A Gauss-Kuzmin Theorem for Continued Fractions Associated with Nonpositive Integer Powers of an Integer m ≥ 2

We consider a family {τ m : m ≥ 2} of interval maps which are generalizations of the Gauss transformation. For the continued fraction expansion arising from τ m, we solve a Gauss-Kuzmin-type problem.


Introduction
Chan considered some continued fraction expansions related to random Fibonacci-type sequences [1,2]. In [1], he studied the continued fraction expansions of a real number in the closed interval [0, 1] whose digits are differences of consecutive nonpositive integer powers of 2 and solved the corresponding Gauss-Kuzmin-Lévy theorem. In fact, Chan has studied the transformation related to this new continued fraction expansion and the asymptotic behaviour of its distribution function. Giving a solution to the Gauss-Kuzmin-Lévy problem, he showed in [1, Theorem 1.1] that the convergence rate involved is O( ) as → ∞ with 0 < < 1.
The purpose of this paper is to prove a Gauss-Kuzmintype problem for the continued fraction expansions of real numbers in [0, 1] whose digits are differences of consecutive nonpositive integer powers of an integer ≥ 2. In this section, we show our motivation and main theorems.
Roughly speaking, the metrical theory (or, as called by Khintchine, the measure theory) of continued fraction expansions is about properties of the sequence ( ) ∈N + . It started on October 25, 1800, with a note by Gauss in his mathematical diary (entry 113) [3]. Define the regular continued fraction transformation on the closed interval := [0, 1] by where ⌊⋅⌋ denotes the floor (or entire) function. In modern notation, Gauss wrote that "for very simple argument" we have where denotes the Lebesgue measure on and is the th iterate of . Nobody knows how Gauss found (3), and his achievement is even more remarkable if we realize that modern probability theory and ergodic theory had started almost a century later. In general, finding the invariant measure is a difficult task.
Twelve years later, in a letter dated January 30, 1812, Gauss wrote to Laplace that he did not succeed in solving satisfactorily "a curious problem" and that his efforts "were 2 The Scientific World Journal unfruitful. " In modern notation, this problem is to estimate the error This has been called Gauss' Problem. It received a first solution more than a century later, when Kuzmin [4] showed in 1928 that as → ∞, uniformly in with some (unspecified) 0 < < 1. This has been called the Gauss-Kuzmin theorem or the Kuzmin theorem.
By such a development, generalizations of these problems for nonregular continued fractions are also called the Gauss-Kuzmin problems.

Chan's Continued Fraction Expansions.
In this paper, we consider a generalization of the Gauss transformation and prove an analogous result. Especially, we will solve its Gauss-Kuzmin problem in Theorem 3.
This transformation was studied in detail by Chan in [2] and Lascu in [6].
Fix an integer ≥ 2. In [2], Chan shows that any ∈ [0, 1) can be written as the form where 's are nonnegative integers. Such 's are also called incomplete quotients (or continued fraction digits) of with respect to the expansion in (7) in this paper. This continued fraction is treated as the following dynamical systems.
(ii) In addition to (i), one writes ( , B , , ) as ( , B , ) with the following probability measure on ( , B ): where Define the quantized index map : → N by By definition, ( − ) = ⌊ ⌋. By using and , the sequence ( ) ∈N + in (7) is obtained as follows: with 0 ( ) = . In this way, gives the algorithm of Chan's continued fraction expansion (7).

Known Results and
Applications. For Chan's continued fraction expansions, we show known results and their applications in this subsection.
In [8], Chan proved a Gauss-Kuzmin-Lévy theorem for the transformation 2 . He showed that the convergence rate of the th distribution function of 2 to its limit is O( ) as → ∞ with ≤ 0.880555 uniformly in .
In [9,10], Sebe investigated the Perron-Frobenius operator of 2 by replacing a probability measure of the measurable space ( , B ). Especially, Sebe studied the Perron-Frobenius operator of ( , B , 2 , 2 ); that is, is a unique operator on 1 ( , 2 ) satisfying The Scientific World Journal 3 The asymptotic behavior of was shown by using well-known general results [11,12]. By a Wirsing-type approach [13], Sebe obtained a better estimate of the convergence rate involved [9]. In fact, its upper and lower bounds of the convergence rate were obtained as O( ) and O(V ), respectively, when → ∞, with < 0.209364308 and V > 0.206968896 ([9, Theorem 4.3]). They provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem.
Furthermore, by restricting the Perron-Frobenius operator to the Banach space of functions : → C of bounded variation, Iosifescu and Sebe [14] proved that the exact optimal convergence rate of Here is the inverse of the golden ratio; that is, we have For ≥ 0, define the sequence ( , ) ∈N recursively by where is the probability measure on ( , B ) defined as the following distribution function: For in (13), let denote its restriction on ∞ ( ) ⊂ 1 ( , 2 ). From [12, Proposition 2.1.10], we see that is the transition operator of the Markov chain ( , ) ∈N + on ( , B , ) for any ≥ 0.

Known Results for
in Definition 1(ii), recall the main results in [6,15].
In [6], Lascu proved a Gauss-Kuzmin theorem for the transformation . In order to solve the problem, he applied the theory of random systems with complete connections (RSCC) by Iosifescu and Grigorescu [11]. We remind that a random system with complete connections is a quadruple where : × N → , and is the transition probability function from ( , B ) to (N, P(N)) given by Also, the associated Markov operator of RSCC (16) is denoted by and has the transition probability function where ( , ) = { ∈ N : , ( ) ∈ }. Using the asymptotic and ergodic properties of operators associated with RSCC (16), that is, the ergodicity of RSCC, he obtained a convergence rate result for the Gauss-Kuzmintype problem.
By a Wirsing-type approach [13] to the Perron-Frobenius operator of the associated transformation under its invariant measure, Sebe [15] studied the optimality of the convergence rate. Actually, Sebe obtained upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem. In the case = 3, the upper and lower bounds of the convergence rate were obtained as ( 3 ) and (V 3 ), respectively, when → ∞, with V 3 > 0.262765464 and 3 < 0.264687208.

Application to the Asymptotic Growth Rate of a Fibonacci-Type Sequence.
We explain an application of to a Fibonacci-type sequence here. As it is known, the Fibonacci sequence ( ) is recursively defined as follows: Equivalently, ( ) is also defined by Binet's formula = ( +1 − −( +1) )/ √ 5 for ≥ 0 where := (1 + √ 5)/2 is the golden ratio. By this formula, the asymptotic growth rate of ( ) is obtained as follows: A random Fibonacci sequence ( ) is defined as (with fixed 1 and 2 ) where ( ) and ( ) are random coefficients. For such ( ), the quest for its asymptotic growth rate is more difficult. We show two examples of random Fibonacci sequences as follows.
(i) Define the random Fibonacci sequence ( ) as where the signs in (23) are chosen independently and with equal probabilities. Recently, Viswanath [21] has proved that its asymptotic growth rate is given as with probability 1.

4
The Scientific World Journal (ii) Fix an integer ≥ 2. Define the random Fibonacci sequence ( ) as where 's are as in (7). By using the ergodicity of ( , B , , ) (Proposition 2(i)), Chan proved that its asymptotic growth rate is given as follows [2]: where is as in (10).

A Khintchine-Type Result and Entropy.
In probabilistic number theory, statistical limit theorems are established in problems involving "almost independent" random variables. The nonnegative integers , ∈ N + , define random variables on the measure space ( , B , P), where P is a probability measure on . Continued fraction expansions of almost all irrational numbers are not periodic. Nevertheless, we readily reproduce another famous probabilistic result. It is the asymptotic value of the geometric mean of 1 , 2 , . . . , ; that is, where 's are given in (12). This is a Khintchine-type result and we obtain for almost all real numbers = [ 1 ( ), 2 ( ), 3 ( ), . . . ] ∈ (0, 1). As it can be seen, is a constant independent of the value of .
As it is well known, entropy is an important concept of information in physics, chemistry, and information theory [22]. The connection between entropy and the transmission of information was first studied by Shannon in [23]. The entropy can be seen as a measure of randomness of the system or the average information acquired under a single application of the underlying map. Entropy also plays an important role in ergodic theory. Thus in 1958 Kolmogorov [24] imported Shannon's probabilistic notion of entropy into the theory of dynamical systems and showed how entropy can be used to tell whether two dynamical systems are nonconjugate. Like Birkhoff 's ergodic theorem [22] the entropy is a fundamental result in ergodic theory. For a measure preserving transformation, its entropy is often defined by using partitions, but in 1964 Rohlin [25] showed that the entropy of a -measure preserving operator : [ , ] → [ , ] is given by the beautiful formula From Rohlin's formula it follows that the entropy of the operator in (8) on the unit interval with respect to the measure in (9) is given by where 1 , , and are given in (12), (26), and (27), respectively.

Main Theorem.
We show our main theorems in this subsection. Fix an integer ≥ 2. Let be as in (10) and let ( , B ) be as in Section 1.2. If has the expansion in (7) and is as in (8), then the question about the asymptotic distribution of appears. If we know this, then the corresponding probability that +1 = is simply written as prob( −( +1) < < − ). We will show that the event ≤ has the following asymptotic probability: This result allows us to say that the probability density function is invariant under : if a random variable in the unit interval has the density , and then so does . The reason for this invariance is that, for 0 ≤ < + ℎ ≤ 1, lies between and + ℎ if and only if there exists ≥ 1, so that lies between 1/( + + ℎ) and 1/( + ). Thus Taking the limit as ℎ → ∞ gives that, for an arbitrary probability density function for , the corresponding density for is given a.e. in by the equation The Scientific World Journal 5 Clearly, the operator : 1 → 1 admits the density function as an eigenfunction corresponding to the eigenvalue 1; that is, = . Here 1 denotes the Banach space of all complex functions : → C for which ∫ | | < ∞.
The only eigenvalue of modulus 1 of is 1 and this eigenvalue is simple.
From another perspective, the operator is an ergodic operator on the unit interval [2], is the density of the invariant measure, and is called transfer operator for [6]. The transfer operator has the same analytical expression as the Perron-Frobenius operator of under the Lebesgue measure [6].
Our main result is the following theorem.
In turn, -mixing implies lots of limit theorems in both classical and functional versions. To form an idea of the results to be expected it is sufficient to look at the corresponding results for the regular continued fraction expansions [12].
(iii) In (37) we emphasized the probabilistic nature of Gauss' result. Khintchine [26] and Doeblin [27] found new probabilistic results on the regular continued fraction transformation. These types of results were established also for the transformation [2,6]. These results establish, among other properties, that the map is ergodic (Proposition 2 (i)). Kuzmin's theorem may then be rephrased by saying that the convergence encountered in the mixing process (the "approach to equilibrium") is in fact exponential. If we define the linear operator Π 1 by then there exists 0 < < 1 such that The norm ‖ ⋅ ‖ V is defined by ‖ ‖ V = ‖ ‖ + "total variation of [12].
Problem 5. (i) Solve the Gauss-Kuzmin-Lévy problem of for ≥ 3. For example, study the optimality of the convergence rate. Use the same strategy as in [14].
(ii) It is known that the Riemann zeta function is written by using a kind of Mellin transformation of the Gauss transformation in (2) as follows [28]: This is derived by using the Euler-Maclaurin summation formula ( [29, page 14]) and the definition of . Then, by replacing with in (8), can we regard as a new zeta function? The rest of the paper is organised as follows. In Section 2, we prove Theorem 3. In Section 2.1, we give the necessary results used to prove the Gauss-Kuzmin theorem for the continued fractions presented in Section 1. The essential argument of the proof is the Gauss-Kuzmin-type equation. We will also give some results concerning the behavior of the derivative of { , } in (35) which will allow us to complete the proof of Theorem 3 in Section 2.2.

Proof of Theorem 3
In this section, we will prove Theorem 3 applying the method of Rockett and Szüsz [30]. Fix an integer ≥ 2.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.