arXiv subjects
Publications and source records attributed to Ion Coltescu.
The nearest integer continued fraction of a real number $x$ from $[-1/2, 1/2)$ is defined. Some metrical properties of these expansions are presented. We define the approximation coefficients and give an important result on them. The main result consists in obtaining a stationary state for the transformation $τ_{1/2}$ which is absolutely continuous with respect to the Lebesgue measure.
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 \[x= \lim_{n\rightarrow\infty}[b_1(x), b_2(x), ..., b_n(x)]_m. \]
The present paper deals with Gauss' problem on continued fractions. We present a new proof of a theorem which Szüsz applied in order to solve this problem. To be noted, that we obtain the value $0.7594...$ for $q$, which has been optimized by Szüsz in his 1961 paper "Über 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 $τ$ under $γ$, where $τ$ is the continued fraction transformation, and $γ$ is the Gauss' measure.
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 measurable map $T$ on a measure space induces a representation $Π_{T}$ of a Cuntz algebra ${\cal O}_{N}$ when $T$ satisfies a certain condition. For such two maps $τ$ and $σ$ and representations $Π_τ$ and $Π_σ$ associated with them, we show that $Π_τ$ is the restriction of $Π_σ$ when $τ$ is a jump transformation of $σ$. Especially, the Gauss map $τ_1$ and the Farey map $σ_1$ induce representations $Π_{τ_1}$ of ${\cal O}_{\infty}$ and that $Π_{σ_1}$ of ${\cal O}_{2}$, respectively, and $Π_{τ_1}=Π_{σ_1}|_{{\cal O}_{\infty}}$ with respect to a certain embedding of ${\cal O}_{\infty}$ into ${\cal O}_{2}$.