scholarly journals Continued Fractions Related to $(t,q)$-Tangents and Variants

10.37236/2014 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
General Version ◽  
Continued Fraction Expansion ◽  
Special Cases ◽  
Tangent Function

For the $q$-tangent function introduced by Foata and Han (this volume) we provide the continued fraction expansion, by creative guessing and a routine verification. Then an even more recent $q$-tangent function due to Cieslinski is also expanded. Lastly, a general version is considered that contains both versions as special cases.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

scholarly journals Continued fractions, the Chen–Stein method and extreme value theory

2019 ◽  
Vol 41 (2) ◽  
pp. 461-470
Author(s):  
ANISH GHOSH ◽  
MAXIM SØLUND KIRSEBOM ◽  
PARTHANIL ROY
Keyword(s):  
Rate Of Convergence ◽  
Extreme Value Theory ◽  
Upper Bound ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Real Analysis ◽  
Continued Fraction Expansion ◽  
Stein Method

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

scholarly journals Maximizing Bernoulli measures and dimension gaps for countable branched systems

2020 ◽  
pp. 1-19
Author(s):  
SIMON BAKER ◽  
NATALIA JURGA
Keyword(s):  
Probability Measure ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Random Variables ◽  
General Setting ◽  
Continued Fraction Expansion ◽  
Bernoulli Measures ◽  
Independent And Identically Distributed

Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists $c_{0}>0$ , such that $\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure $\unicode[STIX]{x1D707}$ , which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

scholarly journals On Absolutely Normal and Continued Fraction Normal Numbers

2018 ◽  
Vol 2019 (19) ◽  
pp. 6136-6161 ◽  
Author(s):  
Verónica Becher ◽  
Sergio A Yuhjtman
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Main Difficulty ◽  
Normal Numbers ◽  
Continued Fraction Expansion ◽  
Mathematical Operations ◽  
Construction Works

Abstract We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n4 mathematical operations. The construction works by defining successive refinements of appropriate subintervals to achieve, in the limit, simple normality to all integer bases and continued fraction normality. The main difficulty is to control the length of these subintervals. To achieve this we adapt and combine known metric theorems on continued fractions and on expansions in integers bases.

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 Non-periodic continued fractions in hyperelliptic function fields

2001 ◽  
Vol 64 (2) ◽  
pp. 331-343 ◽  
Author(s):  
Alfred J. van der Poorten
Keyword(s):  
Exponential Growth ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Function Fields ◽  
Square Root ◽  
90Th Birthday ◽  
Continued Fraction Expansion ◽  
Generic Polynomial ◽  
Partial Quotients ◽  
Hyperelliptic Function

Dedicated to George Szekeres on his 90th birthdayWe discuss the exponential growth in the height of the coefficients of the partial quotients of the continued fraction expansion of the square root of a generic polynomial.

A dimensional result in continued fractions

2014 ◽  
Vol 10 (04) ◽  
pp. 849-857 ◽  
Author(s):  
Yu Sun ◽  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Dimensional Theory ◽  
Cambridge Philos ◽  
Dimension Formula ◽  
Growth Properties

Given x ∈ (0, 1), let [a1(x), a2(x), a3(x),…] be the continued fraction expansion of x and [Formula: see text] be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {an(x), n ≥ 1} and proved that for any β > 0, the set [Formula: see text] is of Hausdorff dimension [Formula: see text]. In this paper, we consider, for any β > 0, the set [Formula: see text] and show that the Hausdorff dimension of F(β) is [Formula: see text].

Some properties of the continued fraction expansion of (m/n) e1/q

1970 ◽  
Vol 67 (1) ◽  
pp. 67-74 ◽  
Author(s):  
K. R. Matthews ◽  
R. F. C. Walters
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Positive Integers

Introduction. Continued fractions of the form are called Hurwitzian if b1, …, bh, are positive integers, ƒ1(x), …, ƒk(x) are polynomials with rational coefficients which take positive integral values for x = 0, 1, 2, …, and at least one of the polynomials is not constant. f1(x), …, fk(x) are said to form a quasi-period.

scholarly journals Continued fractions and diophantine equations in positive characteristic

2021 ◽  
Vol 109 (123) ◽  
pp. 143-151
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Ring Of Polynomials ◽  
Algebraic Power Series

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

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