A gauss-kuzmin theorem and related questions for $\theta$-expansions

Using the natural extension for $\theta$-expansions, we give an infinite-order-chain representation of the sequence of the incomplete quotients of these expansions. Together with the ergodic behavior of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.


Introduction
During the last fifty years a large amount of research has been devoted to the study of various algorithms for the representation of real numbers by means of sequences of integers. Motivated by problems in random continued fraction expansions (see [1]), Chakraborty and Rao [4] have initiated a systematic study of the continued fraction expansion of a number in terms of an irrational θ ∈ (0, 1). This new expansion of positive reals, different from the regular continued fraction expansion is called θ-expansion.
The purpose of this paper is to solve a Gauss-Kuzmin problem for θexpansions. In order to solve the problem, we apply the theory of random systems with complete connections extensively studied by Iosifescu and Grigorescu [7]. First we outline the historical framework of this problem. In Section 1.2, we present the current framework. In Section 1.3, we review known results.

Gauss' Problem
One of the first and still one of the most important results in the metrical theory of continued fractions is the so-called Gauss-Kuzmin theorem. Any irrational 0 < x < 1 can be written as the infinite regular continued fraction where a n ∈ N + := {1, 2, 3, . . .} [8]. Such integers a 1 , a 2 , . . . are called incomplete quotients (or continued fraction digits) of x. The metrical theory of continued fraction expansions started on 25th October 1800, with a note by Gauss in his mathematical diary [3]. Define the regular continued fraction (or Gauss) transformation τ on the unit interval I := [0, 1] by where ⌊·⌋ denotes the floor (or entire) function. With respect to the asymptotic behavior of iterations τ n = τ • · · · • τ (n-times) of τ , Gauss wrote (in modern notation) that where λ denotes the Lebesgue measure on I. In 1812, Gauss asked Laplace [3] to estimate the n-th error term e n (x) defined by e n (x) := λ(τ −n [0, x]) − log(1 + x) log 2 , n ≥ 1, x ∈ I. (1.4) This has been called Gauss' Problem. It received first solution more than a century later, when R.O. Kuzmin [10] showed in 1928 that e n (x) = O(q √ n ) as n → ∞, uniformly in x with some (unspecified) 0 < q < 1. This has been called the Gauss-Kuzmin theorem or the Kuzmin theorem.
One year later, using a different method, Paul Lévy [12] improved Kuzmin's result by showing that |e n (x)| ≤ q n for n ∈ N + , x ∈ I, with q = 3.5 − 2 √ 2 = 0.67157.... For such historical reasons, the Gauss-Kuzmin-Lévy theorem is regarded as the first basic result in the rich metrical theory of continued fractions. An advantage of the Gauss-Kuzmin-Lévy theorem relative to the Gauss-Kuzmin theorem is the determination of the value of q.
To this day the Gauss transformation, on which metrical theory of regular continued fraction is based, has fascinated researchers from various branches of mathematics and science with many applications in computer science, cosmology and chaos theory [5]. In the last century, mathematicians broke new ground in this area. Apart from the regular continued fraction expansion, very many other continued fraction expansions were studied [13,15].
By such a development, generalizations of these problems for non-regular continued fractions are also called as the Gauss-Kuzmin problem and the Gauss-Kuzmin-Lévy problem [9,11,16,17,18].

θ-expansions
For a fixed θ ∈ (0, 1), we start with a brief review of continued fraction expansion with respect to θ, analogous to the regular continued fraction expansion which corresponds to the case θ = 1.
If r 1 = a 1 θ, then we write i.e., If a 1 θ < r 1 , define r 2 by where 0 < 1/r 2 < θ. So, r 2 > 1/θ ≥ θ and let In this way, either the process terminates after a finite number of steps or it continues indefinitely. Following standard notation, in the first case we write x := [a 0 θ; a 1 θ, . . . , a n θ] (1.5) and we call this the finite continued fraction expansion of x with respect to θ (terminating at the n-stage). In the second case, we write x := [a 0 θ; a 1 θ, a 2 θ, . . .] (1.6) and we call this the infinite continued fraction expansion of x with respect to θ. When 0 < x < θ, we have a 0 = 0 and instead of writing we simply write x := [a 1 θ, a 2 θ, . . .] (1.8) which is the same in the usual notation Such a n 's are also called incomplete quotients (or continued fraction digits) of x with respect to the expansion in (1.9).
In general, the θ-expansion of a number x > 0 is where a 0 = ⌊x/θ⌋. For x ∈ [0, θ], the θ-continued fraction expansion of x in (1.9) leads to an analogous transformation of Gauss map τ in (1.2). A natural question is whether this new transformation admits an absolutely continuous invariant probability like the Gauss measure in the case θ = 1. Until now, the invariant measure was identified only in the particular case θ 2 = 1/m, m a positive integer [4].
Motivated by this argument, since the invariant measure is a crucial tool in our approach, in the sequel we will consider only the case θ 2 = 1/m with m a positive integer. Then [a 1 θ, a 2 θ, a 3 θ, . . .] is the θ-expansion of any x ∈ [0, θ] if and only if the following conditions hold: (i) a n ≥ m for any m ∈ N + ; (ii) in case when x has a finite expansion, i.e., x = [a 1 θ, a 2 θ, a 3 θ, . . . , a n θ], then a n ≥ m + 1.
This continued fraction is treated as the following dynamical system. (1.12) By using T θ , the sequence (a n ) n∈N + in (1.9) is obtained as follows: a n = a n (x) = a 1 T n−1 with T 0 θ (x) = x and (1.14) In this way, T θ gives the algorithm of θ-expansion.

Known results and applications
In this subsection we recall known results and their applications for θexpansions.
(1. 15) In what follows the stated identities hold for all n in case x has an infinite θ-expansion and they hold for n ≤ k in case x has a finite θ-expansion terminating at the k-th stage.

Application to ergodic theory
Similarly to classical results on regular continued fractions, using the ergodicity of T θ and Birkhoff's ergodic theorem [6], a number of results were obtained. For q n in (1.17), its asymptotic growth rate β is defined as This is a Lévy result and Chakraborty and Rao [4] obtained that β is a finite number They also give a Khintchin result, i.e., the asymptotic value of the arithmetic mean of a 1 , a 2 , . . . , a n where a 1 and a n are given in (1.14) and (1.13). We have lim n→∞ a 1 + a 2 + . . . + a n n = ∞. (1.25) It should be stressed that the ergodic theorem does not yield any information on the convergence rate in the Gauss problem that amounts to the asymptotic behavior of µ(T −n θ ) as n → ∞, where µ is an arbitrary probability measure on It is only very recently that there has been any investigation of the metrical properties of the θ-expansions. Thus, the results obtained in this paper allow to a solution of a Gauss-Kuzmin type problem. We may emphasize that, to our knowledge, Theorem 6.1 is the first Gauss-Kuzmin result proved for θ-expansions. Our solution presented here is based on the ergodic behavior of a certain random system with complete connections.
The paper is organized as follows. In Section 2, we show the probability structure of (a n ) n∈N + under the Lebesgue measure by using the Brodén-Borel-Lévy formula. In Section 3, we consider the so-called natural ex- [14]. In Section 4, we derive its Perron-Frobenius operator under different probability measures on ([0, θ], B [0,θ] ). Especially, we derive the asymptotic behavior for the Perron-Frobenius operator of ( In Section 5, we study the ergodicity of the associated random system with complete connections (RSCC for short). In Section 6, we solve a variant of Gauss-Kuzmin problem for θ-expansions. By using the ergodic behavior of the RSCC introduced in Section 5, we determine the limit of the sequence ( µ(T n θ < x) ) n≥1 of distributions as n → ∞.

Prerequisites
Roughly speaking, the metrical theory of continued fraction expansions is about the sequence (a n ) n∈N + of incomplete quotients and related sequences [8]. As remarked earlier in the introduction we will adopt a similar strategy to that used for regular continued fractions. We begin with a Brodén-Borel-Lévy formula for θ-expansions. Then some consequences of it to be used in the sequel are also derived.
For any n ∈ N + and i (n) = (i 1 , . . . , i n ) ∈ N n m , define the fundamental interval associated with i (n) by where I(i (0) ) = [0, θ]. For example, for any i ∈ N m we have We will write I(a 1 , . . . , a n ) = I a (n) , n ∈ N + . If n ≥ 1 and i n ∈ N m , then we have I(a 1 , . . . , a n ) = I i (n) .
From the definition of T θ and (1.19), we have where u(a (n) ) and v(a (n) ) are defined as and v a (n) := where p n := p n (x) and q n := q n (x) are defined in (1.16) .
. For any n ∈ N + , the conditional probability λ θ (T n θ < x|a 1 , . . . , a n ) is given as follows: where s n is defined in (2.8) and a 1 , . . . , a n are as in (1.13) and (1.14).
The Brodén-Borel-Lévy formula allows us to determine the probability structure of incomplete quotients (a n ) n∈N + under λ θ . Proposition 2.2. For any i ∈ N m and n ∈ N + , we have where (s n ) n∈N + is defined in (2.8), and . (2.14) 2 is the starting point of an approach to the metrical theory of θ-expansions via dependence with complete connections (see [7], Section 5.2) , λ θ ) with the following transition mechanism: from state s the possible transitions are to any state 1/(s + iθ) with corresponding transition probability P i (s), i ∈ N m .

An infinite-order-chain representation
In this section we introduce the natural extension T θ of T θ in (1.11) and define extended random variables according to Chap.1.3.3 of [8]. Then we give an infinite-order-chain representation of the sequence of the incomplete quotients for θ-expansions. Definition 3.1. The natural extension [14] of ([0, θ],
(ii) From Proposition 3.2, the doubly infinite sequence (a l (x, y)) l∈Z is strictly stationary (i.e., its distribution is invariant under a shift of the indices) under γ θ .
The following theorem will play a key role in the sequel. Proof. Let I n denote the fundamental interval I(a 0 , a −1 , . . . , a −n ) for n ∈ N.
We have (3.16) and x(y n θ + 1) (xy n + 1)θ , the proof is completed. The probability structure of (a l ) l∈Z under γ θ is given as follows. where a = [a 0 θ, a −1 θ, . . .] and the functions P i , i ∈ N m , are defined by (2.14).
Proof. Let I n be as in the proof of Theorem 3.4. We have We have for some y n ∈ I n . From (3.18), the proof is completed.
Remark 3.6. The strict stationarity of (a l ) l∈Z , under γ θ implies that for any i ∈ N m and l ∈ Z, where a = [a l θ, a l−1 θ, . . .]. The last equation emphasizes that (a l ) l∈Z is an infinite-order-chain in the theory of dependence with complete connections (see [7], Section 5.5).
The Perron-Frobenius operator of T θ under µ is defined as the bounded linear operator U µ which takes the Banach space L 1 µ into itself and satisfies the equation where P i , i ≥ m, is as in (2.14) and u i , i ≥ m, is defined by Proof. (i) Let T θ,i denote the restriction of T θ to the subinterval I(i) : For any i ≥ m, by the change of variable x = (T θ,i ) −1 (y) = u i (y), we successively obtain (4.7) Now, (4.2) follows from (4.6) and (4.7).
(ii) We will use mathematical induction. For n = 0, the equation (4.5) holds by definitions of f and h. Assume that (4.5) holds for some n ∈ N. Then By the very definition of the Perron-Frobenius operator U we have Therefore,  Proof. The transition operator of (s n,a ) n∈N + takes f ∈ B([0, θ]) to the function defined by (4.11) where E a stands for the mean-value operator with respect to the probability measure γ θ,a , whatever a ∈ [0, θ].
A similar reasoning is valid for the case of the Markov chain (s l ) l∈Z .

Ergodicity of the associated RSCC
The facts presented in the previous sections lead us to a certain random system with complete connections associated with the θ-expansion. To study the ergodicity of this RSCC it becomes necessary to recall some definitions and results from [7]. According to the general theory we have the following statement. (ii) u : W × X → W is a (W ⊗ X , W)-measurable map; (iii) P is a transition probability function from (W, W) to (X, X ).
For any n ∈ N + , consider the maps u (n) : W × X n → W , defined by where x (n) = (x 1 , . . . , x n ) ∈ X n . We will simply write wx (n) for u (n) (w, x (n) ). For every w ∈ W , r ∈ N + and A ∈ X r , define where χ A is the indicator function of the set A. Obviously, for n ∈ N + fixed, P r is a transition probability function from (W, W) to (X r , X r ).
By virtue of the existence theorem ( where B w = {x ∈ X : wx ∈ B}, w ∈ W , B ∈ W. The iterates of the operator U are given by It follows that the n-step transition probability function is given by Hence the transition operator associated with the Markov chain with state space (W, W) and transition probability function Q is defined by Its iterates are given by where Q n is the n-step transition probability function. Putting for all n ∈ N + , w ∈ W and B ∈ W, it is clear that Q n is a transition probability function on (W, W). Let U n be the Markov operator associated with Q n . Let (W, d) be a metric space and let L(W ) denote the Banach space of all complex-valued Lipschitz continuous functions on W with the following norm: f into L(W ) boundedly with respect to · L and there exist k ∈ N + , r ∈ [0, 1) and R < ∞ such that Alternatively, the Markov chain itself is said to be a Doeblin-Fortet chain.
The definition below isolates a class of RSCCs, called RSCCs with contraction, for which the associated Markov chains are Doeblin-Fortet chains.
Definition 5.4. An RSCC {(W, W) , (X, X ) , u, P } is said to be with contraction if and only if (W, d) is a separable metric space, r 1 < ∞, R 1 < ∞ and there exists j ∈ N + such that r j < 1. Here and where P j is a transition probability function from (W, W) to X j , X j defined in (5.1).
We shall also need the following results. This immediately implies the validity of Moreover, Q ∞ (·, B) ∈ E(1) for any B ∈ W and for any w ∈ W , Q ∞ (w, ·) is a stationary probability for the chain. (Here E(1) is the set of the eigenvalues of modulus 1 of the operator U ).
A criterion of regularity for a compact Markov chain is expressed in Theorem 5.9 in terms of the supports σ n (w) of the transition probability functions Q n (w, ·), n ∈ N + . Theorem 5.9. A compact Markov chain is regular with respect to L(W ) if and only if there exists a point w * ∈ W such that lim n→∞ d (σ n (w), w * ) = 0, w ∈ W. (5.12) The application of this criterion is facilitated by the inter-relationship among the sets σ n (w), n ∈ N + , which is given in the next lemma. Then Q n (·, ·) will denote the n-step transition probability function of the same Markov chain.
Proposition 5.11. RSCC (5.14) is regular with respect to L([0, θ]). Moreover there exist a stationary probability measure Q ∞ = γ θ and two positive constants q < 1 and K such that where for any s ∈ We have w n ∈ [0, θ] and letting n → ∞ in (5.18) we get .
Clearly, w n+1 ∈ σ 1 (w n ) and Lemma 5.10 and an induction argument show that w n ∈ σ n (s), n ∈ N + . Thus where d stands for the Euclidian distance on the line. Now, the regularity of U with respect to L([0, θ]) follows from Theorem 5.9. From (5.10) and Theorem 5.8 there exist a stationary probability measure Q ∞ and two constants q < 1 and K such that U n f − U ∞ f L ≤ Kq n f L , n ∈ N + , f ∈ L([0, θ]) (5.19) where U n f is as in (5.17) and Here Q ∞ = γ θ is the invariant probability measure of the transformation T θ in (1.11), i.e., Q ∞ has the density ρ θ = 1/(x + mθ), x ∈ [0, θ], with the normalizing factor 1/ log(1 + θ 2 ).
Remark 5.12. Another way to put this is that ρ θ is the eigenfunction of eigenvalue 1 of the Perron-Frobenius operator U .