On the CLT for rotations and BV functions

On the CLT for rotations and BV functions

Jean-Pierre Conze  and  Stéphane Le Borgne Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Abstract.

Let be a rotation on the circle and let be a step function. We denote by the corresponding ergodic sums . Under an assumption on , for example when has bounded partial quotients, and a Diophantine condition on the discontinuity points of , we show that is asymptotically Gaussian for in a set of density 1. An important point is the control of the variance for belonging to a large set of integers. When is a quadratic irrational, this information can be improved. The method is based on decorrelation inequalities for the ergodic sums taken at times , where the ’s are the denominators of .

IRMAR - UMR 6625, F-35000 Rennes, France

1. Introduction

Let us consider an irrational rotation on . By the Denjoy-Koksma inequality, the ergodic sums of a centered BV (bounded variation) function are uniformly bounded along the sequence of denominators of . But, besides, one has a stochastic behaviour at a certain scale along other sequences . In a sense the process defined by the above sums presents more complexity than the ergodic sums under the action of hyperbolic maps for which a central limit theorem is usually satisfied. We propose a quantitative analysis of this phenomenon.

Several papers have been devoted to this topic. M. Denker and R. Burton (1987), M. Lacey (1993), M. Weber (2000) and other authors proved the existence of functions whose ergodic sums over rotations satisfy a CLT after self-normalization. D. Volný and P. Liardet in 1997 showed that, when has unbounded partial quotients, for a dense set of functions in the class of absolutely continuous, or Lipschitz continuous or differentiable functions, the distributions of the random variables , and , are dense in the set of all probability measures on the real line.

Most often in these works the functions that are dealt with are not explicit. Here we consider ergodic sums of simple functions such as step functions. Let us mention the following related papers. For , F. Huveneers [Hu09] studied the existence of a sequence such that after normalization is asymptotically normally distributed. In [CoIsLe17] it was shown that, when has unbounded partial quotients, along some subsequences the ergodic sums of some step functions can be approximated by a Brownian motion.

Here we will use as in [Hu09] a method based on decorrelation inequalities which applies in particular to the bounded type case (bpq), i.e., when the sequence of partial quotients of is bounded. It relies on an abstract central limit theorem valid under some suitable decorrelation conditions. If is a step function, we give conditions which insure that for in a set of density 1, the distribution of is close to a normal distribution. (Theorem 3.4). Beside the remarkable recent “temporal” limit theorems for rotations (see [Be10], [BrUl17], [DoSa16]), this shows that a “spatial” asymptotic normal distribution can also be observed, for times restricted to a large set of integers.

An important point is the control of the variance for belonging to a set of density 1, at least in the case of with bounded partial quotients. In the special case where is a quadratic irrational, this information can be improved. A result for vectorial step functions can also be proven (Theorem 3.9). To apply the results to a step function, a Diophantine type condition is needed (cf. Condition (15)), which holds generically with respect to the discontinuities.

The results are presented in Section 2 and 3. There are based on the decorrelation of the ergodic sums taken at times and on an abstract central limit theorem, whose proofs are given in Sections 4 and 5.

2. Variance of the ergodic sums

Notation The uniform measure on identified with is denoted by . The arguments of the functions are taken modulo 1. For a 1-periodic function , we denote by the variation of computed for its restriction to the interval and use the shorthand BV for “bounded variation”. Let be the class of centered BV functions. By we denote a numerical constant whose value may change from a line to the other or inside a line.

The number is an irrational number in , with partial quotients and denominators : If belongs to , its Fourier coefficients satisfy:

 (1) cr(φ)=γr(φ)r,r≠0, with K(φ):=supr≠0|γr(φ)|<+∞.

The class contains the step functions with a finite number of discontinuities.

The ergodic sums are denoted by . Their Fourier series are

 (2) φN(x) :=N−1∑j=0φ(x+jα)=∑r≠0γr(φ)reπi(N−1)rαsinπNrαsinπrαe2πirx.

If belongs to , then so do the sums and we have

 |cr(φN)|≤K(φ)|r||sinπNrα||sinπrα|≤NK(φ)|r|.

If is a BV function, then so is and .

2.1. Reminders on continued fractions

For , denotes its distance to the integers: . Recall that

 2∥x∥≤|sinπx|≤π∥x∥, ∀x∈R.

Let be an irrational number. Then, for each , we write , where and are the numerators and denominators of . Recall that

 (3) 1an+1+2≤qnqn+1+qn≤qn∥qnα∥=qn|θn|≤qnqn+1=qnan+1qn+qn−1≤1an+1,
 (4) ∥kα∥≥∥qn−1α∥≥ 1qn+qn−1≥ 12qn, for 1≤k
 (5) ∥qnα∥=(−1)n(qnα−pn), θn=(−1)n∥qnα∥, α=pnqn+(−1)n∥qnα∥qn.

For we put

 (6) m=m(n):=ℓ,  if n∈[qℓ,qℓ+1[.

If has bounded partial quotients (), then there is a constant such that .

Thereafter we will need the following assumption which is satisfied by a.e. :

Hypothesis 1.

There are two constants such that

 (7) an≤Anp,∀n≥1.

Recall that every integer can be represented as follows (Ostrowski’s expansion):

 (8) n=m(n)∑k=0bkqk,with 0≤b0≤a1−1, 0≤bk≤ak+1 for 1≤k

Indeed, if , we can write , with , . and by iteration, we get (8).

In this way, we can associate to every its coding, which is a word , with , . Let us call “admissible” a finite word , , with , such that for two consecutive letters , we have , with .

Let us show that the Ostrowski’s expansion of an integer is admissible. The proof is by induction. Let be in . We start the construction of the Ostrowski’s expansion of as above. Now the following steps of the algorithm yield the Ostrowski’s expansion of (excepted some zero’s which might be added at the end). Since , the Ostrowski’s expansion of is admissible. It remains to check that, if , then . If , we would have , a contradiction.

Therefore, if we associate to an admissible word the integer , there is a 1 to 1 correspondence between the Ostrowski’s expansion of the integers , when runs in , and the set of (finite) admissible words starting with .

For given by (8), putting , , for and

 (9) fk(x):=bk−1∑i=0φqk(x+(nk−1+iqk)α),

 φn(x) =m(n)∑k=0nk−1∑j=nk−1φ(x+jα)=m(n)∑k=0bkqk−1∑j=0φ(x+nk−1α+jα) (10) =m(n)∑k=0bk−1∑i=0φqk(x+(nk−1+iqk)α)=m(n)∑k=0fk(x).

By convention, the expression is taken to be 0, if .

If is a denominator of and is a BV function, one has (Denjoy-Koksma inequality) :

 (11) ∥φq∥∞=supx|q−1∑i=0φ(x+iα)|≤V(φ).

One can also show that if satisfies (1) then . By Denjoy-Koksma inequality, we have .

2.2. Lower bound for the variance

Let be in . Keeping only indices that belong to the sequence , the variance at time is bounded from below as follows, with the constant ,

 ∥φn∥22 = 2∑k>1|ck(φ)|2(sinπnkα)2(sinπkα)2≥2ℓ∑j=1|cqj(φ)|2(sinπnqjα)2(sinπqjα)2≥cℓ∑j=1|cqj(φ)|2∥nqjα∥2∥qjα∥2.

Therefore we have, for :

 (12) ∥φn∥22 ≥ cℓ∑j=1|γqj(φ)|2a2j+1∥nqjα∥2≥cδ2ℓ∑j=1|γqj(φ)|2a2j+11∥nqjα∥≥δ.

Bounds for the mean variance

An upper bound for the variance and a lower bound for the mean of the variance are shown in [CoIsLe17]):

Proposition 2.1.

There are constants such that, if satisfies (1), then

 (13) ∥φn∥22≤CK(φ)2ℓ∑j=0a2j+1, ∀n∈[qℓ,qℓ+1[,

and the mean of the variances satisfies:

 (14) ⟨Dφ⟩n≥cℓ−1∑j=0|γqj(φ)|2a2j+1, ∀n∈[qℓ,qℓ+1[.

The proposition implies that, if is an integer such that , then . If the partial quotients of are bounded, under a condition on the sequence , using the results below, it can be deduced that the behaviour of for the indices giving the record variances is approximately Gaussian. But our goal is to obtain such a behaviour for a large set of integers . To do it, we need to obtain a lower bound of the variance for such a large set.

Bounds for the variance for a large set of integers

We will assume that satisfies the condition:

 (15) ∃N0,η,θ0>0 such that 1NCard{j≤N:|γqj(φ)|≥η}≥θ0∀N≥N0.

This condition is clearly satisfied by , since in this case . The validity of (15) for different step functions is discussed in subsection 3.2.

For , we estimate how many times for , we have . Since or , depending whether or , we have .

Remark: We are looking for values of such that is small. Of course there are special values of , like , such that this quantity is big, as shown by Denjoy-Koksma inequality Let us check it directly.

Recall that for any irrational , there are and such that

 (16) qn/qm≤Cλm−n, ∀m≥n.

We have , for , and , for . By (16), this implies ; hence, for , if .

For the complementary of on the circle, it follows: . This gives a bound for when varies, as expected.

Nevertheless, we will show that is small for a large set of values of .

Lemma 2.2.

For every and every interval of integers of length , we have

 (17) N2−1∑n=N11Iδ({nqjα})≤20(δ+q−1j+1)L,∀j such that qj+1≤2L.

Proof. For a fixed and , let us describe the behaviour of the sequence .

Recall that (modulo 1) we have , with (see (5)). We treat the case even (hence ). The case odd is analogous.

Therefore the problem is to count how many times, for even, we have or .

We start with . Putting , we have , for , where is such that .

Putting , the previous inequality shows that .

Starting now from , we have for , where is such that:

 w(j,2)+(n3−1−n2)θj<1

Again we put .

We iterate this construction and obtain a sequence (with ) such that

 {nθj}=w(j,i)+(n−ni)θj, ∀n∈[ni,ni+1[, w(j,i)+(ni+1−1−ni)θj<1

where is defined recursively by and satisfies , for every .

Since for and , we have , for each . This implies .

For each , the number of integers such that is bounded by . (This number is less than 2 if .)

Altogether, the number of integers such that is bounded by

 2R(j)(1+δθ−1j)≤2(Lθ−1j−1+2)(1+δθ−1j) ≤4(L+θ−1j)(δ+θj)≤4(L+2qj+1)(δ+q−1j+1).

If , the previous term at right is less than . This shows (17). ∎

Lemma 2.3.

Suppose that satisfies Condition (15). Then there are positive constants (not depending on ) such that, if is an interval of length and such that , for every , the subset of

 (18) V(I,δ):={n∈I:∥φn∥2≥c1δ√ℓ)}

has a complementary in satisfying:

 (19) CardV(I,δ)c≤c2(δ+ℓ−1)L.

Proof. Let , where is the constant in (15), and let

 A=A(I,δ):={n∈I:ℓ−1∑j=01Iδ({nqjα})≤ζℓ}.

Let us bound the density of the complementary of in by counting the number of values of in such that in an array indexed by .

By summation of (17) for to and the definition of , the following inequalities are satisfied:

 20(δℓ+∑0≤j≤ℓ−1q−1j+1)L≥∑0≤j≤ℓ−1, n∈I1Iδ({nqjα}) ≥∑n∈Ac∑0≤j≤ℓ−11Iδ({nqjα})≥∑n∈Acζℓ=ζℓCardAc.

Let be the finite constant . Then the set satisfies:

 (20) CardAc≤20ζ−1(δ+Cℓ−1)L, (21) n∈A⇒Card{j≤ℓ−1:∥nqjα∥≥δ}≥(1−ζ)ℓ.

In view of (21) and Condition (15), we have, for :

 Card({j≤ℓ−1:∥nqjα∥≥δ}⋂{j:|γqj(φ)|≥η})≥(1−(ζ+1−η0))ℓ=12η0ℓ.

Putting and ( do not depend on ), this implies by (12):

 (22) ∥φn∥22≥12cδ2η2η0ℓ=c21δ2ℓ, ∀n∈A, and CardA≥1−c2(δ+ℓ−1)L;

hence and therefore satisfies (25). ∎

Proposition 2.4.

Suppose that satisfies Condition (15). The subset of defined by

 (23) W:={n∈N:∥φn∥2≥c1√m(n)/√lnm(n)}

satisfies

 (24) limN→∞Card(W∩[0,N[)N=1.

Proof. By (16), there is a fixed integer such that , for .

Let us take and let . By this choice of , the condition of Lemma (2.3) is satisfied by and for big enough, since

 qm(N)≤2(qm(N)−qm(N)−uN)≤2(N−N1).

If , then .

Let .

If is in the complementary of in , then we have:

 ∥φn∥2≤c2√m(n)/√lnm(n)≤c2√m(N)/√lnm(N1).

Therefore, by Lemma (2.3) with , we have

 (25) CardBc≤c2((lnm(N1))−12+m(N)−1)(N−N1)

Now, , hence: given , for big enough,

 Card(Wc∩[0,N[)N ≤N1N+c2((lnm(N1))−12+m(N)−1) =qm(N)−uNN+c2((ln(m(N)−uN))−12+m(N)−1)≤ε.\qed

Remark: if is bpq, then is of order and the density of satisfies, for some constant ,

 (26) Card(W∩[0,N[)N≥1−C(lnlnN)−12.

2.3. A special case: the quadratic numbers

If is a quadratic number, the variance has the right order of magnitude for in a big set of integers. More precisely, the following bounds hold for the ergodic sums of under the rotation by :

Theorem 2.5.

If is a quadratic number and satisfies Condition (15), there are positive constants such that, for big enough:

 (27) #{n≤N: η1lnn≤∥φn∥22≤η2lnn}≥N(1−RN−ζ).

Proof. For every , the upper bound follows from (13). By (12), the variance is bounded from below by and we would like to bound from below these sums for a large set of integer .

We will first make a few remarks on the Ostrowski expansion in base , then study the quantity before going back to the previous sum.

First step: Remarks on Ostrowski expansions.

The sequence in this case is ultimately periodic: there exist integers and such that

 ∀n≥n0  an+p=an.

The denominators of are defined recursively: : , and for . Using matrices we can write:

 (qnqn+1)=(011an+1)(qn−1qn)

Let be the matrix

 (011an0+ℓ)(011an0+ℓ−1)…(011an0+1).

For every we have

 (qn0+kp+ℓ−1qn0+kp+ℓ)=AℓAkp(qn0−1qn0).

The matrix is a matrix with determinant and non negative integer coefficients (positive if ). Without loss of generality we may suppose that is even (if not, just replace it by ). The matrix has two different eigenvalues that will be denoted by and (its inverse) . The number also is a quadratic number and is diagonal in a base of with coordinates in . We deduce that there exist integers such that, for every , ,

 (28) qn0+kp+ℓ=1r[(sℓ+tℓλ)λk+(uℓ+vℓλ)λ−k]

We will use a subshift of finite type defined by the admissible words corresponding to the periodic part of the sequence . A finite word is admissible if , , with , such that for two consecutive letters , we have , with .

Let us consider the sequences of infinite admissible sequences corresponding to the Ostrowski expansions for the periodic part of the sequence (indices larger than ).

 X:={x=(xi)i∈N such that ∀i xixi+1≠uan0+i+1 with u≠0}.

The space is not invariant under the action of the left shift but it is invariant under the action of (because is periodic of period for ). We define an irreducible, aperiodic subshift of finite type as follows: the state space of is the set of words of , a transition between two such words and can occur if the concatenation is the beginning of length of a sequence in .

The integers between 0 and are coded by the Ostrowski expansions of length at most . From (28) we see that the exponential growth rate of the number of Ostrowski expansions of length at most