1 -1 -n i ) θ j < 1 < w(j, i) + (n i+1 -n i ) θ j ,
with w(j, i) dened recursively by w(j, i + 1) = {w(j, i) + (n i+1 -n i ) θ j } and satisfying w(j, i) < θ j , for i = 1, ..., R(j).

Since (n i+1 -n i + 1) θ j ≥ w(j, i) + (n i+1 -n i ) θ j > 1 for i = 1 and i = R(j), we have

n i+1 -n i ≥ θ -1 j -1, for each i = 1, R(j). This implies R(j) ≤ L θ -1 j -1 + 2.
For each i, the number of integers n ∈

[n i , n i+1 -1[ such that {nθ j } ∈ [0, δ[ ∪ ]1 -δ, 1[ is bounded by 2(1 + δ θ -1 j )
. (This number is less than 2 if δ < θ j .) Altogether, using (3) and the assumption 2L ≥ q j+1 , the number of integers n ∈

I such that {nθ j } ∈ [0, δ[ ∪ ]1 -δ, 1[ is bounded by 2R(j)(1 + δ θ -1 j ) ≤ ( 2L θ -1 j -1 + 4) (1 + δ θ -1 j ) ≤ 4 (L + θ -1 j ) (δ + θ j ) ≤ 4 (L + 2q j+1 ) (δ + q -1 j+1 ) ≤ 20 (δ + q -1 j+1 ) L.
Card {n ∈ [N 1 , N 1 + L] : d(nq j α, Z/r) ≤ δ} ≤ 20r (δ + q -1 j+1 )L, if q j+1 ≤ 2L.

Lemma 1.2. Let I = [N 1 , N 1 + L] be an interval and such that q ≤ 2L.

a) For all δ ∈]0, 1 2 [ and ζ ∈]0, 1[, the density of set

A := {n ∈ I : Card (j < : d(nq j α, Z) ≤ δ) ≤ ζ } (15)
satises

Card(A) ≥ (1 -20 ζ -1 (δ + C -1 )) L. (16)
b) Under Condition (12) on ϕ, there are positive constants η 0 , c (not depending on δ) such that, for every δ ∈]0, 1 2 [, the subset V (I, δ, ) := {n ∈ I :

ϕ n 2 ≥ η 0 δ √ } satises: Card (V (I, δ, )) ≥ (1 -c (δ + -1 )) L. (17)
Proof. a) Let A c = I \ A be the complementary of A. We will nd an upper bound of the density L -1 Card(A c ) by counting the number of values of n in I such that nq j α < δ in an array indexed by (j, n).

By summing (13) from j = 0 to j = -1 and using the denition of A, we get:

20 (δ + 0≤j≤ -1 q -1 j+1 ) L ≥ 0≤j≤ -1 n∈I 1 I δ ({n q j α}) ≥ n∈A c 0≤j≤ -1 1 I δ ({n q j α}) ≥ n∈A c ζ = ζ Card(A c ). With C := ∞ j=0 q -1 j , we have Card(A c ) ≤ 20 ζ -1 (δ + C -1 ) L, so (16) is shown. b) With ζ = 1 2 θ
, where θ is the constant in [START_REF] Feller | An introduction to probability theory and its application[END_REF], in view of the denition of A and (12), we have, for n ∈ A:

Card {j ≤ -1 : nq j α ≥ δ} {j : |γ q j (ϕ)| ≥ η} ≥ (1 -(ζ + 1 -θ)) = 1 2 θ . Putting c := 20 ζ -1 max(1, C) and η 0 = ( 1 2 c 0 η 2 θ) 1 2
, this implies by ( 9) and ( 16):

ϕ n 2 2 ≥ 1 2 c 0 δ 2 η 2 θ = η 2 0 δ 2 , ∀n ∈ A, and Card (A) ≥ 1 -c (δ + -1 ) L; (18)
hence A ⊂ V (I, δ, ) and therefore V (I, δ, ) satises [START_REF] Kitchens | Symbolic dynamics. One-sided, two-sided and countable state Markov shifts Universitext[END_REF].

The constants c and η 0 below are those of Lemma 1.2.

Theorem 1.3. Under Condition [START_REF] Feller | An introduction to probability theory and its application[END_REF] on ϕ, the density of the subset

W := {n ∈ N : ϕ n 2 ≥ η 0 m(n) ln m(n) 1 2 } satises for every N ≥ 1: Card W ∩ [0, N [ N ≥ 1 -2c (ln m(N )) -1 2 .
Proof. Since t/ ln t is increasing for t ≥ e, we have, after the rst terms, for n in W c ∩[0, N [:

ϕ n 2 < η 0 m(n) ln m(n) 1 2 ≤ η 0 m(N ) ln m(N ) 1 2 . Therefore, by b) of Lemma 1.2 with I = [0, N [, L = N , δ = (ln m(N )) -1 2 and = m(N ), it follows Card W c ∩ [0, N [ N ≤ c (ln m(N )) -1 2 + c m(N ) -1 ≤ 2c (ln m(N )) -1 2 .
1.3. A counter-example.

In the next sections we will show that, under a Diophantine condition on α, for a big set of n, the distribution of ϕ n / ϕ n 2 is approximately Gaussian. In particular, by [START_REF] Doob | Stochastic Processes[END_REF], if n ≤ q +1 is an integer such that ϕ n 2 = max k<q +1 ϕ k 2 , then we have the lower bound ϕ n

2 2 ≥ c -1 j=0 |γ q j | 2 a 2
j+1 . Under Condition (12), it can be shown that, for these indices n giving the record variances, when the partial quotients of α are bounded, the distribution of ϕ n / ϕ n 2 is asymptotically Gaussian.

Let us show by a counter-exemple that this is not true without a condition on α.

For a parameter γ > 0, let the sequence (a n ) n≥1 be dened by

a n = n γ if n ∈ {2 k : k ≥ 0}, = 1 if n / ∈ {2 k : k ≥ 0}.
Let α be the number which has (a n ) n≥1 for sequence of partial quotients. For simplicity, let us take for ϕ the sawtooth function ϕ 0 dened above for which γ k = -1 2πi , ∀k.

For > 1, let n := max{n < q +1 : ϕ n 2 = max k<q +1 ϕ k 2 }. We have

ϕ n 2 2 ≥ c -1 j=0 |γ q j | 2 a 2 j+1 ≥ c log 2 ( ) s=0
2 2γs ≥ c 2γ .
In the sum ϕ n (x) = [START_REF] Conze | Diusive behaviour of ergodic sums over rotations[END_REF], we can isolate the indices k for which k + 1 is a power of 2 (for the other indices a k+1 = 1) and write

k=0 b k q k -1 j=0 ϕ(x + N k-1 α + jα) dened in
ϕ n (x) = U + V with U = log 2 ( ) p=1 b 2 p q 2 p -1 j=0 ϕ(x + N 2 p -1 α + jα), V = k∈[0, ]∩ {a k+1 =1} b k q k -1 j=0 ϕ(x + N k-1 α + jα).
We will see in (49) that the variance of a sum where b k equals 0 or 1 is bounded as follows

k∈[0, ]∩ {a k+1 =1} b k q k -1 j=0 ϕ(x + N k-1 α + jα) 2 2 ≤ C log( ).
On the other side, we also have

| log 2 ( ) p=1 b 2 p -1 q 2 p -1 -1 j=0 ϕ(x + N 2 p -2 α + jα)| ≤ log 2 ( ) p=1 a 2 p V (ϕ) ≤ C log 2 ( ) p=1
2 γp ≤ C γ .
The previous bounds imply ϕn

ϕn 2 = U ϕn 2 + V ϕn 2 with ϕ n 2 ≥ c γ , U ∞ ≤ C γ , V 2 ≤ (C log( )) 1/2 . Thus, if γ > 1/2, one has U ϕn 2 ∞ ≤ C c , V ϕn 2 2
→ 0 and the limit points of the distributions of

ϕ n ϕ n 2 have all their supports included in [-C c , C c ],
hence are not Gaussian.

1.4. A special case: quadratic numbers.

When α is a quadratic number, using the ultimate periodicity of the sequence (a n (α)) n≥1

and the good properties of the associated Ostrowski's expansion of the integers, it is possible to improve the result of Theorem 1.3 on the variance. In this subsection we show that the variance ϕ n 2 2 of the ergodic sums of ϕ under the rotation by a quadratic number α is of order ln n for n in a big set of integers whose size is precisely estimated. For example, if we take ϕ(x) = {x} -1 2 , Theorem 1.5 shows that there are positive constants

η 1 , η 2 , R and ξ ∈]0, 1[ such that, 1 N Card {n ≤ N : η 1 ln n ≤ ϕ n 2 2 ≤ η 2 ln n} ≥ (1 -R N -ξ ). (19)
The main step in the proof is the following proposition showing that, in case of a quadratic number, for most of the integers n (in a set whose size is precisely estimated), nq j α is far from 0 for a big proportion of j's:

Proposition 1.4. If α is a quadratic number, for every ε 0 ∈]0, 1 2 [, there are δ ∈]0, 1 2 [ and positive constants C and ξ such that for every ≥ 1:

Card {n < q +1 : Card (j < : d(nq j α, Z) ≥ δ) ≥ (1 -ε 0 ) } ≥ (1 -Cq -ξ +1 ) q +1 . ( 20
)
The proof of Proposition 1.4 is given in Appendix.

Theorem 1.5. If α is a quadratic number and if ϕ satises Condition [START_REF] Feller | An introduction to probability theory and its application[END_REF], there are positive constants η 1 , η 2 , R and ξ ∈]0, 1[ such that, for N big enough, it holds :

Card {n ≤ N : η 1 ln n ≤ ϕ n 2 2 ≤ η 2 ln n} ≥ N (1 -R N -ξ ). (21)
Proof. There is η 2 > 0 such that the upper bound in ( 21) holds for every n ≥ 1: indeed, when α is quadratic, as

(q k ) is equivalent to a geometric sequence, m(n) is equivalent to ln(n) up to a multiplicative constant factor. Therefore, for n ∈ [q , q +1 [ (i.e., m(n) = ), ( 10
) implies ϕ n 2 2 ≤ CK(ϕ) 2 j=0 a 2 j+1 ≤ η 2 ln(n), for some positive constant η 2 .
For the lower bound, by [START_REF] De La Rue | On the central limit theorem for aperiodic dynamical systems and applications[END_REF] we have ϕ n 12) is satised and, for ε 0 = 1 2 θ, let δ = δ(ε 0 ) be given by Proposition 1.4. According to ( 12) and ( 20), for big enough, the set of integers n < q +1 such that simultaneously nq j α ≥ δ and |γ q j (ϕ)| ≥ η, for at least 1 2 θ dierent indices j, has a cardinal bigger than q +1 (1 -C q -ξ +1 ) for some constants C > 0, ξ ∈]0, 1[.

2 2 ≥ c 0 δ 2 j=1 |γ q j (ϕ)| 2 a 2 j+1 1 nq j α ≥δ . Let ϕ in C be such that (
Therefore we have ϕ n 2 2 ≥ c 0 2 η 2 δ 2 θ = η 1 for more than q +1 (1 -Cq -ξ +1
) values of n between 1 and q +1 . This shows that, for N ∈ [q , q +1 [, the cardinal of the set {n < N :

ϕ n 2 2 ≤ η 1 } is less than Cq 1-ξ +1 ≤ C N 1-ξ (
because for a quadratic number sup q +1 /q < +∞). Hence, the result.

A central limit theorem and its application to rotations

2.1. Decorrelation and CLT.

An abstract CLT under a decorrelation property

Below Y 1 denotes a r.v. with a normal distribution N (0, 1). Recall that, if X, Y are two real random variables, their mutual (Kolmogorov) distance in distribution is dened by:

d(X, Y ) = sup x∈R |P(X ≤ x) -P(Y ≤ x)|.
The notation C denotes an absolute constant whose value may change from a line to the other.

Proposition 2.1. Let N be a positive integer. Let (q k ) 1≤k≤N be an increasing sequence of positive integers such that for a constant ρ ∈]0, 1[

q k /q m ≤ C ρ m-k , 1 ≤ k < m ≤ N. (22)
Let (f k ) 1≤k≤N be real centered BV functions such that for constants u k

f k ∞ ≤ u k , V(f k ) ≤ C u k q k , 1 ≤ k ≤ N. (23)
Moreover assume that, for some constant θ, the following decorrelation properties hold:

| X ψ f k dµ| ≤ C V(ψ) u k k θ q k , 1 ≤ k ≤ N, ∀ψ BV, ( 24
) | X ψ f k f m dµ| ≤ C V (ψ) u k u m m θ q k , 1 ≤ k ≤ m ≤ N, ∀ψ BV centered, (25)
| X ψ f k f m f t dµ| ≤ C V (ψ) u k u m u t t θ q k , 1 ≤ k ≤ m ≤ t ≤ N, ∀ψ BV centered. ( 26
)
Then, putting w

N := max N j=1 u j , S N := f 1 + • • • + f N , there is for every δ > 0 a constant C(δ) > 0 (depending only on δ) such that the condition w N S N 2 ≤ N p-1 2 , with p ∈ [0, 1 8 [, (27) implies d
( S N S N 2 , Y 1 ) ≤ C(δ)N -1-8p 12 +δ . ( 28
)
The proposition is proved in Section 3. We apply it to an irrational rotation by taking for q k 's the denominators of α (they satisfy [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF]) and for f k the ergodic sums ϕ b k q k of a function ϕ (composed by a translation), where the b k 's (b k ≤ a k+1 ) are given by the Ostrowski's expansion described above.

Decorrelation between partial ergodic sums

In order to apply the previous proposition we will prove decorrelation properties between the ergodic sums of ϕ ∈ C at time q n under the following assumption on α:

Hypothesis 1. There are two constants A ≥ 1, p ≥ 0 such that a n ≤ A n p , ∀n ≥ 1. (29)
Remark 2. a) The case α of bounded type, i.e., with bounded partial quotients, corresponds to p = 0. In this case, as we have seen, m(n) is of order ln n.

b) Observe that m(n) can be smaller, but at least of order ln n ln ln n up to a bounded factor, under the more general assumption 1. Lemma 2.2. a) For every p > 1, for a.e. α, there is a nite constant A(α, p) such that

a n ≤ A(α, p) n p , ∀n ≥ 1. (30) b) If α satises (29), then there is c > 0 such that kα ≥ c |k| (log k) p , ∀k > 1. (31) Proof. a) We have a n+1 (α) = 1/θ n (α) where θ is the Gauss map. Let γ > 1. Since α → (a 1 (α)) 1 γ is integrable for the θ-invariant measure dx 1+x on ]0, 1], we have, for a constant A(γ): µ{α : a n (α) > n s } ≤ A(γ) n -s γ .
By the Borel-Cantelli lemma, it follows that for a.e α there is

C(α, γ) such that, if s > γ, a n (α) ≤ C(α, γ) n s , ∀n ≥ 1.
b) For every irrational α, there are C > 0 and λ > 1 such that the denominators of α satisfy q ≥ Cλ , for every ≥ 1.

For k ≥ 2, let n be such that q n-1 ≤ k < q n . Since Cλ n-1 ≤ q n-1 ≤ k, it follows that n ≤ C log k, for some constant C . By (29), we have a n ≤ An p ≤ A(C log k) p . Since kα > q n-1 α ≥ 1 2q n ≥ 1 4a n q n-1 ≥ 1 4a n k , this implies (31).
As a corollary, using Theorem 1.3, it follows that for a.e. α, under the rotation by α, for a function ϕ ∈ C satisfying [START_REF] Feller | An introduction to probability theory and its application[END_REF], the growth of the variance ϕ n 2 2 is roughly of order ln n. In Subsection 2.2 we will see that, if α satises (29) with p < 1 4 , ( 12) itself is a generic condition for some class of step functions ϕ.

Proposition 2.3. Let ψ and ϕ be BV centered functions. Suppose that α satises Hypothesis 1. Then there are constants C, θ 1 , θ 2 , θ 3 such that, for every

1 ≤ k ≤ m ≤ : | X ψ ϕ b k q k dµ| ≤ C V(ψ) V(ϕ) k θ 1 q k b k , (32)
| X ψ ϕ b k q k ϕ bmqm dµ| ≤ C V(ψ) V(ϕ) 2 m θ 2 q k b k b m , (33)
| X ψ ϕ b k q k ϕ bmqm ϕ b q dµ| ≤ C V(ψ) V(ϕ) 3 θ 3 q k b k b m b . ( 34
)
The proposition is proved in Section 4. From the propositions 2.1 and 2.3 we will deduce a convergence toward a Gaussian distribution under a variance condition, by bounding the distance to the normal distribution.

Theorem 2.4. Let ϕ be in C satisfying [START_REF] Feller | An introduction to probability theory and its application[END_REF].

1) The set dened (cf. Theorem 1.3) by

W := {n ∈ N : ϕ n 2 ≥ η 0 (log m(n)) -1 2 m(n) 1 2 } (35)
has density 1 in N.

Suppose that α satises Hypothesis 1 (i.e., for constants A ≥ 1, p ≥ 0, a n ≤ A n p , ∀n ≥ 1)

with p < 1 8 . Then, for δ ∈]0, 1-8p 12 [, there is a constant C(δ) such that, for n in W , d( ϕ n ϕ n 2 , Y 1 ) ≤ C(δ) m(n) -1-8p 12 +δ -→ n∈W, n→∞ 0. (36)
In particular when α has bounded partial quotients, we have p = 0 and m(n) can be replaced by log n.

2) Suppose that α is a quadratic irrational. With the notation of Theorem 1.5, let

V := {n ≥ 1 : η 1 log n ≤ ϕ n 2 ≤ η 2 log n}.
Then, there are two constants R, ξ > 0 such that -the density of V satises:

Card(V [1, N ]) ≥ N (1 -R N -ξ ), for N ≥ N 0 ; (37)
-for δ ∈]0, 1 12 [, there is a constant C(δ) such that, for n ∈ V :

d( ϕ n ϕ n 2 , Y 1 ) ≤ C(δ) (log n) -1 12 +δ -→ n∈V, n→∞ 0. (38)
Proof. 1) The result on the density of the set W follows from Theorem 1.3.

For (36), we use Proposition 2.

1 with N = m(n) (where m(n) is such that n ∈ [q m(n) , q m(n)+1 [),
f k dened by [START_REF] Conze | On recurrence and ergodicity for geodesic ows on non-compact periodic polygonal surfaces[END_REF] and the decomposition of the ergodic sums given by ( 6), i.e.,

ϕ n (x) = m(n) k=0 f k (x), where f k (x) := b k -1 i=0 ϕ q k (x + (n k-1 + iq k )α) = ϕ b k q k (x + n k-1 α).
The decorrelation inequalities in Proposition 2.3 are obtained for functions of the form ϕ b k q k . But in the proof of the decorrelation inequalities, one sees that they remain valid for f k , since translations on the variable do not change the modulus of the Fourier coecients.

As

f k ∞ ≤ b k V (ϕ) ≤ a k+1 V (ϕ)
, up to a xed factor the constant u k in the statement of Proposition 2.3 can be taken to be a k+1 ≤ k p , for some constant p > 0, by Hypothesis 1.

With the notation of Proposition 2.1, we have w

N := max N j=1 b j , ϕ n = S N = f 1 + • • • + f N . For n ∈ W and under Hypothesis 1, we have w N S N 2 ≤ CN p-1 2 (log N ) 1 2 .
The factor (log N )

1
2 can be absorbed in the factor N p-1 2 by taking p larger and we have (27). By (28) it follows:

d( ϕ n ϕ n 2 , Y 1 ) ≤ C(δ) m(n) -1-8p 12 +δ .
2) In the quadratic case, p = 0 and the property of the set V is given by Theorem 1. [START_REF] Feller | An introduction to probability theory and its application[END_REF] on the coecients γ q k (ϕ), i.e.:

∃M, η, θ > 0 such that 1 N Card {j ≤ N : a j+1 |γ q j (ϕ)| ≥ η} ≥ θ, ∀N ≥ M.
The functions {x} - 12) is satised, because two consecutive q k 's are relatively prime and therefore cannot be both even.

1 2 = -1 2πi r =0 1 r e 2πirx and 1 [0, 1 2 [ -1 [ 1 2 ,1[ = r 2 πi(2r+1) e 2πi(2r+1
γ q k = 0 if q k is even, = 2 πi if q k is odd. Clearly, (
In general, for a step function, Condition [START_REF] Feller | An introduction to probability theory and its application[END_REF] (and therefore a lower bound for the variance ϕ n 2 2 for a large set of integers n) is related to the Diophantine properties of its discontinuities with respect to α. We discuss now this point.

Let us consider a centered step function ϕ on [0, 1[ taking a non null constant value v j ∈ R on the interval [u j , u j+1 [, j = 0, 1, ..., s, with u 0 = 0 < u 1 < ... < u s < u s+1 = 1:

ϕ = s j=0 v j 1 [u j ,u j+1 [ -c. (39)
The constant c above is such that ϕ is centered, but it plays no role below. Lemma 2.5. If ϕ is given by (39), there is a continuous periodic function H ϕ (u 1 , ..., u s ) ≥

0 such that |γ r (ϕ)| 2 = π -2 H ϕ (ru 1 , ..., ru s ). (40) Proof. Since ϕ(r) = s j=0 v j πr e -πir(u j +u j+1 ) sin πr(u j+1 -u j ), r = 0, H ϕ (u 1 , ..., u s ) is [ s j=0 v j cos π(u j + u j+1 ) sin π(u j+1 -u j )] 2 + [ s j=0 v j sin π(u j + u j+1 ) sin π(u j+1 -u j )] 2 . Examples: 1) ϕ = ϕ(u, • ) = 1 [0,u[ -u, H ϕ (u) = sin 2 (πu). 2) ϕ = ϕ(w, u, • ) = 1 [0, u] -1 [w, u+w] , H(ϕ) = 4 sin 2 (πu) sin 2 (πw).
We show now that ( 12) is satised generically by the family of step functions parametrised by (u 1 , ..., u s ) dened by (39). Corollary 2.6. 1) Suppose that ϕ is a step function given by (39) for s ≥ 1, with parameter (u 1 , ..., u s ). Then Condition ( 12) is satised if (u 1 , ..., u s ) is such that the sequence (q k u 1 , ..., q k u s ) k≥1 is uniformly distributed in T s .

2) This latter condition holds for a.e. value of (u 1 , ..., u s ) in T s .

Proof. 1) If the sequence (q k u 1 , ..., q k u s ) k≥1 is uniformly distributed in T s , we have with the notation of Lemma 2.5:

lim N 1 N N k=1 |γ q k (ϕ)| 2 = lim n 1 n n k=1 H ϕ (q k u 1 , ..., q k u s ) = T s H(x 1 , ..., x s ) dx 1 ...dx s > 0, for a.e. (u 1 , ..., u s ) ∈ T s . (41)
Let N 0 and δ > 0 be such that, for N ≥ N 0 , 1

N N k=1 |γ q k (ϕ)| 2 ≥ δ. The sequence (|γ q k (ϕ)| 2 , k ≥ 1) is bounded by K := π -2 H ϕ ∞ . Therefore, we have, for N ≥ N 0 , δ ≤ 1 N N k=1 |γ q k (ϕ)| 2 ≤ K N N k=1 1 |γq j (ϕ)|≥η + η 2 N N k=1 1 |γq j (ϕ)|<η ≤ K N N k=1 1 |γq j (ϕ)|≥η + η 2 .
This shows:

1 N N k=1 1 |γq j (ϕ)|≥η ≥ K -1 (δ -η 2 ), for N ≥ N 0 .
It follows that ( 12) is satised with M = N 0 , η = ( δ 2 )

1 2 , θ = K -1 δ 2 .
2) To prove the uniform distribution for a.e. value of (u 1 , ..., u s ) in T s , by Weyl equirepartition criterium it suces to show, for all integers r 1 , ..., r s not all 0,

lim k 1 N N k=1 e 2iπq k (r 1 u 1 +...+rsus) = 0, for a.e. (u 1 , ..., u s ) ∈ T s . (42)
Since (q k ) is a strictly increasing sequence of integers, (42) follows from the law of large numbers for orthogonal bounded variables (Rajchman's theorem) which is recalled in Appendix 2 in a slightly more general formulation (Proposition 6.1).

Besides a generic result, there are also specic values of the parameter (u 1 , ..., u s ) for which (12) holds. A simple example (for s = 1) is:

Example 3: ϕ( r 1 r 2 , • ) = 1 [0, r 1 r 2 [ -r 1 r 2 , for r 1 , r 2 ∈ N, 0 < r 1 < r 2 .
We will give another example of special values related to the rectangular billiard model in example 4 below.

Remark 4. For the case of example 1, let us make some remarks about the degeneracy of the variance.

It is known that if α is bpq and if lim k | sin(πq k u)| = 0, where q k are the denominators of α, then u ∈ Zα + Z (cf. for instance [START_REF] Conze | Recurrence, ergodicity and invariant measures for cocycles over a rotation[END_REF]). But it is easily seen that there is an uncountable set of u's such that lim N Observe also that, if α is not bpq, there are many u's for which the sequence (q k u mod1)

does not satisfy the equidistribution property in a strong sense and (41) fails.

Indeed, let u = n≥0 b n q n α mod 1, b n ∈ Z, 0 ≤ b n ≤ a n+1
, be the so-called Ostrowski expansion of u associated to the denominators of α. It can be shown that, if lim n |bn| a n+1 = 0, then lim k q k u = 0 (Proposition 1 in [START_REF] Guenais | Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier[END_REF]). There is an uncountable set of u's satisfying the condition lim n |bn| a n+1 = 0 if α is not bpq. For these values of u, we have lim k γ q k (ϕ(u, .)) = 0. Therefore Condition [START_REF] Feller | An introduction to probability theory and its application[END_REF], which is used to get a lower estimate of the variance, fails, although, if u is not in the countable set Zα + Z, ϕ(u, .) is not a coboundary (and even generates an ergodic cocycle). Remark 5. Another remark is about the generic validity of estimates of the variance.

As previous remarked, in Theorem 2.4 the CLT is written with self-normalisation (by ϕ n 2 ). In Theorem 1.3 the lower bound given for the variance ϕ n 2 2 for n in the set W can be smaller than the mean of the variance.

Inequalities [START_REF] Dolgopyat | Temporal distributional limit theorems for dynamical systems[END_REF] and [START_REF] Doob | Stochastic Processes[END_REF] give a precise estimation of the variance in the mean when an information is available on γ q i (ϕ).

For example in the case of the saw-tooth function, we get the estimate m(n) k=1 a 2 k for the mean of the variance.

If we consider Example 1 or more generally ϕ = ϕ(u, .) given by (39), the same estimate is valid generically with respect to u under a condition on α. This is a consequence of the equidistribution argument used previously and of Proposition 6.1. Namely, using this proposition and an approximation by trigonometric polynomials, we get:

If 1 ≤ a n ≤ n p , with p < 1 4 , if H ϕ (u 1 , ..., u s ) is a continuous periodic function on the torus T s , s ≥ 1, then: lim N N k=1 a 2 k H(q k u 1 , ..., q k u s ) N k=1 a 2 k = T s
H(x 1 , ..., x s )dx 1 ...dx s , for a.e. u.

(

) 43
For instance, in Example 1, lim

N N k=1 a 2 k sin 2 (πq k u) N k=1 a 2 k = 1 2
, for a.e. u.

By [START_REF] Doob | Stochastic Processes[END_REF], it follows that the mean of the variance,

1 n n-1 k=0 ϕ k (u, .) 2 2 , is of order m(n) k=1 a 2 k
generically with respect to u, if α satises Hypothesis 1, i.e., a n = O(n p ), ∀n ≥ 1, with p < 1 4 .

Vectorial case

For simplicity, we consider the case of two components. Let be given a vectorial function Φ = (ϕ 1 , ϕ 2 ), where ϕ 1 , ϕ 2 are two centered step functions with respectively s 1 , s 2 discontinuities: This is done in the next proposition.

ϕ i = s i j=0 v i j 1 [u i j ,u i j+1 [ -c i , for i = 1, 2. Let the matrix Γ n be dened by Γ n (a, b) := (log n) -1 aϕ 1 n + bϕ 2
Proposition 2.8. Let Λ be a compact space and (F λ , λ ∈ Λ) be a family of nonnegative non identically null continuous functions on T d depending continuously on λ.

If a sequence (z n ) is equidistributed in T d , then ∃ N 0 , η > 0 such that Card{n ≤ N : F λ (z n ) ≥ η} ≥ θ N, ∀N ≥ N 0 , ∀λ ∈ Λ. ( 44
) Proof. For λ ∈ Λ, let u λ ∈ T d be such that F λ (u λ ) = sup u∈T d F λ (u). We have F λ (u λ ) > 0 and there is η λ > 0 and an open neighborhood U λ of u λ such that F λ (u) > 2η λ for u ∈ U λ .
Using the continuity of F λ with respect to the parameter λ, the inequality

F ζ (u) > η λ holds for u ∈ U λ and ζ in an open neighborhood V λ of λ. By compactness of Λ, there is a nite set (λ j , j ∈ J) such that (V λ j , j ∈ J) is an open covering of Λ. Let θ := 1 2 inf j∈J Leb(U λ j ). By equidistribution of (z n ), there is N 0 such that 1 N N n=1 1 U λ j (z n ) ≥ θ, ∀N ≥ N 0 , ∀j ∈ J. Let η := inf j∈J η λ j . For every λ ∈ Λ, there is j ∈ J such that λ ∈ V λ j and therefore F λ (z n ) ≥ η λ j ≥ η, if z n ∈ U λ j . This implies: Card{n ≤ N : F λ (z n ) ≥ η} ≥ Card{n ≤ N : z n ∈ U λ j } ≥ θN, ∀N ≥ N 0 .
A generic result By Proposition 2.8 applied for (a, b) in the unit sphere, for a.e. values of the parameter (u 1 1 , ..., u 1 s 1 , u 2 1 , ..., u 2 s 2 ), the functions aϕ 1 + bϕ 2 satisfy Condition [START_REF] Feller | An introduction to probability theory and its application[END_REF] uniformly in (a, b) in the unit sphere. Hence Theorem 2.7 applies generically with respect to the discontinuities.

Special values: an application to the rectangular billiard in the plane

Example 4 Now, for an application to the periodic billiard, we consider the vectorial

function ψ = (ϕ 1 , ϕ 2 ) with ϕ 1 = 1 [0, α 2 ] -1 [ 1 2 , 1 2 + α 2 ] = 2 π r∈Z e -πi(2r+1) α 2 sin(π(2r + 1) α 2 ) 2r + 1 e 2πi(2r+1) • , ϕ 2 = 1 [0, 1 2 -α 2 ] -1 [ 1 2 ,1-α 2 ] = -2i π r∈Z e πi(2r+1) α 2 cos(π(2r + 1) α 2 ) 2r + 1 e 2πi(2r+1) • .
The Fourier coecients of ϕ 1 and ϕ 2 of order r are null for r even.

Let us consider a linear combination ϕ a,b = aϕ 1 + bϕ 2 . For r = 2t + 1 odd, we have:

c 2t+1 (aϕ 1 + bϕ 2 ) = 2 π 1 2t + 1 e -πi(2t+1) α 2 [a sin(π(2t + 1) α 2 ) -ib cos(π(2t + 1) α 2 )].
If q j is even, γ q j (ϕ a,b ) is null. If q j is odd, we have

|γ q j (ϕ a,b )| 2 = |a sin(πq j α 2 ) -ib sin(π( 1 2 + q j α 2 ))| 2 ,
For q j odd, we have by (2),

q j α 2 = p j 2 + θ j 2 , hence q j α 2 - p j 2 ≤ θ j 2 ≤ 1 2q n+1 , 1 2 + q j α 2 = q j 2 - p j 2 - θ j 2 , hence q j β 2 - 1 2 + p j 2 ≤ θ j 2 ≤ 1 2q n+1 ,
This implies, for q j odd:

γ q j (ϕ a,b ) = a(1 + O( 1 q j+1 )), if p j is odd, = b(1 + O( 1 q j+1 )), if p j is even.
The computation shows that, if α is such that, in average, there is a positive proportion of pairs (p j , q j ) which are (even, odd) and a positive proportion of pairs (p j , q j ) which are (odd, odd), then the condition of Theorem 2.7 is fullled by the vectorial step function ψ = (ϕ 1 , ϕ 2 ).

For an application to the model of rectangular periodic billiard in the plane described in [START_REF] Conze | On recurrence and ergodicity for geodesic ows on non-compact periodic polygonal surfaces[END_REF], we refer to [START_REF] Conze | Diusive behaviour of ergodic sums over rotations[END_REF].

Proof of Proposition 2.1 (CLT)

The dierence H X,Y (λ) := |E(e iλX ) -E(e iλY )| can be used to get an upper bound of the distance d(X, Y ) thanks to the following inequality ( [START_REF] Feller | An introduction to probability theory and its application[END_REF], Chapter XVI, Inequality (3.13)): if X has a vanishing expectation, then, for every U > 0,

d(X, Y ) ≤ 1 π U -U H X,Y (λ) dλ λ + 24 π 1 σ √ 2π 1 U . (45)
Using (45), we get an upper bound of the distance between the distribution of X and the normal law by bounding |E(e iλX ) -e -1 2 σ 2 λ 2 |.

We will use the following remarks:

V(f g) ≤ f ∞ V(g) + g ∞ V(f ), ∀f, g BV, ( 46
) if g ∈ C 1 (R, R) and u is BV, then V(g • u) ≤ g ∞ V(u). (47)
Let w k := max k j=1 u j , where u j is larger than

f j ∞ (see Proposition 2.1). Since V(f k ) ≤ Cu k q k , (24) implies | X f k f m dµ| ≤ C q k q m m θ w 2 m , for k ≤ m. (48)
Bounding the moments

The following inequalities are implied by (49):

(52)

v 2 k = v 2 n,k = F n,k 2 2 ≤ Cn τ ln n w 2 n , G n,k 2 2 ≤ C n δ ln n w 2 n , 0 ≤ k < p(n).
Since q 1 + q 2 + ... + q n ≤ Cq n+1 , ∀n ≥ 1, by [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF], it follows by (47) and hypothesis (23):

V(e iζ(F n,0 +•••+F n,k-1 ) ) ≤ C|ζ| w n q (k-1) ν+n 1 .

(

) Lemma 3.2. | S n 2 2 - p(n)-1 k=0 v 2 k | = | S n 2 2 - p(n)-1 k=0 F n,k 2 2 | ≤ C n 1-τ -δ 2 ln n w 2 n , 53
S n -S n

2 2 = p(n)-1 k=0 G k 2 ≤ C n 1-τ +δ ln n w 2 n . (55)
Proof. It follows from ( 48) and ( 22), with

C 0 = Cρ (1-ρ) 2 , | X ( b u=a f u ) ( d t=c f t ) dµ| ≤ C 0 ρ c-b d θ w 2 d , ∀a ≤ b < c ≤ d.
Therefore, we have, with

C 1 = C 0 i≥0 ρ iν , writing simply F k , G k instead of F n,k , G n,k , 0≤j<k<p(n) | F j F k dµ| ≤ C 0 n θ w 2 n 0≤j<k<p(n) ρ kν+1-(jν+n 1 ) ≤ C 0 n θ w 2 n ρ n 2 0≤j<k<p(n) ρ (k-1)ν-jν ≤ C 0 ρ n 2 n θ w 2 n p(n) i≥0 ρ iν ≤ C 1 n 1 2 -δ+θ w 2 n ρ n δ .
The LHS of (54) is less than the sum for k = 0 to p(n) -1 of

G 2 k dµ + | G k F k dµ| + | G k F k+1 dµ| +2 0≤j<k [| F j (F k + G k ) dµ| + | G j G k dµ|] + 2 0≤j<k-1 | G j F k dµ|.
The rst term is bounded by C n δ ln n w 2

n , the second one and the third one bounded by C n δ+τ 2 ln n w 2 n are the biggest. The other terms are negligible as shown by the preliminary computation: n θ ρ n δ is small compared to a power of n, for n big.

Therefore the LHS of (54) is less than:

C 1 n 1-τ n δ+τ 2 ln n w 2 n = C 1 n 1-τ -δ 2 ln n w 2 n .
An analogous computation shows that the LHS of (55) behaves like p(n)-1 k=0

G 2 k dµ which gives the bound C n 1-τ +δ ln n w 2 n of (55).

Approximation of the characteristic function of the sum S n by a product

For ζ ∈ R, let I n,-1 (ζ) := 1, I n,k (ζ) := X e iζ (F n,0 +•••+F n,k ) dµ, 0 ≤ k < p(n).
then, by (61) and (56) of Lemma 3.3, we get

|J n (ζ) - p(n) k=1 (1 -1 2 ζ 2 v 2 k )| ≤ |J n (ζ) -I n,p(n) (ζ)| + p(n)-1 k=0 |I n,k (ζ) -(1 -ζ 2 2 v 2 k ) I n,k-1 (ζ)| ≤ C [|ζ| w n n 1-τ +δ 2 (ln n) 1/2 + n 1-τ |ζ| 3 w 3 n n τ ln 2 (n) + n 1-τ ζ 4 w 4 n n 2τ ln(n) 2 ≤ C [|ζ| w n n 1-τ +δ 2 (ln n) 1 2 + |ζ| 3 w 3 n n (ln n) 2 + ζ 4 w 4 n n 1+τ (ln n) 2 ]. (62)
Approximation of the exponential by a product

Below, ζ will be such that |ζ| v n,k ≤ 1. This is satised if

|ζ| n τ 2 w n (log n) 1 2 ≤ 1. ( 63
) Lemma 3.4. If (ρ k ) k∈J is a nite family of real numbers in [0, 1[, then 0 ≤ e -k∈J ρ k - k∈J (1 -ρ k ) ≤ k∈J ρ 2 k , if 0 ≤ ρ k ≤ 1 2 , ∀k. (64)
Proof. We have ln(1

-u) = -u -u 2 v(u), with 1 2 ≤ v(u) ≤ 1, for 0 ≤ u ≤ 1 2 and 1 -e -ε k ≤ ε k , if k ε k ≥ 0. Writing 1 -ρ k = e -ρ k -ε k , with ε k = -(ln(1 -ρ k ) + ρ k ), the previous inequality implies 0 ≤ ε k ≤ ρ 2 k , if 0 ≤ ρ k ≤ 1 2
. Therefore, under this condition, we have:

0 ≤ e -J ρ k -J (1 -ρ k ) = e -J ρ k 1 -e -J ε k ≤ e -ρ k ε k ≤ ε k ≤ ρ 2 k .
We apply (64

) with ρ k = 1 2 ζ 2 v 2
n,k , under Condition (63). In view of (52) it follows:

|e 1 2 ζ 2 v 2 k - p(n)-1 k=0 (1 - 1 2 ζ 2 v 2 k )| ≤ 1 4 ζ 4 p(n)-1 k=0 v 4 k ≤ Cζ 4 w 4 n n 1+τ ln 2 (n). ( 65
)
The bound is like the last term in (62).

Conclusion

From ( 62) and (65), it follows:

|J n (ζ) -e -1 2 ζ 2 p(n)-1 k=0 v 2 k | ≤ C [|ζ| w n n 1-τ +δ 2 (ln n) 1 2 + |ζ| 3 w 3 n n (ln n) 2 + ζ 4 w 4 n n 1+τ (ln n) 2 ].
We replace ζ by λ Sn 2 ; hence Condition (63) becomes

λ S n 2 n τ 2 w n (log n) 1 2 ≤ 1.
(66)

We get:

| X e iλ Sn(x) Sn 2 dµ(x) -e -1 2 λ 2 Sn 2 2 k v 2 k | ≤ C [|λ| w n S n 2 n 1-τ +δ + |λ| 3 w 3 n S n 3 2 n ln 2 (n) + λ 4 w 4 n S n 4 2 n 1+τ ln 2 (n)].
Since |e -a -e -b | ≤ |a -b|, for any a, b ≥ 0, we have, by (54):

|e -1 2 λ 2 -e -1 2 λ 2 Sn 2 2 p(n)-1 k=0 v 2 k | ≤ 1 2 λ 2 S n 2 2 | S n 2 2 - p(n)-1 k=0 v 2 k | ≤ Cλ 2 w 2 n S n 2 2 n 1-τ -δ 2 ln n.
Let us call respectively E 1 the error in neglecting the sums on the small blocks, E 2 the error in the replacement of e -1 2 λ 2 by exp(-

1 2 λ 2 v 2 k Sn 2
2

), E 3 the error of order 3 in the expansion, E 4 the approximation error of the exponential by the product.

Finally we get the bound | X e iλ Sn(x)

Sn 2 dµ(x) -e -1 2 λ 2 | ≤ E 1 + E 2 + E 3 + E 4 ≤ C [|λ| w n S n 2 n 1-τ +δ 2 ln 1 2 (n) + λ 2 w 2 n S n 2 2 n 1-τ -δ 2 ln n + |λ| 3 w 3 n S n 3 2 n ln 2 (n) + λ 4 w 4 n S n 4 2 n 1+τ +δ ln 2 (n)].
Denote by Y 1 a r.v. with N (0, 1)-distribution. Putting R n := wn Sn 2 , the bound reads:

C [|λ| R n n 1-τ +δ 2 + λ 2 R 2 n n 1-τ -δ 2 ln n + |λ| 3 R 3 n n ln 2 (n) + λ 4 R 4 n n 1+τ +δ ln 2 (n)]. ( 67
)
Notice that δ can be taken arbitrary small. A change of its value modies the generic constant C in the previous inequalities. Therefore we take δ = 0 in the optimisation below, keeping in mind that the constant factor in the inequalities depends on δ. Likewise the ln n factors can be neglected.

We have an inequality of the form

H Sn Sn 2 ,Y 1 (λ) ≤ C 4 i=1 |λ| α i R α i n n γ i
, where the exponents are given by the previous inequality. In view of (45), it follows that, up to a constant factor, d( Sn Sn 2 , Y 1 ) ≤

1 U n + 4 i=1 α -1 i U α i n R α i n n γ i ≤ 1 U n + U n R n n 1-τ 2 + 1 2 U 2 n R 2 n n 1-τ 2 + 1 3 U 3 n R 3 n n + 1 4 U 4 n R 4 n n 1+τ .
Now, we optimize the choice of U = U n . As R n is less than n -β for some β > 0, if we take U n = n γ with γ > 0, then the previous inequality gives inside the bracket the bound:

n 1-τ 2 -β+γ + 1 2 n 1-τ 2 -2β+2γ + 1 3 n 1-3β+3γ + 1 4 n 1+τ -4β+4γ .
We choose U n such that 1/U n is of the same order as the second term, i.e., we take n

-γ = n 1-τ 2 -2β+2γ , i.e., γ = τ 2 +2β-1 3 . If τ = 1 2 and if β = 1 2 -p with p > 0, then it gives: γ = 1 -8p 12 > 0 if p < 1 8 .
The four terms in the bound are respectively:

(A) = - 1 6 + p 3 , (B) = -γ = - 1 12 + 2p 3 , (C) = - 1 4 + p, (D) = - 1 6 + 4p 3 .
We check that (B) is the biggest term: (B) -

(A) = 1 12 + p 3 > 0, (B) -(C) = 1 6 -p 3 > 0, if p < 1 2 , (B) -(D) = 1 12 -2p 3 > 0, if p < 1 8 .
This gives the bound stated in Proposition 2.1 for the distance to the normal law:

For every δ > 0, for N big enough, there is a constant C(δ) > 0 (depending only on δ)

such that, if w N S N 2 ≤ N p-1 2 with p ∈ [0, 1 8 [, then d( S N S N 2 , Y 1 ) ≤ C(δ) N -1-8p 12 +δ << 1.
To conclude, observe that, if w n ≤ n p with p < 1 8 and S n 2 ≥ Cn

1 2 /(log n) 1 2 , (66) is satised for |λ| ≤ U n = n γ , since n γ n τ 2
w n (log n)

1 2 S n 2 ≤ Cn 1-8p 12 n p-1 4 (log n) 1 2 = Cn -1 6 + p 3 (log n) 1 2 ≤ 1, for n big.

Proof of Proposition 2.3 (decorrelation)

For the proof of Proposition 2.3, by homogeneity, we may assume that ψ and ϕ are BV centered functions with variation ≤ 1. Moreover, we may also assume b i = 1, ∀i. Indeed, the decorrelation inequalities will follow from bounds on sums of products of quantities

like | ϕ bnqn (j)| ≤ b n | ϕ qn (j)| or ϕ bnqn 2 ≤ b n ϕ qn 2 .
First we truncate the Fourier series of the ergodic sums ϕ q . For functions in C, the remainders are easily controlled and it suces to treat the case of trigonometric polynomials.

For ϕ ∈ C, the Fourier coecients of order j = 0 of the ergodic sum ϕ n satisfy:

| ϕ n (j)| = |γ j (ϕ)| |j| | sin πnjα| | sin πjα| ≤ π 2 K(ϕ)| |j| njα jα . ( 68
)
Recall also (cf. ( 8)) that, if q is a denominator of α, then

ϕ q ∞ = sup x | q-1 =0 ϕ(x + α)| ≤ V (ϕ) and ϕ q 2 ≤ 2π K(ϕ).
We will use the notations: S L f for the partial sum of order L ≥ 1 of the Fourier series of f ∈ L 2 (T), R L f := f -S L f for the remainder and, q n denoting the denominators of α,

a n := q n+1 q n ≤ a n+1 + 1, c n := q n+1 q n ln q n+1 . ( 69
)

Preliminary inequalities and truncation

We begin by some inequalities which are valid for any irrational number α.

Lemma 4.1. There is a constant C such that, if q is a denominator of α,

|j|≥q 1 j 2 Ljα 2 jα 2 ≤ C L q , ∀L ∈ [1, q]. (70)
Proof. If f is a non negative BV function with integral µ(f ), by Denjoy-Koksma inequality applied to f -µ(f ), we have

∞ j=q f (jα) j 2 ≤ ∞ i=1 1 (iq) 2 q-1 r=0 f ((iq + r)α) ≤ 1 q 2 ( ∞ i=1 1 i 2 ) (q µ(f ) + V (f )) = π 2 6 ( µ(f ) q + V (f ) q 2 ).
Taking for f (x) respectively 1 [0, 1 L ] (|x|) and 1

x 2 1 [ 1 L , 1
2 [ (|x|), we obtain:

j: jα ≤ 1 L , j≥q 1 j 2 ≤ C ( 1 q 2 + 1 Lq ), j: jα ≥ 1 L , j≥q 1 j 2 1 jα 2 ≤ C ( L 2 q 2 + L q ).
This implies (70), since for L ≤ q:

1 2 |j|≥q 1 j 2 Ljα 2 jα 2 ≤ L 2 jα ≤ 1 L , j≥q 1 j 2 + jα > 1 L , j≥q 1 j 2 1 jα 2 ≤ 2C ( L q + L 2 q 2 ) ≤ 4C L q .
We will use the good equirepartition of the numbers kα when k varies between 1 and q n through two inequalities given in the following lemma, which will be used several times. Lemma 4.2. We have

q t+1 -1 j=qt 1 jα ≤ q t+1 -1 j=1 1 jα ≤ Cq t+1 ln q t+1 , ∀t ≥ 0, (71)
1≤j<q r+1 1 j jα ≤ C r t=0 q t+1 q t ln q t+1 = C r t=0 c t , ∀r ≥ 0. (72)
Proof. There is exactly one element of the set {jα mod 1, j = 1, ..., q t+1 -1} in each interval [ q t+1 , +1 q t+1 [, = 1, ..., q t+1 -1. Moreover, for 1 ≤ j < q t+1 , one has jα ≥ 1 2q t+1 .

This implies:

q t+1 -1 j=1 1 jα ≤ 2q t+1 + q t+1 -1 =1 1 /q t+1 ≤ Cq t+1 ln q t+1 .
From (71) applied for t = 1, ..., r, we deduce (72):

1≤j<q r+1 1 j jα = r t=0 qt≤j<q t+1 1 j jα ≤ r t=0 1 q t qt≤j<q t+1 1 jα ≤ C r t=0 q t+1 q t ln q t+1 . Lemma 4.3. For ϕ ∈ C, it holds S qr ϕ qn ∞ ≤ C V (ϕ) ln(q r ). (73)
Proof. Using the Fejér kernel, we get

S qr ϕ qn ∞ ≤ ϕ qn ∞ + 1 q r |j|<qr |j ϕ qn (j)| ≤ ϕ qn ∞ + CK(ϕ) 1 q r qr-1 j=1 1 ||jα|| .
(73) follows by ( 8) and (71).

Truncation

Now we bound the truncation error for the Fourier series of the ergodic sums ϕ bn qn .

Lemma 4.4. If ψ is bounded and ϕ ∈ C, with C 1 = V (ϕ) 2 ψ ∞ , C 2 = V (ϕ) 3 ψ ∞ , up
to a numerical factor, we have, for q n ≤ q m ≤ q r ≤ q :

| ψ [ϕ qn ϕ qm -S q ϕ qn S q ϕ qm ] dµ|

≤ C 1 ( q m q ) 1 2 , (74)
| ψ [ϕ qn ϕ qm ϕ qr -S q ϕ qn S q ϕ qm S q ϕ qr ] dµ| ≤ C 2 ( q r q )

1 2 ln 2 (q ). (75)
Proof. We use the bound (70) which gives, for

q n ≤ q , R L ϕ qn 2 2 = |j|≥q | ϕ qn (j)| 2 = |j|≥q |γ j (ϕ)| 2 j 2 q n jα 2 jα 2 ≤ C 2 K(ϕ) 2 q n q .
For ψ bounded, as ϕ qn 2 ≤ CK(ϕ), this implies:

| ψ [ϕ qn ϕ qm -S q ϕ qn S q ϕ qm ] dµ| ≤ ψ ∞ [ ϕ qn 2 R q ϕ qm 2 + R q ϕ qn 2 ϕ qm 2 ] ≤ C V (ϕ) 2 ψ ∞ [( q n q ) 1 2 + ( q m q ) 1 2 ].
This proves (74). For (75), in each term of the expansion of (S q ϕ qn + R q ϕ qn ) (S q ϕ qm + R q ϕ qm ) (S q + R q ) -S q ϕ qn S q ϕ qm S q ϕ qr , we bound one factor in L 2 -norm and the others in uniform norm using (73).

Inequalities under Hypothesis 1

Recall that the decorrelation inequalities of Lemma 4.5 are based on Hypothesis 1 on α.

From (29) in Hypothesis 1, one deduces: for constants B, C, the coecients in Ostrowski's expansion satisfy b n ≤ B n p and, since q n ≤ B n (n!) p , ln q n ≤ C n ln n, c n ≤ C n p+1 ln n.

(

The case when α has bounded partial quotients corresponds to p = 0 and we have then ln q n ≤ C n.

Let us mention that Hardy and Littlewood in [START_REF] Hardy | Some problems of diophantine approximation: a series of cosecants[END_REF] considered quantities similar to that in the lemma below. One of their motivations was to study asymptotically the number of integral points contained in homothetic triangles.

Lemma 4.5. If a k+1 ≤ Ak p , ∀k ≥ 1 and n ≤ m ≤ , we have for every Λ ≥ 1:

∞ j=1 q n jα j 2 jα ≤ C n p+2 ln n q n+1 , (77)
1≤j,k<q Λ , j =k

q n jα q m kα |k -j| k j jα kα ≤ C q n+1 Λ 2p+4 (ln Λ) 2 , (78)
-q Λ <i,j,k<q Λ , i+j+k =0

q n iα q m jα q kα |i + j + k| i j k iα jα kα

≤ C q n+1 Λ 3p+8 . (79)
Proof. 1) Proof of (77)

We use the inequalities:

jq k α j ≤ q k α ≤ 1 q k+1
for j < q k+1 , jq k α ≤ 1 for j ≥ q k+1 . For > n, we write

q -1 j=1 q n jα j 2 jα = (A) + (B), with (A) := q n+1 -1 j=1 1 j q n jα j jα ≤ 1 q n+1 q n+1 -1 j=1 1 j 1 jα ≤ 1 q n+1 n k=0 1 q k q k+1 -1 j=q k 1 jα ≤ C 1 q n+1 n k=0 q k+1
q k ln q k+1 , by (72);

(B) := q -1 j=q n+1 q n jα j 2 jα ≤ -1 k=n+1 q k+1 -1 j=q k 1 j 2 jα ≤ -1 k=n+1 1 q 2 k q k+1 -1 j=q k 1 jα ≤ C -1 k=n+1 1 q k q k+1 q k ln q k+1 , by (71)
.

By (4), we know that

q n+1 q k ≤ Cρ k-n , with ρ < 1, for k ≥ n+1. By hypothesis, a k+1 ≤ Ak p .
It follows with the notation (69):

(A) ≤ C q n+1 n k=0 c k ≤ C n p+2 ln n q n+1
and for (B), with a bound which doesn't depend on ≥ n:

1 q n+1 -1 k=n+1 q n+1 q k q k+1 q k ln q k+1 ≤ C 1 q n+1 ∞ j=0 ρ j (j + n + 1) p+1 ln(j + n + 1) ≤ C n p+1 ln n q n+1 .
2) Proof of (78)

To bound the sum in (78), we cover the square [1, q Λ [×[1, q Λ [ in N × N by rectangles R r,s = [q r , q r+1 [×[q s , q s+1 [ for r and s varying between 0 and Λ -1 and then we bound the sum on each of these rectangles (minus the diagonal if r = s).

Distinguishing dierent cases according to the positions of r and s with respect to n + 1 and m + 1, we have, for j ∈ [q r , q r+1 [, k ∈ [q s , q s+1 [, j = k. q n jα q m kα |k -j| j k jα kα ≤ 1 q max(r,n+1) q max(s,m+1) 1 |k -j| jα kα .

By (71) and (72), using (k -j)α ≤ jα + kα ), we have

(j,k)∈Rr,s 1 |k -j| jα kα ≤ (j,k)∈Rr,s 1 |k -j| (k -j)α jα + 1 |k -j| (k -j)α kα ≤ q max(r,s)+1 ln(q max(r,s)+1 ) max(r,s) t=0 c t .
It follows (j,k)∈Rr,s, j =k q n jα q m kα |k -j| k j jα kα ≤ q max(r,s)+1 q max(r,n+1) q max(s,m+1) ln(q max(r,s)+1 ) max(r,s)

t=0 c t ≤ 1 q n+1 ln(q Λ+1 ) Λ t=0 c t max k=1,...,Λ a k . The square [1, q Λ [×[1, q Λ [ is covered by Λ 2 rectangles R r,
q n jα q m kα |k -j| k j jα kα ≤ Λ 2 C q n+1 ln(q Λ+1 ) Λ t=0 c t max k=1,...,Λ a k ≤ C q n+1 Λ 2p+5 ln(Λ) 2 .
3) Proof of (79)

Here we consider sums with three indices i, j, k. Though we do not write it explicitly, these sums are to be understood to be taken on non zero indices i, j, k such that i + j + k = 0.

We cover the set of indices by sets of the form

R ±r,±s,±t = {(i, j, k) : ±i ∈ [q r , q r+1 [, ±j ∈ [q s , q s+1 [, ±k ∈ [q t , q t+1 [}
Distinguishing dierent cases according to the positions of r, s and t with respect to n+1, we get: if (i, j, k) ∈ R ±r,±s,±t and n ≤ m ≤ , q n iα q m jα |q kα |i| |j| |k| ≤ 1 q max(r,n+1) q max(s,n+1) q max(t,n+1) .

(

) 80
We have

1 iα jα kα ≤ 1 (i + j + k)α [ 1 jα kα + 1 iα kα + 1 iα jα
].

We then use (71) and (72) three times, sum over R ±r,±s,±t and get:

(i,j,k)∈R ±r,±s,±t

1 |i + j + k| iα jα kα ≤ ( 3 max(r,s,t) v=0 c v ) ln 2 (q max(
r,s,t)+1 ) (q s+1 q t+1 + q r+1 q t+1 + q r+1 q s+1 ) .

By (80) we then have: R ±r,±s,±t q n iα q m jα |q kα |i + j + k| |i| |j| |k| iα jα kα ≤ C

( 3 max(r,s,t) v=0 c v ) ln 2 (q max(r,s,t) ) q max(r,n+1) q max(s,n+1) q max(t,n+1) (q s+1 q t+1 + q r+1 q t+1 + q r+1 q s+1 )

≤ C q n+1 ( 3Λ v=0 c v ) ln 2 (q Λ+1 )( max k=1,...,Λ a k ) 2 .
One needs 8Λ 3 boxes R ±r,±s,±t to cover the set {-q Λ < i, j, k < q Λ , i + j + k = 0}. This implies for a constant C:

-q Λ <i,j,k<q Λ , i+j+k =0

q n iα q m jα |q kα |i + j + k| |i| |j| |k| iα jα kα ≤ C q n+1 Λ 3p+8 .

Proof of Proposition 2.3

By (68) we have | ψ ϕ qn dµ| ≤

j =0 | ϕ qn (j)| | ψ(-j)| ≤ K j≥1
q n jα j 2 jα and (32) follows from (77):

j≥1 q n jα j 2 jα ≤ C n p+2 ln n q n+1 .
We prove now (33). With L = q Λ , we have:

ψ S L ϕ qn S L ϕ qm dµ = |j|,|k|≤L,j =k ϕ qn (j) ϕ qm (k) ψ(j -k).
In what follows, the constant C is equal to V (ψ)V (ϕ) 2 (up to a factor not depending on ψ and ϕ) which may change.

Recall that, by ( 4), there is a constant B such that m ≤ B ln q m , ∀m ≥ 1. q n jα q m kα |k -j| j k jα kα ≤ C q n+1 Λ 2p+4 (ln Λ) 2 .

Putting it together with the truncation error term (74) and replacing q n+1 by q n , we get

| X ψ ϕ qn ϕ qm dµ| ≤ C [ Λ 2p+4 (ln Λ) 2 q n + ( q m q Λ ) 1 2 ], for n ≤ m ≤ Λ. (81)
Recall that ( qm q Λ )

1 2 ≤ ρ Λ-m 2 .
Let us take Λ -m of order 2(ln 1 ρ ) -1 ln q n , i.e., such that the second term in the bracket of the RHS of (81) is of order 1/q n . We have then Λ ≤ max(m, C 1 log q n ) and with Hypothesis 1 the rst term in the bracket is less than

C 1 q n max( (ln q n ) 2p+5 , m 2p+5 ) ≤ C 2 q n max( (ln q n ) 2p+5 , (ln q m ) 2p+5 ) ≤ C 2 q n (ln q m ) 2p+5 ≤ C 3 m 2p+5 q n .
This shows (33) with θ 2 = 2p + 5.

In the same way, (34) follows from (75) and (79).

As recalled in Subsection 1.1, every n < q m+1 is coded by an admissible word b 0 ...b m , with b 0 ∈ {0, 1, ..., a 1 -1}, b j ∈ {0, 1, ..., a j+1 }, j = 1, ..., m, where for two consecutive letters b j-1 , b j , if b j = a j+1 , then b j-1 = 0.

For α the golden mean, a nite word b 0 ...b m is admissible if it is composed of 0's and 1's and two consecutive letters b i , b i+1 cannot be both 1. We denote by X the space of one-sided innite admissible sequences, that is sequences of 0, 1 without two consecutive 1's. For simplicity the letter b will denote an admissible word, either nite or innite. The context will make clear if b is nite or not.

If b = b 0 ...b is an admissible word, we put n b, := i=0 b i q i .

When α is the golden mean, we use the sub-shift (X, σ), where σ = σ X is the shift on X. Let µ be the σ-invariant probability measure on X of maximal entropy. Let C x 0 ...xn denote the cylinder composed of sequences starting with x 0 . . . x n . For n ≥ 1, depending whether x 0 and x n are both equal to 1, or only one of them, or none, we have

µ(C x 0 ...xn ) = 1 α + 2 α -n , α α + 2 α -n or α 2 α + 2 α -n .
If E ⊂ X is a union of cylinders of length n, its measure can be compared to the number of cylinders which compose it:

α + 2 α + 1 µ(E) ≤ α -n Card {cylinder W of length n : W ⊂ E} ≤ (α + 2) µ(E). (85)
♦ In the general case, let us consider the set of innite admissible sequences corresponding to the Ostrowski expansions for the periodic part of the sequence (a n ):

X := {x = (x i ) i∈N such that ∀i x i-1 x i = ua n 0 +i+1 with u = 0}.
The space X is invariant under the action of σ p X (because (a n ) is p-periodic for n ≥ n 0 ).

We dene an irreducible aperiodic sub-shift of nite type as follows: the state space of Y is the set of words x 0 . . . x p-1 of X, a transition between two such words w 1 and w 2 is allowed if the concatenation w 1 w 2 is the beginning of length 2p of a sequence in X.

From (84) we see that the exponential growth rate of the number of Ostrowski expansions of length at most n 0 + pk is ln λ (with respect to k). It is also the growth rate of the number of words of length pk of X. As these words correspond to the words of length k in Y , the topological entropy of (Y, σ) is ln λ (where σ = σ Y is the shift to the left on Y ). There is a unique invariant probability measure µ on (Y, σ) with entropy ln λ. This measure can be constructed as follows. Let B be the matrix with entries 0 and 1 that gives the allowed transitions between elements of the alphabet of Y . As the topological entropy of Y is the logarithm of the spectral radius of B, this spectral radius is λ. Let U and V be two positive vectors such that BU = λU, t BV = λV, t U V = 1. The measure µ is the Markovian measure determined by its values on cylinders given by µ(C y 0 y 1 ...yn ) = V y 0 U yn λ -n , when y 0 y 1 . . . y n is an admissible word (see [START_REF] Kitchens | Symbolic dynamics. One-sided, two-sided and countable state Markov shifts Universitext[END_REF] pp.21-23 and p.166 for more details on this classical construction). As there are only nitely many products V y 0 U yn , there exists a constant c > 0 such that, if a subset E of Y is a union of cylinders of length n, then (86)

1 c µ(E) ≤ Card {W cylinders of length n : W ⊂ E} λ -n ≤ c µ(E).

Lemma of large deviations

We will use the following inequality of large deviations for irreducible Markov chains with nite state space (see [START_REF] Lezaud | Cherno-type bound for nite Markov chains[END_REF], Theorem 3.3):

Lemma 5.1. Let A be nite union of cylinders. For every ε ∈]0, 1[, there are two positive constants R(ε), ξ(ε)

depending on A such that µ{x ∈ X : 1 L L-1 k=0 1 A (σ k x) ≤ µ(A)(1 -ε)} ≤ R(ε) e -ξ(ε)L , ∀L ≥ 1. (87)
2) Reduction of the Ostrowski expansion to a window By ( 2) and ( 4) we have, for a constant ρ < 1, q i q j α ≤ Cρ |j-i| . Hence, for 0

≤ j ≤ , if κ is such that 0 ≤ j -κ ≤ j + κ ≤ : | i=0 b i q i q j α - j+κ-1 i=j-κ b i q i q j α | ≤ i=j+κ b i q i q j α + j-κ-1 i=0 b i q i q j α ≤ C ρ -κ . (88)
It means that i=0 b i q i q j α = n b, q j α is well approximated by j+κ-1 i=j-κ b i q i q j α which depends on a word with indices belonging to a window around j, with a precision depending on the size of the window. This is valid for any irrational α.

The quantity introduced in the next denition can be viewed as a function of an innite word b or of a nite word b j-κ 0 , ..., b j+κ 0 . We put Γ(b, j) := 1 5

j+κ 0 i=j-κ 0 (-1) i b i α j-i + (-α) i-j α. (89)
A simple computation shows that Γ(b, j + 1) = -Γ(σb, j). Therefore we have:

Γ(σ k b, κ 0 ) = (-1) k Γ(b, k + κ 0 ). (90)
Lemma 5.2. Let α be the golden mean For every δ > 0, there is κ

0 = κ 0 (δ) such that d(n b, q j α -Γ(b, j), Z/5) ≤ δ, if j ≥ κ 0 . (91)
Proof. We can restrict the sum n b, q j α = i=0 b i q i q j α to the sum j+κ 0 i=j-κ 0 b i q i q j α, since their distance modulo 1 is small for κ 0 big enough by (88). By (82), we have q i q j = 1 + α

5 α i+j + 2 -α 5 (-α) -(i+j) + (-1) i 5 α j-i + (-α) i-j ; hence: j+κ 0 i=j-κ 0 b i q i q j α = 1 5 j+κ 0 i=j-κ 0 [b i (1 + α)α i+j+1 + b i (-1) i+j (2 -α)α 1-(i+j) ] + Γ(b, j).
The distance to Z of the rst sum above at right is small by (83).

The lemma shows that for the golden mean the distance to Z/5 of j=0 b i q i q j α is almost the distance to Z/5 of Γ(b, j), which depends on the short word b j-κ 0 ...b j+κ 0 (reduction to a window of width 2κ 0 of the long word b 0 ...b ) in such a way that its values, when j varies, are the values of a xed function computed for shifted words.

♦ The lemma extends to a general quadratic number. We need some notation.

For an integer i, we write i = i + pη i + n 0 , where i is the class of i -n 0 modulo p and η i the integer part of (i -n 0 )/p. The classes mod p are identied with the integers 0, ..., p -1.

With the notation introduced in (84), we put

T (i, j) := α r 2 (s i + t i λ)(u j + v j λ), U (i, j) := α r 2 (u i + v i λ)(s j + t j λ). Lemma 5.3. Let δ ∈]0, 1 2r [. There is κ 0 = κ 0 (δ) such that, if j ≥ n 0 + κ 0 p, (92)
d(n b, q j α -

n 0 +(η j +κ 0 )p-1 i=n 0 +(η j -κ 0 )p b i T (i, j)λ η i -η j + U (i, j)λ η j -η i , Z/r) ≤ δ.
Proof. Recall that (a n ) is p-periodic for n ≥ n 0 . We consider indices j ≥ n 0 and take sums on windows union of blocks of length p, hence of the form n 0 + mp, . . . , n 0 + qp -1.

Using (84), the product q n 0 +kp+m q n 0 +k p+m is equal to

1 r 2 (s m + t m λ)(s m + t m λ)λ k+k + (u m + v m λ)(u m + v m λ)λ -(k+k ) + 1 r 2 (s m + t m λ)(u m + v m λ)λ k-k + (u m + v m λ)(s m + t m λ)λ k -k .
Still using (84), we have

s m r (s m + t m λ) r λ k+k = s m r (q n 0 +(k +k)p+m - 1 r (u m + v m λ)λ -(k +k) ).
From this (and a similar equality) we obtain 2), the distance of the left side term above to Z/r is bounded by Cλ -(k +k) . It follows:

q n 0 +kp+m q n 0 +k p+m α - 1 r 2 (s m + t m λ)(u m + v m λ)λ k-k + (u m + v m λ)(s m + t m λ)λ k -k α = s m r q n 0 +(k +k)p+m α + t m r q n 0 +(k +k+1)p+m α -[ s m r 2 (u m + v m r 2 λ)λ -(k +k) α + t m r 2 (u m + v m λ)λ -(k +k+1) α]. Since d(q n 0 +(k +k)p+m α, Z/r) ≤ d(q n 0 +(k +k)p+m α, Z) ≤ Cλ -(k +k) by (
d(q i q j α -T (i, j)λ η i -η j + U (i, j)λ η j -η i , Z/r) ≤ Cλ -p(η j +η i ) , ∀i, j ≥ n 0 .

Thus, using (88), for κ 0 large enough and if j ≥ n 0 + κ 0 p, we have:

d(n b, q j α - n 0 +(η j +κ 0 )p-1 i=n 0 +(η j -κ 0 )p b i T (i, j)λ η i -η j + U (i, j)λ η j -η i , Z/r) ≤ Cλ -κ 0 p + C n 0 +(η j +κ 0 )p-1 i=n 0 +(η j -κ 0 )p λ -η j -η i ≤ Cλ -κ 0 p + 2Cκ 0 λ κ 0 -2η j ≤ δ.
3) From long words to short words We can take c = 4 for the golden mean, c = u -2 0 , with u 0 = inf k>1 q k-1 -1 q k -1 , in the general case.

Proof. The proof is given for the golden mean. The general case is analogous.

The ways of completing a short word into a long one depend only on the rst letter b 0 and the last letter b 1 : if b 0 = 1, any admissible beginning ts; if b 0 = 1, then only the admissible beginnings nishing by 0 t; if b 1 = 0 then any admissible ending ts; if b 1 = 1, only endings with 0 as rst letter t.

The number of admissible words of length r is q r+1 , the number of admissible words of length r beginning (or ending) by 0 is q r . Let denote S i , i = 1, ..., 4, the set of short words b 0 . .

. b 1 such that b 0 = b 1 = 0, b 0 = 0 and b 1 = 1, b 0 = 1 and b 1 = 0, b 0 = b 1 = 1, respectively.
Depending on the set S i to which Λ(w) belongs, the cardinal of Card Λ

-1 (Λ(w)) is D 1 = q 0 q -1 +1 , D 2 = q 0 q -1 , D 3 = q 0 -1 q -1 +1 , or D 4 = q 0 -1 q -1 respectively. Since, 1
2 ≤ q r /q r+1 ≤ 1, for all r, we have

D 1 = max i D i , D 4 = min i D i , D 4 ≤ D 1 ≤ 4D 4
and nally

Card (P) = 4 i=1 Card (P ∩ S i ) = 4 i=1 1 D i Card {w ∈ L : Λ(w) ∈ P ∩ S i } ≤ 1 D 4 4 i=1 Card {w ∈ L : Λ(w) ∈ P ∩ S i } = 1 D 4 Card {w ∈ L : Λ(w) ∈ P}, Card (S) = 4 i=1 Card (S i ) = 4 i=1 1 D i Card {w ∈ L : Λ(w) ∈ S i } ≥ 1 D 1 Card (L).
4a) End of the proof of Proposition 1.4 when α is the golden mean Let δ be a small positive number. Its value will be chosen later. It follows from ( 14) for big enough that, if q -1 j+1 < δ:

Card {n ∈ [1, q +1 [ : d(nq j α, Z/5) ≤ 3δ} ≤ C 1 δq +1 , ∀j ≤ . ( 93
)
If κ 0 is big enough, from (91) in Lemma 5.2, we have with Γ(b, j) dened in (89):

d(n b, q j α, Z/5) ≥ 3δ ⇒ d(Γ(b, j), Z/5) ≥ 2δ ⇒ d(n b, q j α, Z/5) ≥ δ.

(94)

By taking κ 0 large enough, we can suppose q -1 κ 0 +1 < δ. By (93) (translated in terms of words) for each j ∈ [κ 0 , ], the proportion of words b = b 0 . . . b of length + 1 for which d(n b, q j α, Z/5) ≥ 3δ, is smaller than C 1 δ. Therefore, if ≥ κ 0 , we get Using ⇒ in (94), we have therefore, taking L = -κ 0 , for -κ 0 ≥ j ≥ κ 0 , µ{b ∈ X : Card {j ∈ [κ 0 , [: d(n b, q j α, Z/5) ≥ δ} ≤ µ(A δ )(1 -ε)( -κ 0 )} ≤ R e -ξ ( -κ 0 ) . By (85), the previous inequality translated in terms of cardinal yields for a constant C 5 : Card {b 0 . . . b : Card {j < : d(n b, q j α, Z/5) ≥ δ} ≤ (1 -C 4 δ) 2 } ≤ C 5 e -ξ q +1 . If δ is taken small enough to get (1 -ε 0 ) ≤ (1 -C 4 δ) 2 and using that e ξ is equivalent to a power of q +1 (because (q ) is equivalent to a geometric progression), the previous inequality shows (20) of Proposition 1.4.

The same computation can be done for n 0 +(η j +κ 0 )p-1 i=n 0 +(η j -κ 0 )p b i U (i, j)λ η i -η j . Taking the sum for the T 's and U 's, we get Γ j (y) = Γ 0 j (σ η j -κ 0 y).

(99)

From ( 98), ( 97) and (99), it follows, if

≥ n 0 + 2κ 0 p and j ≥ n 0 + κ 0 , Card{b 0 . . . b : d(Γ j (σ η j -κ 0 y), Z/r) ≤ 2δ} ≤ C 1 δq +1 .

But Γ 0 j (σ η j -κ 0 y) depends only on the short word b n 0 +(η j -κ 0 )p . . . b n 0 +(η j +κ 0 )p-1 , which is a sub-word of the long word b 0 . . . b . By Lemma 5.4 we obtain for constants C 2 , C 3 > 0:

Card{b n 0 +(η j -κ 0 )p . . . b n 0 +(η j +κ 0 )p-1 : d(Γ 0 j (σ η j -κ 0 y), Z/r) ≤ 2δ} ≤ C 2 δλ 2κ 0 . b i T (i, m)λ η i -k-κ 0 + U (i, m)λ k+κ 0 -η i , and, if j = (k + κ 0 )p + m ∈ [n 0 + κ 0 , ] (i.e., η j = k + κ 0 , j = m), d(Γ 0 m (σ k y), Z/r) ≥ 2δ ⇒ d(n b, q j α, Z/r) ≥ δ.

In particular:

p Card{k < η -κ 0 : d(Γ 0 m (σ k y), Z/r) ≥ 2δ, m = 0, ..., p -1} ≤ Card {j < (η -κ 0 )p : d(n b, q j α, Z/r) ≥ δ}.

By Lemma 5.1, for the Markov chain deduced from Y with state space the set of words of length 2κ 0 in Y , we get from (87): µ{y : Card {j < (η -κ 0 )p : d(n b, q j α, Z/r) ≥ δ} ≤ µ(A δ )(1 -ε)p(η -κ 0 )} ≤ Re -ξ(η -κ 0 ) . This can be translated in terms of cardinal using (86):

Card {y 0 . . . y η -κ 0 : Card {j < (η -κ 0 )p : d(n b, q j α, Z/r) ≥ δ} ≤ µ(A δ )(1 -ε)p(η -κ 0 )} is smaller than C 4 e -ξ(η -κ 0 ) λ η -κ 0 . It implies Card {b 0 . . . b : Card {j < : d(n b, q j α, Z/r) ≥ δ} ≤ µ(A δ )(1-ε)p(η -κ 0 )} ≤ C 5 e -ξ(η -κ 0 ) λ η -κ 0 .

If η > (κ 0 + 1)/ε (that is ≥ p(κ 0 + 2)/ε), then p(η -κ 0 ) ≥ (1 -ε) and, for some C 6 > 0, there are less than C 6 e -ξη λ η words b of length such that (102) Card {j ≤ : d( i=0 b i q i q j α, Z/r) ≥ δ} ≤ µ(A δ )(1 -ε) 2 . As 1 -2γ > 0, we can choose p such that p(1 -2γ) > 1. We have then: lim n R n p (x) = 0, for a.e. x.

1 . 1 .

 11 n-1)rα sin πnrα sin πrα e 2πirx . If ϕ ∈ C, then V(ϕ n ) ≤ nV (ϕ) and | ϕ n (r)| = |γ r (ϕ)| |r| | sin πnrα| | sin πrα| ≤ n K(ϕ) |r| , r = 0.Reminders on continued fractions.

Remark 1 .

 1 For every δ ∈]0, 1 2r [ and every interval [N 1 , N 1 + L], we have by a slight extension of Lemma 1.1:

sin 2 (

 2 πq k u) = 0 and thus for which Condition[START_REF] Feller | An introduction to probability theory and its application[END_REF] does not hold.

n 2 2

 2 and denote by I 2 the 2-dimensional identity matrix. Theorem 2.7. If α has bounded partial quotients and if the condition (12) is satised uniformly with respect to (a, b) in the unit sphere, there are 0 < r 1 , r 2 < +∞ two constants such that for a large set of integers n as in Theorem 2.4: -Γ n satises inequalities of the form r 1 I 2 ≤ Γ n (a, b) ≤ r 1 I 2 ; -the distribution of Γ -1 n Φ n converges to the standard 2-dimensional normal law.Proof. We only sketch the proof. The classical method of proof of a CLT for a vectorial function is to show a scalar CLT for all linear combinations of the components of the function. So the proof is like that of Theorem 2.1, but taking care of the bound from below of the variance for aϕ 1 n + bϕ 2 n : (12) should be uniform for (a, b) on the unit sphere.

  The functions ψ, ϕ are real valued. By (78), it holds | ψ S L ϕ qn S L ϕ qm dµ| ≤ |j|,|k|≤L,j =k | ϕ qn (j)| | ϕ qm (k)| | ψ(j -k)| ≤ C 1≤j,k≤L

Lemma 5 . 4 .

 54 Let 1 ≤ 0 ≤ 1 ≤ be three integers and let Λ : b 0 . . . b → b 0 . . . b 1 be the restriction map from the set L of admissible words to shortened words. There is a constant c > 0 such that, if S is the image of Λ, for any subset P of S, we have Card (P) Card (S) ≤ c Card {w ∈ L : Λ(w) ∈ P} Card (L) .

  Card {b 0 . . . b : d(Γ(b, j), Z/5) ≤ 2δ} ≤ C 1 δq , ∀j ∈ [κ 0 , ].

  Γ(b, j) depends only on the short word b j-κ 0 . . . b j+κ 0 , part of the long word b = b 0 . . . b . It follows, using Lemma 5.4 that Card {b j-κ 0 . . . b j+κ 0 : d(Γ(b, j), Z/5) ≤ 2δ} ≤ C 2 δq 2κ 0 +2 , ∀j ∈ [κ 0 , ].

1 k=0 1

 11 A δ := {b : d(Γ(b, κ 0 ), Z/5) ≥ 2δ}, it follows from (96) and (85):µ(A c δ ) ≤ α -2κ 0 -2 Card {b j-κ 0 . . . b j+κ 0 : d(Γ(b, j), Z/5) ≤ 2δ} ≤ C 2 δq 2κ 0 +2 α -2κ 0 -2 ≤ C 3 δ.Let C 4 be a constant > C 3 and ε = C 4 δ. Observe that we can chose large enough so that µ(A δ ) ( -κ 0 ) ≥ (1 -ε) : indeed, we have µ(A δ ) -(1 -ε) > 0 and by taking > µ(A δ )κ 0 /(µ(A δ ) -(1 -ε)) we obtain the required inequality. Now we use L-A δ (σ k b) = Card {k < L : d(Γ(σ k b, κ 0 ), Z/5) ≥ 2δ} and (90). According (94) with j = k + κ 0 and Lemma 5.1 with A = A δ and ε = C 4 δ (we assume δ < C -1 4 ), there are two positive constants R = R(ε), ξ = ξ(ε) such that µ{b ∈ X : Card {j ∈ [κ 0 , L + κ 0 [: d(Γ(b, j), Z/5) ≥ 2δ} ≤ µ(A δ )(1 -ε)L} ≤ R(ε) e -ξL .

1 k=0 1 A

 11 100) and (86) imply that(101) µ(A c δ ) = µ{y ∈ Y : d(Γ 0 m (y), Z/r) < 2δ, m = 0, ..., p -1} ≤ C 3 δ. Now, we have nδ (σ k y) = Card{k < n : d(Γ 0 m (σ k y), Z/r) ≥ 2δ, m = 0, ..., p -1}, Γ 0 m (σ k y) = n 0 +(k+2κ 0 )p-1 i=n 0 +kp

Proof. 1 )

 1 Setting R N (x) := N k=1 u k g k (x) N k=1 u k, by orthogonality and the conditions on u k , there is a constant C such thatX |R N (x)| 2 dµ ≤ C N 2γ-1 , which implies ∞ n=1 R n p 2 2 < +∞, if p(1 -2γ) > 1.

2 ) 2 (≤p j=1 u j 2 ]u k g k | n p k=1 u k 2 ]

 2222 Therefore, it suces to show that: lim n sup ∈Jn |R n p + (x) -R n p (x)| = 0, where J n = {0, 1, ..., (n + 1) p -n p -1}. For reals A, C, B , D , ∈ J n , with C, D > 0, it holds: ∈Jn |B | + |A| C .This implies, withA = n p k=1 u k g k , B = n p + k=n p +1 u k g k , C = n p k=1 u k , D = n p + k=n p +1 u k , max ∈Jn |R n p + (x) -R n p (x)| ≤ max ∈J | n p + k=n p +1 u k g k | n p k=1 u k + | n p k=1 u k g k | n p k=1 u k . (108)By a lemma of Rademacher-Mensov ([11], p. 156), if Y 1 , ..., Y L are mutually orthogonal functions in a probability space (X, µ) with nite variances σ 2 1 , ..., σ 2 we put M n,p := max ∈Jn | n p + k=n p u k g k |, then by (109) we haveE(M 2 n,p ) ≤ C(log(4 p n p-1 )) C (log n) 2 n p-1 n 2pγ = C (log n) 2 n p(2γ+1)-1 .It follows:E[ max ∈J | n p + k=n p u k g k | n ≤ C (log n) 2 n p(2γ+1)-1 n 2p = C (log n) 2 n p(2γ-1)-1 .Therefore, since 2γ -1 < 0, we haven E[ max ∈J | n p + k=n p < +∞, so that lim n max ∈J | n p + k=n p u k g k | n p k=1 u k = 0, a.e.Both terms in the right side of (108) converge a.e. to 0, which implies a.e.: lim n max ∈Jn |R n p + (x) -R n p (x)| → 0.

  ϕ belongs to the class C of centered BV functions, with Fourier series r =0

	5. Remark 3. The previous result is written with a self-normalisation. If α is quadratic, let us consider the ergodic sums normalised by √ ln n: (ϕ n / √ r e 2πir. , ln n) γr(ϕ) to apply Theorem 2.4 we have to check Condition

n≥1 . Then, for n ∈ V , the accumulation points of the sequence of distributions are Gaussian non degenerated with a variance belonging to a compact interval. 2.2. Application to step functions, examples. If

  ). are immediate examples where this condition is satised. In the second case, one observes that

  s and the sums on these rectangles are bounded by the same quantity. It follows, with Hypothesis 1, 1≤j,k<q Λ , j =k

JEAN-PIERRE CONZE AND STÉPHANE LE BORGNE

(51)

Proof. We show (50) and (51). The proof of (49) is the same.

1) For (50), it suces to bound the sums

Replacing f k by w -1 m+ f k , we will deduce the bound (50) from the inequalities ( 23), ( 24), (25) when w k ≤ 1, for 1 ≤ k ≤ m + . By [START_REF] Thouvenot | Limit laws for ergodic processes[END_REF] and (46), we have

From (24) and [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF], then from (25) and [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF], we obtain

Set κ = θ+3 ln(1/ρ) ln(m + ). If t -s or u -t ≥ κ, the previous inequalities imply:

It implies:

Now the result follows from:

2) For (51), we bound the sums [START_REF] Ostrowski | Bemerkungen zur Theorie der Diophantischen Approximationen[END_REF] and successively (24), (25), (26).

We obtain (because f v is centered for the rst inequality):

Putting κ = θ+4 ln(1/ρ) log(m + ), we get by the previous inequalities, for constants C, C 2 , C 3 :

The remaining terms give a bound which can be absorbed in the previous one, namely:

Proof of Proposition 2.1

The proof is given in several steps.

Dening blocks

We split the sum S n := f 1 + • • • + f n into small and large blocks. The small ones will be removed, providing gaps and allowing to take advantage of the decorrelation properties assumed in the statement of the proposition.

Let τ, δ be parameters (δ close to 0) such that 0 < δ < 1 2 and δ < τ . We set for n ≥ 1:

We put S n :=

Proof. We use

We have

For the rst term, using (53), we have:

Similarly, for the second term we apply (25) and ( 53) and we get:

Likewise (26) and Lemma 3.

At last, by (51) we have

From ( 57), ( 58), ( 59) and (60), we deduce that

n n 2τ ln 2 (n). In the sum above, for n big, we keep only the last two terms, since for n big enough the rst terms are smaller than the last ones.

If X and Y are two real square integrable random variables, then |E(e iX ) -E(e iY )| ≤ X -Y 2 . Therefore, using (55), we have for J n (ζ) := X e iζ Sn dµ: The proof consists in several steps. To bound from below d(nq j α, Z), successively we code n as an admissible word (Ostrowski's coding), reduce long words to short words, then interpret cardinals in terms of cylinders and invariant measure for a subshift. Finally we use a result of large deviations recalled in Lemma 5.1.

For the reader's convenience, at each step we will consider rst the simpler special case of the golden mean α = √ 5+1 2 (the corresponding rotation number is

Then the general case is treated between the signs ♦ and and may be skipped if α is the golden mean.

When α is the golden mean, its partial quotients are equal to 1 and (q n ) (the Fibonacci sequence with q -1 = 0, q 0 = 1, q 1 = 1, ...) is almost a geometric sequence of ratio α. We have

♦ For a general quadratic number α, the sequence (a n ) is ultimately periodic: there are integers n 0 , p such that a n+p = a n , ∀n ≥ n 0 .

, for i > 1.

From the recursive relation

q n , between the denominators (q n ) of α, it follows, ∀k ≥ 1,

, p.

The matrix A p is a 2 × 2 matrix with determinant (-1) p and non negative integer coefcients (positive if p > 1). It has two distinct eigenvalues λ > 1 and -λ -1 (where λ is a quadratic number) and it is diagonal in a basis of R 2 with coordinates in Q[λ]. We have λ p + (-λ -1 ) p ∈ Z.

Without loss of generality we may suppose that p is even (otherwise, we replace it by 2p).

Therefore there are integers r, s , t , u , v for ∈ {0, . . . , p -1} such that

For every , (q n 0 +kp+ ) k≥1 behaves like a geometric progression with ratio λ.

For the golden mean, (84) corresponds to (82) for n even and r = 5.

1) Ostrowski's coding, invariant measure for a subshift of nite type and counting ♦ 4b) End of the proof of Proposition 1.4 for a general quadratic number

As for the golden number, we take a positive number δ whose value will be xed later. By [START_REF] Hardy | Some problems of diophantine approximation: a series of cosecants[END_REF], if κ 0 is large enough, we have for some

Let Γ j , Γ 0 j be the functions on Y

Remark that the sums on the right can be viewed as functions of y through the b i 's. Letting y k := b n 0 +kp . . . b n 0 +kp+p-1 , we see that the sum inside the denition of Γ j is a function of y η j -κ 0 . . . y η j +κ 0 -1 .

Let A δ be the subset of Y dened by A δ := {y : d(Γ 0 j (y), Z/r) ≥ 2δ, for j = 0, . . . , p -1}.

By (92) in Lemma 5.3, if κ 0 is suciently large, we have the implication d(n b, q j α, Z/r) ≥ 3δ ⇒ d(Γ j (b), Z/r) ≥ 2δ.

(98)

As j = j -p, η j+p = η j + 1 and η i+p = η i + 1, we obtain by η j -κ 0 iterations:

Since η j-(η j -κ 0 )p = κ 0 , the last quantity reduces to

By (101), for ε 0 > 0, we can choose ε and δ such that µ(A δ )(1 -ε) 2 = 1 -ε 0 . On the other hand, since c -1 q +1 ≤ λ η ≤ cq +1 for some c > 0, C 6 e -ξη λ η ≤ C 7 q 1-ζ +1 , for some positive constants ζ, C 7 . Finally, we have obtained [START_REF] Lezaud | Cherno-type bound for nite Markov chains[END_REF] (in terms of number of admissible words):

6. Appendix 2: weighted orthogonal functions Let (g n ) be a sequence of orthogonal real functions in L 2 of a probability space (X, µ) and (u n ) be a sequence of positive constants. By the Lebesgue dominated convergence theorem, if the functions g n are uniformly bounded, the following condition

(103) is satised if the following condition holds:

1 ≤ u n ≤ n γ , ∀n ≥ 1, with 0 ≤ γ < 1. ≤ N γ-1 → 0.

But (103) and the result of Proposition 6.1 can fail if the parameter γ in (105) is taken ≥ 1. Indeed, suppose that g k 2 = 1, and let us take u k = k if k is a power of 2, else u k = 1.

Then, we have 1 ≤ u k ≤ k, 2 n k=1 u 2 k ≥ 4 3 2 2n and 2 n k=1 u k = 2 n+1 -(n + 1), so that (107)