Abstract
We consider a family of interval maps which are generalizations of the Gauss transformation. For the continued fraction expansion arising from , we solve a Gauss-Kuzmin-type problem.
1. 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 whose digits are differences of consecutive nonpositive integer powers of 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 as with .
The purpose of this paper is to prove a Gauss-Kuzmin-type problem for the continued fraction expansions of real numbers in whose digits are differences of consecutive nonpositive integer powers of an integer . In this section, we show our motivation and main theorems.
1.1. Gauss’ Problem and Its Progress
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 .
Roughly speaking, the metrical theory (or, as called by Khintchine, the measure theory) of continued fraction expansions is about properties of the sequence . 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 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 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) . This has been called the Gauss-Kuzmin theorem or the Kuzmin theorem.
One year later, using a different method, Lévy [5] improved Kuzmin's result by showing that , , with . The Gauss-Kuzmin-Lévy theorem is the first basic result in the rich metrical theory of continued fractions.
By such a development, generalizations of these problems for nonregular continued fractions are also called the Gauss-Kuzmin problems.
1.2. 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 . In [2], Chan shows that any 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.
Definition 1. Fix an integer .(i)The measure-theoretical dynamical system is defined as follows: , where denotes the -algebra of all Borel subsets of and is the transformation where for .(ii)In addition to (i), one writes as with the following probability measure on : where
Define the quantized index map by By definition, . By using and , the sequence in (7) is obtained as follows: with . In this way, gives the algorithm of Chan's continued fraction expansion (7).
Proposition 2. Let be as in Definition 1(ii).(i) is ergodic.(ii)The measure is invariant under ; that is, for any .
By Proposition 2(ii), is a “dynamical system” in the sense of [7, Definition 3.1.3].
1.3. Known Results and Applications
For Chan’s continued fraction expansions, we show known results and their applications in this subsection.
1.3.1. Known Results for Case
For in Definition 1(ii), assume ; that is, we consider only in here.
In [8], Chan proved a Gauss-Kuzmin-Lévy theorem for the transformation . He showed that the convergence rate of the th distribution function of to its limit is as with uniformly in .
In [9, 10], Sebe investigated the Perron-Frobenius operator of by replacing a probability measure of the measurable space . Especially, Sebe studied the Perron-Frobenius operator of ; that is, is a unique operator on satisfying 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 and , respectively, when , with and ([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 of bounded variation, Iosifescu and Sebe [14] proved that the exact optimal convergence rate of to is as uniformly in . Here is the inverse of the golden ratio; that is, we have For , define the sequence recursively by , , with . Then it is an -valued Markov chain on where is the probability measure on defined as the following distribution function: For in (13), let denote its restriction on . From [12, Proposition 2.1.10], we see that is the transition operator of the Markov chain on for any .
1.3.2. Known Results for Case
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 , and is the transition probability function from to given by Also, the associated Markov operator of RSCC (16) is denoted by and has the transition probability function where .
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-Kuzmin-type problem.
For more details about using RSCC in solving the Gauss-Kuzmin-Lévy-type theorems, see [11, 16–20].
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 , the upper and lower bounds of the convergence rate were obtained as and , respectively, when , with and .
1.3.3. 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 for where 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 and ) 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 .(ii)Fix an integer . Define the random Fibonacci sequence as where ’s are as in (7). By using the ergodicity of (Proposition 2(i)), Chan proved that its asymptotic growth rate is given as follows [2]: where is as in (10).
1.3.4. 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 , , define random variables on the measure space , where 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 ; that is, where ’s are given in (12). This is a Khintchine-type result and we obtain for almost all real numbers . 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 , , and are given in (12), (26), and (27), respectively.
1.4. Main Theorem and Its Consequences
1.4.1. Main Theorem
We show our main theorems in this subsection. Fix an integer . Let be as in (10) and let 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 is simply written as . 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 , lies between and if and only if there exists , so that lies between and . Thus Taking the limit as gives that, for an arbitrary probability density function for , the corresponding density for is given a.e. in by the equation Clearly, the operator admits the density function as an eigenfunction corresponding to the eigenvalue ; that is, . Here denotes the Banach space of all complex functions for which .
The only eigenvalue of modulus of is 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.
Theorem 3 (the Gauss-Kuzmin theorem). Let and be as in (8) and (31), respectively. When a nonatomic probability measure on is given, define functions () on by for . Then there exists a constant such that is written as
Remark 4. (i) From (36), we see that
where is the measure defined in (9). In fact, the Gauss-Kuzmin theorem estimates the error
(ii) The solution of this problem implies that in (12) is exponentially -mixing under (and under many other probability measures including ) [11, 12]; that is,
for any (the -algebra generated by the random variables ), , and , with suitable positive constants and .
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 by
then there exists such that
The norm is defined by [12].
Problem 5. (i) Solve the Gauss-Kuzmin-Lévy problem of for . 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.
2. Proof of Theorem 3
In this section, we will prove Theorem 3 applying the method of Rockett and Szüsz [30]. Fix an integer .
2.1. Necessary Lemmas
In this subsection, we show some lemmas. First, we show that in (35) satisfy a Gauss-Kuzmin-type equation.
Lemma 6. For functions in (35), the following Gauss-Kuzmin-type equation holds: for and where .
Proof. Let , , and .
From (8) and (12), we see that
From the definition of and (45) it follows that, for any , . From this and using the -additivity of , we have
Then (44) holds because and
Remark 7. Assume that, for some , the derivative exists everywhere in and is bounded. Then it is easy to see by induction that exists and is bounded for all . This allows us to differentiate (44) term by term, obtaining
We introduce functions as follows:
Then (48) is
where is given in (18).
For , define and by
Then we get
Lemma 8. For in (49), define . Then where
Proof. We have
Now from (50) and by calculus, we have
where
Now (56) implies
We now must calculate the maximum value of the sums in this expression.
First, we note that
where we use and .
Next, observe that the function
is decreasing for and . Hence, . This leads to
The relations (58), (59), and (61) imply (53) and (54).
2.2. Proof of Theorem 3
Introduce a function such that
Because and , we have . To prove Theorem 3, we have to show the existence of a constant such that
If we can show that , then its integration will show (36).
To demonstrate that has this desired form, it suffices to prove the following lemma.
Lemma 9. For any and , there exists a constant such that
Proof. Let be as in (54). Using Lemma 8, to show (64) it is enough to prove that . To this end, for , observe that Therefore for any , .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to express sincere thanks to the anonymous referees.