On the Continued Fractions of Conjugate Quadratic Irrationalities

1980 ◽  
Vol 23 (2) ◽  
pp. 199-206
Author(s):  
Fritz Herzog
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Quadratic Irrationality ◽  
Partial Quotients ◽  
Image Position ◽  
The Right ◽  
Lagrange's Theorem

Let1be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on.

scholarly journals Range-renewal structure in continued fractions

2016 ◽  
Vol 37 (4) ◽  
pp. 1323-1344
Author(s):  
JUN WU ◽  
JIAN-SHENG XIE
Keyword(s):  
Continued Fractions ◽  
Level Sets ◽  
Continued Fraction ◽  
Irrational Number ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Image Position ◽  
Almost All

Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$, $$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$ The Hausdorff dimensions of certain level sets about $R_{n}$ are discussed.

scholarly journals CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

2002 ◽  
Vol 45 (3) ◽  
pp. 653-671 ◽  
Author(s):  
J. L. Davison
Keyword(s):  
Rate Of Convergence ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Subject Classification ◽  
Specific Sequence ◽  
Secondary 11B37 ◽  
Quadratic Irrational ◽  
Infinite Word ◽  
Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

2021 ◽  
pp. 1-29
Author(s):  
LINGLING HUANG ◽  
CHAO MA
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Natural Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Nonnegative Number ◽  
Partial Quotients

Abstract This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

Ramanujan's Continued Fraction and the Bauer-Muir Transformation

1961 ◽  
Vol 57 (1) ◽  
pp. 76-79 ◽  
Author(s):  
V. N. Singh
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Basic Analogue ◽  
Ramanujan's Continued Fraction

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

scholarly journals On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

2018 ◽  
Vol 61 (1) ◽  
pp. 283-293
Author(s):  
Poj Lertchoosakul ◽  
Radhakrishnan Nair
Keyword(s):  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Laurent Series ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series ◽  
Almost Everywhere ◽  
Partial Quotients ◽  
Image Position

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

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.

Ramanujan's Continued Fraction

1935 ◽  
Vol 31 (1) ◽  
pp. 7-17 ◽  
Author(s):  
G. N. Watson
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Ramanujan's Continued Fraction

The best of the theorems on continued fractions, to be found in Ramanujan's manuscript note-book may be stated as follows:where there are eight gamma-functions in each product and the ambiguous signs are so chosen that the argument of each gamma-function contains one of the specified numbers of minus signs. Thenprovided that one of the numbers β, γ δ, ε is equal to ± n, where n is a positive integer; and the products and sums on the right range over the numbers α β, γ δ, ε.

Denominator Sequences of continued fractions I

1973 ◽  
Vol 15 (1) ◽  
pp. 112-116 ◽  
Author(s):  
R. T. Worley
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Image Position

Let α be an irrational number with simple continued fraction α = (a0, a1, a2,…). The problem studied is that of whether the sequence (qn) of denominators of the convergents pn/qn to α has a subsequence (Bn) = (qin} which is the sequence of denominators of convergents An/Bn to a different number α′. In other words, does there exist a subsequence {qin} which satisfies qio= 1 and For example, the sequence of denominators of convergents to ½(3 − e) is a subsequence of the sequence of denominators of convergents to ε.

Sign in / Sign up

Export Citation Format

Share Document