On the exceptional sets concerning the leading partial quotient in continued fractions

2021 ◽  
Vol 500 (1) ◽  
pp. 125110
Author(s):  
Mengjie Zhang ◽  
Chao Ma
Keyword(s):  
Continued Fractions ◽  
Partial Quotient ◽  
Exceptional Sets

Metrical property for GCFϵ expansion with the parameter function ϵ(k) = c(k + 1)

2015 ◽  
Vol 11 (07) ◽  
pp. 2065-2072 ◽  
Author(s):  
Ting Zhong ◽  
Quanwu Mu ◽  
Luming Shen
Keyword(s):  
Continued Fractions ◽  
Partial Quotient ◽  
The Real ◽  
Metric Properties ◽  
Metrical Property ◽  
Parameter Function ◽  
Generalized Continued Fractions

This paper is concerned with the metric properties of the generalized continued fractions (GCFϵ) with the parameter function ϵ(kn), where kn is the nth partial quotient of the GCFϵ expansion. When -1 < ϵ(kn) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory128 (2008) 1506–1515] obtained the following metrical properties: [Formula: see text] which are entirely unrelated to the choice of ϵ(kn) ∈ (-1, 1]. Here we deal with the case of ϵ(k) = c(k + 1) with constant c ∈ (0, ∞). It is proved that: [Formula: see text] which change with the real c ∈ (0, ∞). Note that [Formula: see text] as c → 0, it indicates that when c → 0, the GCFϵ has the same metrical property as the case of -1 < ϵ(kn) ≤ 1.

A generalization of the Jarník–Besicovitch theorem by continued fractions

2015 ◽  
Vol 36 (4) ◽  
pp. 1278-1306 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Gauss Map ◽  
Partial Quotient ◽  
Image Position ◽  
Positive Functions ◽  
Besicovitch Sets

We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where $T$ is the Gauss map and $a_{n}(x)$ is the $n$th partial quotient of $x$. In this paper, we consider the size of the generalized Jarník–Besicovitch set $$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$ where ${\it\tau}(x)$ and $f(x)$ are positive functions defined on $[0,1]$.

ON THE LONGEST BLOCK FUNCTION IN CONTINUED FRACTIONS

2020 ◽  
Vol 102 (2) ◽  
pp. 196-206
Author(s):  
TENG SONG ◽  
QINGLONG ZHOU
Keyword(s):  
Growth Rate ◽  
Continued Fractions ◽  
Level Sets ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Partial Quotients

For an irrational number $x\in [0,1)$, let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x):n\geq 1\}$. Given $\unicode[STIX]{x1D6E9}\in \mathbb{N}$, for $n\geq 1$, the $n$th longest block function of $x$ with respect to $\unicode[STIX]{x1D6E9}$ is defined by $L_{n}(x,\unicode[STIX]{x1D6E9})=\max \{k\geq 1:a_{j+1}(x)=\cdots =a_{j+k}(x)=\unicode[STIX]{x1D6E9}~\text{for some}~j~\text{with}~0\leq j\leq n-k\}$, which represents the length of the longest consecutive sequence whose elements are all $\unicode[STIX]{x1D6E9}$ from the first $n$ partial quotients of $x$. We consider the growth rate of $L_{n}(x,\unicode[STIX]{x1D6E9})$ as $n\rightarrow \infty$ and calculate the Hausdorff dimensions of the level sets and exceptional sets arising from the longest block function.

scholarly journals Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction

2019 ◽  
Vol 11 (1) ◽  
pp. 33-41 ◽  
Author(s):  
I.B. Bilanyk ◽  
D.I. Bodnar ◽  
L. Buyak
Keyword(s):  
Recurrence Relation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Linear Recurrence ◽  
Partial Quotient ◽  
Term Linear ◽  
Recurrent Relations ◽  
Multiple Power ◽  
Linear Recurrence Relation ◽  
Branched Continued Fraction

The quotient of two linearly independent solutions of a four-term linear recurrence relation is represented in the form of a branched continued fraction with two branches of branching by analogous with continued fractions. Formulas of partial numerators and partial denominators of this branched continued fraction are obtained. The solutions of the recurrence relation are canonic numerators and canonic denominators of $\mathcal{B}$-figured approximants. Two types of figured approximants $\mathcal{A}$-figured and $\mathcal{B}$-figured are often used. A $n$th $\mathcal{A}$-figured approximant of the branched continued fraction is obtained by adding a next partial quotient to the $(n-1)$th $\mathcal{A}$-figured approximant. A $n$th $\mathcal{B}$-figured approximant of the branched continued fraction is a branched continued fraction that is a part of it and contains all those elements that have a sum of indexes less than or equal to $n$. $\mathcal{A}$-figured approximants are widely used in proving of formulas of canonical numerators and canonical denominators in a form of a determinant, $\mathcal{B}$-figured approximants are used in solving the problem of corresponding between multiple power series and branched continued fractions. A branched continued fraction of the general form cannot be transformed into a constructed branched continued fraction. For calculating canonical numerators and canonical denominators of a branched continued fraction with $N$ branches of branching, $N>1$, the linear recurrent relations do not hold. $\mathcal{B}$-figured convergence of the constructed fraction in a case when coefficients of the recurrence relation are real positive numbers is investigated.

Sign in / Sign up

Export Citation Format

Share Document