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On a Gauss-Kuzmin Type Problem for a Family of Continued Fractions
Dan Lascu
Mathematics , 2010,
Abstract: We study a family of continued fraction expansion of reals from the unit interval. The Perron-Frobenius operator of the transformation which generates this expansion under the invariant measure of this transformation is given. Using the ergodic behavior of homogeneous random system with complete connections associated with this expansion we solve a variant of Gauss-Kuzmin problem for this continued fraction expansion.
On a Gauss-Kuzmin-Type Problem for a Family of Continued Fraction Expansions
Dan Lascu
Mathematics , 2011, DOI: 10.1016/j.jnt.2012.12.007
Abstract: In this paper we study in detail a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of random systems with complete connections by Iosifescu and obtained the solution of its Gauss-Kuzmin type problem.
Metric properties of N-continued fractions
Dan Lascu
Mathematics , 2015,
Abstract: A generalization of the regular continued fractions was given by Burger et al. in 2008 [3]. In this paper we give metric properties of this expansion. For the transformation which generates this expansion, its invariant measure and Perron-Frobenius operator are investigated.
Dependence with complete connections and the Gauss-Kuzmin theorem for N-continued fractions
Dan Lascu
Mathematics , 2015,
Abstract: We consider a family $\{T_N:N \geq 1 \}$ of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from $T_N$, we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu.
A Gauss-Kuzmin theorem for continued fractions associated with non-positive interger powers of an integer $m \geq 2$
Dan Lascu
Mathematics , 2013, DOI: 10.1155/2014/984650
Abstract: We consider a family $\{\tau_m:m\geq 2\}$ of interval maps introduced by Hei-Chi Chan [5] as generalizations of the Gauss transformation. For the continued fraction expansion arising from $\tau_m$, we solve its Gauss-Kuzmin-type problem by applying the method of Rockett and Sz\"usz [18].
On the Metrical Theory of a Generalized Continued Fraction Expansion
Dan Lascu,Katsunori Kawamura
Mathematics , 2010,
Abstract: We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the continued fraction transformations associated with expansions.
On the Szüsz's Solution to Gauss' Problem
Dan Lascu,Ion Coltescu
Mathematics , 2010,
Abstract: The present paper deals with Gauss' problem on continued fractions. We present a new proof of a theorem which Sz\"usz applied in order to solve this problem. To be noted, that we obtain the value $0.7594...$ for $q$, which has been optimized by Sz\"usz in his 1961 paper "\"Uber einen Kusminschen Satz", where the value 0.485 is obtained for $q$. In our proof, we make use of an important property of the Perron-Frobenius operator of $\tau$ under $\gamma$, where $\tau$ is the continued fraction transformation, and $\gamma$ is the Gauss' measure.
Random Systems with Complete Connections and the Gauss Problem for the Regular Continued Fractions
Dan Lascu,Ion Coltescu
Mathematics , 2010,
Abstract: This paper present the important role that random system with complete connections played in solving the Gauss problem associated to the regular continued fractions. Hence, using the ergodic behavior of homogeneous random system with complete connections, we will solve a Gauss - Kuzmin type theorem.
A New Type of Continued Fraction Expansion
Dan Lascu,Ion Coltescu
Mathematics , 2010,
Abstract: In this paper we define a new type of continued fraction expansion for a real number $x \in I_m:=[0,m-1], m\in N_+, m\geq 2$: \[x = \frac{m^{-b_1(x)}}{\displaystyle 1+\frac{m^{-b_2(x)}}{1+\ddots}}:=[b_1(x), b_2(x), ...]_m. \] Then, we derive the basic properties of this continued fraction expansion, following the same steps as in the case of the regular continued fraction expansion. The main purpose of the paper is to prove the convergence of this type of expansion, i.e. we must show that \[x= \lim_{n\rightarrow\infty}[b_1(x), b_2(x), ..., b_n(x)]_m. \]
A Gauss-Kuzmin Theorem for Some Continued Fraction Expansions
Dan Lascu,Katsunori Kawamura
Mathematics , 2011,
Abstract: We consider a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For this expansion, we apply the method of Rockett and Sz\"usz from [6] and obtained the solution of its Gauss-Kuzmin type problem.
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