A Gauss-Kuzmin Theorem and Related Questions for θ-Expansions

Abstract

Using the natural extension for θ-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 previous fraction expansion.

1. 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 [2] have initiated a systematic study of the continued fraction expansion of a number in terms of an irrational . This new expansion of positive reals, different from the regular continued fraction expansion, is called -expansion.

The purpose of this paper is to solve a Gauss-Kuzmin problem for -expansions. In order to solve the problem, we apply the theory of random systems with complete connections extensively studied by Iosifescu and Grigorescu [3]. 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.

1.1. 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 can be written as the infinite regular continued fraction where [4]. Such integers are called incomplete quotients (or continued fraction digits) of . The metrical theory of continued fraction expansions started on October 25, 1800, with a note by Gauss in his mathematical diary [5]. Define the regular continued fraction (or Gauss) transformation on the unit interval by where denotes the floor (or entire) function. With respect to the asymptotic behavior of iterations (-times) of , Gauss wrote (in modern notation) that where denotes the Lebesgue measure on . In 1812, Gauss asked Laplace [5] to estimate the th error term defined by This has been called Gauss’ Problem. It received first solution more than a century later, when Kuzmin [6] showed in 1928 that as , uniformly in with some (unspecified) . This has been called the Gauss-Kuzmin theorem or the Kuzmin theorem.

One year later, using a different method, Lévy [7] improved Kuzmin’s result by showing that for , , with . 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 .

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 [8]. In the last century, mathematicians broke new ground in this area. Apart from the regular continued fraction expansion, many other continued fraction expansions were studied [9, 10].

By such a development, generalizations of these problems for nonregular continued fractions are also called the Gauss-Kuzmin problem and the Gauss-Kuzmin-Lévy problem [11–15].

1.2. -Expansions

For a fixed , we start with a brief review of continued fraction expansion with respect to , analogous to the regular continued fraction expansion which corresponds to the case .

For let If equals , we write If not, define by where . Then and let If , then we write that is, If , define by where . So, 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 and we call this the finite continued fraction expansion of with respect to (terminating at the -stage). In the second case, we write and we call this the infinite continued fraction expansion of with respect to .

When , we have and instead of writing we simply write which is the same in the usual notation Such ’s are also called incomplete quotients (or continued fraction digits) of with respect to the expansion in (17).

In general, the -expansion of a number is where .

For , the -continued fraction expansion of in (17) leads to an analogous transformation of Gauss map in (2). A natural question is whether this new transformation admits an absolutely continuous invariant probability like the Gauss measure in the case . Until now, the invariant measure was identified only in the particular case , a positive integer [2].

Motivated by this argument, since the invariant measure is a crucial tool in our approach, in the sequel we will consider only the case with being a positive integer. Then is the -expansion of any if and only if the following conditions hold:(i) for any ;(ii)in case when has a finite expansion, that is, , then .

This continued fraction is treated as the following dynamical system.

Definition 1. Let and such that . (i)The measure-theoretical dynamical system is defined as follows: denotes the -algebra of all Borel subsets of , and is the transformation (ii)In addition to (i), we write as with the following probability measure on :

By using , the sequence in (17) is obtained as follows: with and

In this way, gives the algorithm of -expansion.

Proposition 2. Let be as in Definition 1(ii).(i) is ergodic.(ii) The measure is invariant under ; that is, for any .

Proof. See Section  8 in [2].

By Proposition 2(ii), is a “dynamical system” in the sense of Definition   in [16].

1.3. Known Results and Applications

In this subsection, we recall known results and their applications for -expansions.

1.3.1. Known Results

Let . Define the th order convergent of by truncating the -expansion in (17). Thus, Chakraborty and Rao proved in [2] that

In what follows the stated identities hold for all in case has an infinite -expansion and they hold for in case has a finite -expansion terminating at the th stage.

To this end, define real functions and , for , by with , , , and . By induction, we have By using (24) and (25), we can verify that By taking in (27), we obtain . Using (26) and (27) we obtain By applying to (28), we can verify that From (25), we have that , . Further, also from (25) and by induction we have that Finally, (23) follows from (29) and (30).

1.3.2. Application to Ergodic Theory

Similar to classical results on regular continued fractions, using the ergodicity of and Birkhoff’s ergodic theorem [17], a number of results were obtained.

For in (25), its asymptotic growth rate is defined as This is a Lévy result and Chakraborty and Rao [2] obtained that is a finite number They also give a Khintchine result, that is, the asymptotic value of the arithmetic mean of , where and are given in (22) and (21). We have

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 as , where is an arbitrary probability measure on .

It is only very recent that there has been any investigation of the metrical properties of the -expansions. Thus, the results obtained in this paper allow a solution of a Gauss-Kuzmin type problem. We may emphasize that, to our knowledge, Theorem 29 is the first Gauss-Kuzmin result proved for -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 under the Lebesgue measure by using the Brodén-Borel-Lévy formula. In Section 3, we consider the so-called natural extension of [18]. In Section 4, we derive its Perron-Frobenius operator under different probability measures on . In particular, 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 (RSCCs for short). In Section 6, we solve a variant of Gauss-Kuzmin problem for -expansions. By using the ergodic behavior of the RSCC introduced in Section 5, we determine the limit of the sequence of distributions as .

2. Prerequisites

Roughly speaking, the metrical theory of continued fraction expansions is about the sequence of incomplete quotients and related sequences [4]. 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 -expansions. Then, some consequences of it to be used in the sequel are also derived.

In this section, let us fix , , .

For , consider , , as in (21) and (22). Putting , , the incomplete quotients , , take positive integer values in .

For any and , define the fundamental interval associated with by where . For example, for any , we have

We will write , . If and , then we have

From the definition of and (27), we have where and are defined as where and are defined in (24) and (25), respectively.

Let denote a Lebesgue measure on . Using (26) we get

To derive the so-called Brodén-Borel-Lévy formula [3, 4] for -expansions, let us define by From (25), for . Hence, for .

Proposition 3 (Brodén-Borel-Lévy-type formula). Let denote the Lebesgue measure on . For any , the conditional probability is given as follows: where is defined in (40) and are as in (21) and (22).

Proof. By definition, we have for any and . Using (27) and (37), we get From this and (39) it follows that for any and .

The Brodén-Borel-Lévy formula allows us to determine the probability structure of incomplete quotients under .

Proposition 4. For any and , one has where is defined in (40), and

Proof. From (35), the case holds. For and , we have . Using (42), we obtain

Remark 5. (i) It is easy to check that
(ii) Proposition 4 is the starting point of an approach to the metrical theory of -expansions via dependence with complete connections (see [3, Section 5.2]).

Corollary 6. The sequence, with is an -Markov chain on with the following transition mechanism: from state the possible transitions are to any state with corresponding transition probability , .

3. An Infinite-Order-Chain Representation

In this section we introduce the natural extension of in (19) and define extended random variables according to Chapter of [4]. Then we give an infinite-order-chain representation of the sequence of the incomplete quotients for -expansions.

3.1. Natural Extension

Let be as in Definition 1(i). Let denote the -algebra of Borel sets in .

Definition 7. The natural extension [18] of is where the transformation of is defined by

This is a one-to-one transformation of with the inverse Iterations of (50) and (51) are given as follows for each : For in (20), define its extended measure on as Then for any .

Proposition 8. The measure is preserved by .

Proof. We show that for any . Since is invertible on , the last equation is equivalent to Recall fundamental interval in (34). Since the collection of Cartesian products , , , , generates the -algebra , it is enough to show that for any , , . It follows from (54) and Proposition 2(ii) that (56) holds for and . If , then it is easy to see that where equals for . Also, if and , then A simple computation yields and then that is, (56) holds.

3.2. Extended Random Variables

With respect to in (50), define extended incomplete quotients , , by with

Remark 9. (i) Since is invertible, it follows that in (61) is also well-defined for . By (52), we have for any and .
(ii) From Proposition 8, the doubly infinite sequence is strictly stationary (i.e., its distribution is invariant under a shift of the indices) under .

The following theorem will play a key role in the sequel.

Theorem 10. For any , one has where (= the -expansion with incomplete quotients ).

Proof. Let denote the fundamental interval for . We have for some . Since the proof is completed.

The probability structure of under is given as follows.

Corollary 11. For any , one has where and the functions , , are defined by (47).

Proof. Let be as in the proof of Theorem 10. We have We have Now for some . From (66), the proof is completed.

Remark 12. The strict stationarity of under implies that for any and , where . The last equation emphasizes that is an infinite-order-chain representation in the theory of dependence with complete connections (see [3, Section 5.5]).

Define extended random variables as , . Clearly, , . It follows from Proposition 8 and Corollary 11 that is a strictly stationary -valued Markov process on with the following transition mechanism: from state the possible transitions are to any state with corresponding transition probability , . Clearly, for any , we have

Motivated by Theorem 10, we will consider the one-parameter family of (conditional) probability measures on defined by their distribution functions Note that .

For any , put

Remark 13. It follows from the properties just described of the process that the sequence is an -valued Markov chain on which starts at and has the following transition mechanism: from state the possible transitions are to any state with corresponding transition probability , .

4. An Operatorial Treatment

Let be as in Definition 1(ii). The Gauss-Kuzmin problem for the transformation can be approached in terms of the associated Perron-Frobenius operator.

Let be a probability measure on such that whenever for any . For example, this condition is satisfied if is -preserving; that is, . Let The Perron-Frobenius operator of under is defined as the bounded linear operator which takes the Banach space into itself and satisfies the equation

Proposition 14. (i) The Perron-Frobenius operator of under the invariant probability measure is given a.e. in by the equation where , , is as in (47) and , , is defined by
(ii) Let be a probability measure on such that is absolutely continuous with respect to the Lebesgue measure and let a.e. in . For any and , one has where , .

Proof. (i) Let denote the restriction of to the subinterval , , that is, Let and for any . Since and is a null set when , we have For any , by the change of variable , we successively obtain Now, (77) follows from (81) and (82).
(ii) We will use mathematical induction. For , (80) holds by definitions of and . Assume that (80) holds for some . Then By the very definition of the Perron-Frobenius operator , we have Therefore, which ends the proof.

Let denote the collection of all bounded measurable functions . A different interpretation is available for the operator restricted to , which is a Banach space under the supremum norm.

Proposition 15. The operator is the transition operator of both the Markov chain on , for any , and the Markov chain on .

Proof. The transition operator of takes to the function defined by where stands for the mean-value operator with respect to the probability measure , whatever .

A similar reasoning is valid for the case of the Markov chain .

Remark 16. In hypothesis of Proposition 14(ii) it follows that for any and , where , . The last equation shows that the asymptotic behavior of as is given by the asymptotic behavior of the th power of the Perron-Frobenius on or on smaller Banach spaces.

5. 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 -expansion. To study the ergodicity of this RSCC, it becomes necessary to recall some definitions and results from [3].

According to the general theory, we have the following statement.

Definition 17. A homogeneous RSCC is a quadruple , where (i) and are arbitrary measurable spaces;(ii) is a -measurable map;(iii) is a transition probability function from to .

For any , consider the maps , defined by where . We will simply write for . For every , and , define where is the indicator function of the set . Obviously, for fixed, is a transition probability function from to .

By virtue of the existence theorem ([3, Theorem ]), for a given RSCC , there exists an associated Markov chain with the transition operator defined by where is the Banach space of all bounded -measurable complex-valued functions defined on . Moreover, the transition probability function of the associated Markov chain is where , , . The iterates of the operator are given by It follows that the -step transition probability function is given by where . Hence, the transition operator associated with the Markov chain with state space and transition probability function is defined by Its iterates are given by where is the -step transition probability function.

Putting for all , and , it is clear that is a transition probability function on . Let be the Markov operator associated with .

Let be a metric space and let denote the Banach space of all complex-valued Lipschitz continuous functions on with the following norm: where

Definition 18. (i) The operator is said to be orderly with respect to if and only if there exists a bounded linear operator on such that
(ii) The operator is said to be aperiodic with respect to if and only if there exists a bounded linear operator on such that
(iii) The operator is said to be ergodic with respect to if and only if it is orderly and the range is one-dimensional.
(iv) The operator is regular with respect to if and only if it is ergodic and aperiodic.

Definition 19. The transition operator of a Markov chain with state space is said to be a Doeblin-Fortet operator if and only if takes into boundedly with respect to and there exist , and 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 20. An RSCC is said to be with contraction if and only if is a separable metric space, , and there exists such that . Here where is a transition probability function from to defined in (89).

We will also need the following results.

Theorem 21. The Markov chain associated with an RSCC with contraction is a Doeblin-Fortet chain.

Lemma 22. Assume that the Markov operator is aperiodic with respect to . Put . Then one has , , , . Moreover, there exist positive constants and such that

This immediately implies the validity of

Definition 23. A Markov chain is said to be compact if and only if its state space is a compact metric space and its transition operator is a Doeblin-Fortet operator.

Theorem 24. A compact Markov chain is orderly with respect to and there exists a transition probability function on such that Moreover, , for any , and, for any , is a stationary probability for the chain. (Here is the set of the eigenvalues of modulus of the operator .)

A criterion of regularity for a compact Markov chain is expressed in Theorem 25 in terms of the supports of the transition probability functions , .

Theorem 25. A compact Markov chain is regular with respect to if and only if there exists a point such that

The application of this criterion is facilitated by the interrelationship among the sets , , which is given in the next lemma.

Lemma 26. For all and , one has where the overline means topological closure in .

Now, we are able to study the following RSCC: where , is given in (78) and the function given in (47) defines a transition probability from to . Here , and denotes the power set of .

Whatever the Markov chain associated with the RSCC (89) has the transition operator , with the transition probability function Then will denote the -step transition probability function of the same Markov chain.

Proposition 27. RSCC (108) is regular with respect to . Moreover, there exist a stationary probability measure and two positive constants and such that where

Proof. Since for any and , it follows that Hence, the requirements of Definition 20 of an RSCC with contraction are met with . By Theorem 21 it follows that the Markov chain associated with this RSCC with contraction is a Doeblin-Fortet chain and its transition operator is a Doeblin-Fortet operator. It remains to prove the regularity of with respect to . For this we have to prove the existence of a point such that , for any , where is the support of measure , .
Let be an arbitrarily fixed number and define We have , and letting in (114), we get Clearly, and Lemma 26 and an induction argument show that , . Thus, where stands for the Euclidian distance on the line. Now, the regularity of with respect to follows from Theorem 25.
From (104) and Theorem 24, there exist a stationary probability measure and two constants and such that where is as in (111) and Here is the invariant probability measure of the transformation in (19); that is, has the density , , with the normalizing factor .

Remark 28. Another way to put this is that is the eigenfunction of eigenvalue 1 of the Perron-Frobenius operator .

6. A Gauss-Kuzmin-Type Theorem

Now, we may determine the limit of the sequence as and give the rate of this convergence.

Theorem 29 (Gauss-Kuzmin-type theorem for ). Let be as in Definition 1(i). (i)For a probability measure on , let assumption (A) as follows:(A) is nonatomic and has a Riemann-integrable density. Then, for any probability measure which satisfies (A), the following holds: (ii)In addition to assumption of in (i), if the density of is Lipschitz continuous, then there exist two positive constants and such that for any and , the following holds:  where with . As a consequence, the th error term of the Gauss-Kuzmin problem is obtained as follows:

Proof. Let be as in (19). By Proposition 14(ii), we have where , for . If , by (118) we have
Taking into account (117), there exist two constants and such that Furthermore, consider the Banach space of all real-valued continuous functions on with the norm . Since is a dense subspace of , we have Therefore, (125) is valid for a measurable function which is -almost surely continuous, that is, for a Riemann-integrable function . Thus, we have Hence, (119) is proved.

Remark 30. Since the Lebesgue measure satisfies assumptions in both (i) and (ii) of Theorem 29, (119) and (120) hold for the case . Hence, Theorem 29 gives the solution of the Gauss-Kuzmin problem for the pair instead of in (3).

Remark 31. Until now, the estimate of the convergence rate remains an open question. To obtain a better estimate of the convergence rate involved, we may use a Wirsing-type approach as in [19–21].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous referee.