A WIRSING-TYPE APPROACH TO SOME CONTINUED FRACTION EXPANSION

Chan (2004) considered a certain continued fraction 
expansion and the corresponding Gauss-Kuzmin-Levy 
problem. A Wirsing-type approach to the Perron-Frobenius operator of the associated 
transformation under its invariant measure allows us to obtain a 
near-optimal solution to this problem.


Introduction
The Gauss 1812 problem gave rise to an extended literature.In modern times, the socalled Gauss-Kuzmin-Lévy theorem is still one of the most important results in the metrical theory of regular continued fractions (RCFs).A recent survey of this topic is to be found in [10].From the time of Gauss, a great number of such theorems followed.See, for example, [2,6,7,8,18].
Taking up a problem raised in [1], we consider another expansion of reals in the unit interval, different from the RCF expansion.In fact, in [1] 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] that the convergence rate involved is O(q n ) as n → ∞ with 0 < q < 1.This unsurprising result can be easily obtained from well-known general results (see [9, pages 202 and 262-266] and [10, Section 2.1.2])concerning the Perron-Frobenius operator of the transformation under the invariant measure induced by the limit distribution function.
Our aim here is to give a better estimation of the convergence rate discussed.First, in Section 2 we introduce equivalent, but much more concise and rigorous expressions than in [1] of the transformation involved and of the related incomplete quotients.Next, in Section 3, our strategy is to derive the Perron-Frobenius operator of this transformation under its invariant measure.In Section 4, we use a Wirsing-type approach (see [22]) to study the optimality of the convergence rate.Actually, in Theorem 4.3 of Section 4 we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-Lévy problem.

Another expansion of reals in the unit interval
In this section we describe another continued fraction expansion different from the regular continued fraction expansion for a number x in the unit interval I = [0,1], which has been actually considered in [1].
Define for any x ∈ I the transformation where {u} denotes the fractionary part of a real u while log stands for natural logarithm.(Nevertheless, the definition of τ is independent of the base of the logarithm used.)Putting with τ 0 (x) = x the identity map and where [u] denotes the integer part of a real u, one easily sees that every irrational x ∈ (0,1) has a unique infinite expansion Here, the incomplete quotients or digits a n (x), n ∈ N + of x ∈ (0,1) are natural numbers.Let Ꮾ I be the σ-algebra of Borel subsets of I.There is a probability measure ν on Ꮾ I defined by

An operator treatment
In the sequel we will derive the Perron-Frobenius operator of τ under the invariant measure ν.
Let µ be a probability measure on Ꮾ I such that µ(τ −1 (A)) = 0 whenever µ(A) = 0, A ∈ Ꮾ I , where τ is the continued fraction transformation defined in Section 2. In particular, Gabriela Ileana Sebe 1945 this condition is satisfied if τ is µ-preserving, that is, µτ −1 = µ.It is known from [10, Section 2.1] that the Perron-Frobenius operator P µ of τ under µ is defined as the bounded linear operator on In particular the Perron-Frobenius operator P λ of τ under the Lebesgue measure λ is Proposition 3.1.The Perron-Frobenius operator P ν = U of τ under ν is given a.e. in I by the equation where ) The proof is entirely similar to that of [10,Proposition 2.1.2].
An analogous result to [10, Proposition 2.1.5]is shown as follows.
In this section we will assume that F 0 ∈ C 1 (I).So, we study the behaviour of U n as n → ∞, assuming that the domain of U is C 1 (I), the collection of all functions f : I → C which have a continuous derivative.
Let f ∈ C 1 (I).Then the series (3.3) can be differentiated term-by-term, since the series of derivatives is uniformly convergent.Putting ∆ k = γ k − γ 2k , k ∈ N we get x ∈ I. Thus, we can write where V : C(I) → C(I) is defined by We are going to show that V n takes certain functions into functions with very small values when n ∈ N + is large.Proof.Let h : R + → R be a continuous bounded function such that lim x→∞ h(x) < ∞.We look for a function g : (0,1] → R such that Ug = h, assuming that the equation holds for x ∈ R + .Then (4.7) yields Gabriela Ileana Sebe 1947 Hence and we indeed have In particular, for any fixed a ∈ I we consider the function h a : R + → R defined by h a (x) = 1/(x + a + 1), x ∈ R + .By the above, the function g a : (0,1] → R defined as satisfies Ug a (x) = h a (x), x ∈ I. Setting ϕ a (x) = g a (x) = 3ax 2 + 4(a + 1)x + 6 (ax + 2) 2 (ax + 1) 2 , ( we have We choose a by asking that (ϕ a /V ϕ a )(0) = (ϕ a /V ϕ a )(1).This amounts to 3a 4 + 12a 3 + 18a 2 − 2a − 17 = 0 which yields as unique acceptable solution a = 0.794741181 ....For this value of a, the function ϕ a /V ϕ a attains its maximum equal to (3/2)(a + 1) 2 = 4.83164386 ... at x = 0 and x = 1, and has a minimum m(a) (ϕ a /V ϕ a )(0.39) = 4.776363306 ....It follows that for ϕ = ϕ a with a = 0.794741181 ..., we have 1948 A Wirsing-type approach to some continued fraction Proof.Since V is a positive operator, we have Noting that α f 0 ≤ ϕ ≤ β f 0 , we can write where α, β, v and w are defined in Proposition 4.1 and Corollary 4.2 and F(x) = (1/ log(4/ 3))log(2(x + 1))/x + 2. In particular, for any n ∈ N + and x ∈ I, Proof.For any n ∈ N and x ∈ I, set d n (F(x)) = µ(τ n < x) − F(x).Then by (4.2) we have Differentiating twice with respect to x yields , n ∈ N, x ∈ I.