A Limit Theorem for Birkhoff sums over Rotations

# A Limit Theorem for Birkhoff Sums of non-Integrable Functions over Rotations

Yakov G. Sinai Mathematics Department
Princeton University
Princeton
New Jersey 08544
USA Landau Institute for Theoretical Physics
Moscow
Russia
and  Corinna Ulcigrai School of Mathematics
University of Bristol
Bristol BS8 1TW
United Kingdom
Dedicated to M. Brin on the occasion of his sixtieth birthday.
###### Abstract.

We consider Birkhoff sums of functions with a singularity of type over rotations and prove the following limit theorem. Let be the non-renormalized Birkhoff sum, where is the rotation number, is the initial point and are uniformly distributed. We prove that has a joint limiting distribution in as tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.

###### Key words and phrases:
Limit theorems, Rotations, Birkhoff Sums, Principal Value, Continued Fraction
###### 2000 Mathematics Subject Classification:
Primary 37A30: Secondary 37E10, 60B10
The first author thanks NSF Grant DMS for the financial support.
The second author thanks the Clay Mathematics Institute, since part of this work was completed while she was supported by a Liftoff Fellowship.

The purpose of this paper is the proof of the following theorem.

###### Theorem 1.

For any Borel-measurable subset there exists

 (0.1) limN→∞Leb{(α,x):1NN−1∑n=011−e2πi(nα+x)∈Ω.}=P(Ω)

Here denotes the two-dimensional Lebesgue measure on and is a probability measure on .

In other words, the trigonometric sums have a limiting distribution.

The theorem follows as a corollary from the following more general theorem.

Let (mod ) be the rotation by on . Let where

• is periodic of period and on ;

• on for some and ;

• is a -periodic function, which extends to a function on .

###### Theorem 2.

For any there exists the limit

 limN→∞Leb{(α,x):a≤1NN−1∑n=0f(Rnαx)≤b}=P(a,b),

where is a probability measure on .

In other words, the random variables considered as functions of and have a limiting distribution.

Using periodicity of and redefining appropriately, we can replace by :

• for .

In what follows we will assume that and satisfy , and .

Theorem 1 follows from Theorem 2. Indeed, splitting into real and imaginary part, we can write

 11−e2πi(nα+x)=12+i2f(Rnαx),f(x):=sin2πx1−cos2πx.

Then satisfies (with ) and . Hence Theorem 1 is a corollary of Theorem 2.

Let us use the notation

 (0.2) SN(α,x,f)=SN(α,x):=N−1∑n=0f(Rnαx)

for the non-normalized Birkhoff sum of the function under . The dependence on will be omitted if there is no ambiguity. Similar theorems can be proved for expressions of the form

 12N+1N∑n=−N1e2πi(nα+x)−1and12N+1N∑n=−Nf(Rnαx).

Another example of Birkhoff sums with this type of singularity is given by the trigonometric series of cosecants, i.e. . This series was investigated by Hardy and Littlewood in [HL30], where they prove in particular that when is a quadratic irrational, the corresponding partial sums are uniformly bounded.

#### Outline of the proof.

The strategy of the proof is the following. For any positive and we construct approximate sums , which are close to in probability, i.e. for all sufficiently large

 Leb{(α,x):∣∣∣Gϵ,δN−1NSN(α,x)∣∣∣≥ϵ}≤δ.

Then we prove that, for each and , has a limiting distribution as and the distributions of are weakly compact in , and . All these statements together allow to prove Theorem 2.

Our strategy is to show that can be expressed as functions of quantities which do have a limiting distribution. In particular, one of the quantities involved is the ratio where are denominators of the continued fraction expansion of and is determined by . We use the renewal-type limit theorem proved in [SU08] which gives the existence of a limiting distribution for the ratio . This theorem is recalled and generalized in §1.2.

The other basic tool is the classical system of partitions of the unit circle induced by the continued fraction expansion (whose definition is recalled in §1.1). Using this system of partitions, the Birkhoff sums in (0.2) are decomposed onto simpler orbit segments, which we call cycles and analyze separately in §2. The key phenomenon which implies the asymptotic behavior of the Birkhoff sums is the cancellation between positive and negative contributions to each cycle (see §2.2) which resemble the existence of the principal value in non-absolutely converging integrals. The decomposition into cycles is explained in §3. The proof of Theorem 2 is given in §4.

## 1. Preliminaries.

### 1.1. Continued fractions and partitions of the interval.

The following system of partitions exists for any with irrational (see, e.g. [Sin94]). Write down the expansion of as a continued fraction:

 α=[a1,a2,…,an,…]

and let be the approximant. Let be the fractional part of . Denote by

 Δ(n):=Δ(n)0={[0,{qnα})ifniseven;[{qnα},1)ifnisodd.

For even, the intervals and are left-most and right-most subintervals of , with endpoints and respectively (see Figure 1, left). Put

 Δ(n)j:=RjαΔ(n)0.

Denote by the length of . Clearly is also the length of any interval .

For any , the intervals , and , are pair-wise disjoint and their union is the whole interval (see Figure 1, left). Denote by the partition of into the intervals with and with . Then in the sense of partitions.

Consider the union . The set , which, as a subset of , is the union of two intervals, can be considered () as a subinterval of the unit circle , with endpoints on the opposite sides of , i. e. when is even, (see Figure 1, right). Consider the induced map obtained as the first return map of on . Then is an exchange of the two intervals and . More precisely, if is even, then

 T(n)(x)={x−λ(n+1)(mod1)ifx∈Δ(n)x+λ(n)(mod1)ifx∈Δ(n+1)

and similarly for odd .

Assume is even. The intervals and can be represented as floors of two towers, on the top of and respectively, where increases with the height of the floor in the tower, as in Figure 1, left. Hence the number of floors in the two towers are and respectively. Let us denote the two towers by111The subscripts l and s stay for large and small respectively, since the tower is both larger and taller than .

 Z(n)l=∪qn+1−1j=0Δ(n)j;Z(n)s=∪qn−1j=0Δ(n+1)j.

Under the action of each point not in the last floor (i.e. not in or ) moves vertically upwards to the next floor. The action on the last floor is determined by : if e.g.  and then .

#### 1.1.1. Recursive structure of the partitions.

Let us also recall how to construct inductively. Given , the partition is obtained from as follows: the intervals , are also elements of the partition . Each is decomposed in subintervals, more precisely in intervals of length and a reminder, which is (see for example Figure 3). If is even, the reminder is the left-most interval of , while the other intervals, from left to right, are with (as in Figure 1, left). Hence, we have the following remark.

###### Remark 1.1.

Each pair of intervals of both belonging to the tower are separated by partition elements belonging to .

Given , consider . Since , is partitioned into elements of . Analyzing the recursive construction of the partitions , we have the following.

###### Remark 1.2.

The partition of into elements of is completely determined by , and , .

### 1.2. The renewal-type limit theorem for denominators.

The existence of the limiting distribution relies on the following limit theorem. Let be the approximants of and the corresponding denominators as functions of .

###### Theorem 3 ([Su08]).

Given , introduce

 (1.1) n(N)=n(N,α)=min{n∈N|qn>Nandniseven}.

Fix also an integer . Then the ratio and the entries for have a joint limiting distribution, as tends to infinity, with respect to the uniform distribution on .

Theorem 3 means that for each there exists a probability measure on such that for all and with ,

 (1.2) limN→∞Leb{α:a

Theorem 3 is a slight modification of Theorem , [SU08]. The differences and a sketch on how to modify the proof of Theorem in [SU08] to obtain Theorem 3 are pointed out in the Appendix §A.2.

As a corollary of Theorem 3, we have the following.

###### Corollary 1.3.

The quantities

 q(n(N))q(n(N)+1),λ(n(N)+1)λ(n(N)),1q(n(N))λ(n(N)+1),1q(n(N)+1)λ(n(N))

have a limiting distribution as tends to infinity.

###### Proof.

Let us recall that and satisfy the following recurrent relations (see [Khi35] and [Sin94] respectively):

 (1.3) qn+1=an+1qn+qn−1,λ(n−1)=an+1λ(n)+λ(n+1),n≥1.

Using them inductively (see [Khi35] or [SU08]), it is easy to show that

 qnqn+1=[an+1,an,…,a1],λ(n+1)λ(n)=[an+2,an+3,…].

Moreover, reasoning as in [SU08], we also have

 ∣∣∣qnqn+1−[an+1,an,…,an−K]∣∣∣≤12K+12; ∣∣∣λ(n+1)λ(n)−[an+2,an+3,…,an+K]∣∣∣≤12K−22,

where the exponential convergence is uniform in . Hence, since by Theorem 3, for each , , have a joint limiting distribution as tends to infinity, and also have a limiting distribution.

For the last two quantities, recall, e.g. from [Khi35], that

 λ(n)=|qnα−pn|=1qn+1+qnαn+1,whereαn+1=Gn+1α=[an+2,an+3,…].

Hence, in particular

 (1.4) 12≤λ(n)qn+1≤1,λ(n+1)qn≤1.

Moreover, since

 1qn(N)+1λ(n(N))=1+qn(N)qn(N)+1[an(N)+2,an(N)+3,…]

the ratio and similarly have limiting distributions. ∎

## 2. Analysis of a cycle.

In this section and in §3, we consider only Birkhoff sums of the function . Since is integrable, Birkhoff sums of are easily controlled in §4 with the help of Birkhoff ergodic theorem.

We first investigate in this section a special type of Birkhoff sum, which is used in §3 as a building block to decompose any other Birkhoff sum. Assume that and if is even or if is odd and consider the Birkhoff sum . We call the orbit segment a cycle and is a sum over a cycle. We remark that all points of a cycle are contained in the same tower and there is exactly one point in each floor of the tower; for this reason, we sometimes refer to as a sum over a tower (see also [Ulc07]). In section §3 we will refer to as the order of the cycle.

To simplify the analysis, we assume in what follows that is even and consider only the partitions with even and their cycles. The following proposition shows that the value of a sum over a cycle is determined essentially by the closest point to the endpoint.

###### Proposition 2.1.

Let be a sum along a cycle, if or if . For each there exist and functions , , which depend only on the following quantities

 (2.1)

and such that, letting if or if , we have

 (2.2) ∣∣∣1qSq(α,x)−gϵn(α,x,q)∣∣∣≤ϵ.

The proof of Proposition 2.1 is given in §2.2. The key ingredient which allows to reduce the sum over a cycle to finitely many terms (and hence to an expression given by depending on the above variables) is that there are cancellations between the two sides, positive and negative, of the singularity. The cancellations occur because the sequence of closest points to is given by a rigid translate of the sequence of closest points to (see Corollary 2.5). In order to prove this fact, we first show, in §2.1. that the partitions have a property of almost symmetry (see Lemma 2.3, in §2.1).

### 2.1. Almost symmetry of the partitions.

Consider the partition and let , for denote the middle points of the intervals , , rearranged in increasing order, so that and similarly let , for be the middle points of the intervals , rearranged in increasing order (see Figure 2).

Since we are interested in comparing the functions and evaluated along orbits segments which contain a point inside each of these intervals, we want to understand what happens to the middle points under the reflection .

Consider the set of reflected points , for and let , for denote its elements rearranged in increasing order. Similarly, let , for be the monotonical rearrangements of the points , for .

###### Lemma 2.2.

Let be even. The two sequences given by the points and the points respectively, excluding the closest point to , i.e. , and the closest point to , i.e. , are rigid translates of each other, i.e. they satisfy:

 (2.3) z′(n)i = z(n)i+1+λ(n+1)−λ(n);i=0,…,qn+1−2 (2.4) z′(n+1)i = z(n+1)i−1+λ(n+1)−λ(n),i=1,…,qn−1.

The restriction on the parity simplify the number of cases in the statement, but similar properties could be proved for odd.

Lemma 2.2 will follow as a corollary of an almost-symmetry property of the partitions (Lemma 2.3 below). Let us consider the following coding of the partitions . The unit interval is decomposed into subintervals which are elements of the partition and either belong to (i.e. are of the form for some ) or to (i.e. are of the form for some ). We will call them intervals of type and type respectively (large or short). Let be a string of letters and , where or according to the type of the interval of (where intervals of the partition are ordered from left to right in ). For example, the string coding the partition in Figure 2 (which is the same that appears also in Figure 1) is .

Let be the reflected string, which encodes the type of intervals after the reflection . Then, the following almost-symmetry property is satisfied by the partitions .

###### Lemma 2.3 (almost symmetry of ξ(n)).

For all , all the letters of the strings and coincide with the exception of the first and last, i.e.

 (2.5) ω′(n)i=ω(n)iforall2≤i≤qn+qn+1−1.

More precisely,

 ifniseven,ω––(n)=(lω(n)2⋯ω(n)qn+qn+1−1s)andω––′(n)=(sω(n)2⋯ω(n)qn+qn+1−1l); ifnisodd,ω––(n)=(sω(n)2⋯ω(n)qn+qn+1−1l)andω––′(n)=(lω(n)2⋯ω(n)qn+qn+1−1s).

Moreover, for , .

###### Proof.

The proof proceeds by induction on . For , where the number of occurrences of is given by . Hence, and there is nothing to prove. Assume that the almost-symmetry is proved for ( for all ) and is even. As it can be seen easily analyzing the recursive construction of in §1.1, the new string is obtained from by substituting each letter with (since which was the shortest length in is now the longest one in ) and substituting each letter with where the number of occurrences is given by , see for example Figure 3.

To verify the desired identities on the letters in and it is enough to verify that the letters occur in the same positions (with the exception the first and last letter of the string). Let denote the number of letters among with (i.e.  the cardinality of with ). Since all in become , the only in the string appear inside each block . Moreover, each occurrence of in generates a string of length in . Hence iff and (for ).

Similarly the string is obtained from by substituting with and substituting each symbol with ( copies of ). If denote the number of letters (i.e. ) among with , then iff and (for ). By the inductive assumption, since and coincide for all but , we have . Hence, for , we have iff and , which implies, as we wanted, that iff . The proof for odd is analogous.

From the definition of we have immediately that , for even and , for odd and the last equalities follow from the (2.5) and the fact that two are never nearby. ∎

###### Proof of Lemma 2.2..

Assume is even. The points and for are middle points of intervals of type respectively before and after the reflection. Let us first prove (2.3) for . The first interval of the partition is of type and hence contains , while belongs to the second interval after the reflection, since . Unless the string has the length and is (in which case there is nothing to prove), by Lemma 2.3 also , so the first two intervals are both of type and we have . Moreover, since by Lemma 2.3, the strings and coincide after the first element, also for all (see Figure 2).

Similarly, the points and for are middle points of intervals of type . In this case, belongs to the first interval after the reflection () and has to be kept aside, while and belong respectively to the interval before reflection and to the after reflection. Since the strings are and respectively, and, again by Lemma 2.3, since the strings then coincide, also for all (see again Figure 2). ∎

### 2.2. Cancellations.

Let be even, and let or according to whether or . Consider the orbit cycle , which is an orbit along a tower of . Let us rename the points of in increasing order, so that

 0

Similarly rearrange in increasing order distances from , i. e. the elements of , renaming them by

 0

From the structure of the partitions described in the second part of Lemma 2.3, one can easily check the following (see also Figure 5).

###### Remark 2.4.

If , and , while if , and .

For the other points, from the partition almost-symmetry expressed by Lemma 2.2, we have the following (see an illustration in Figure 4).

###### Corollary 2.5.

For all

 (2.6) yi−xi+1 = λ(n+1)−2x0,i=0,…,qn+1−2,ifx∈Δ(n); (2.7) yi−xi−1 = 2y0−λ(n),i=1,…,qn−1ifx∈Δ(n+1).
###### Proof.

Assume that (see Figure 5(a)). Since, for some , and both belong to the same , which is a rigid translate of , we have . Similarly, both and belong to the same , which is a rigid translate and a reflection of , hence . Thus, . Using this relation, (2.6) follows from (2.3) in Lemma 2.2. The argument to prove (2.7) when is analogous (see Figure 5(b)) and reduces to (2.4) in Lemma 2.2. ∎

We remark that the points of belong to different floors of the tower of the partition and are in the same relative position inside them. Hence, we have the following.

###### Remark 2.6.

The minimum distance is bounded below by . In particular,