scholarly journals ON GOOD APPROXIMATIONS AND THE BOWEN–SERIES EXPANSION

2021 ◽  
pp. 1-14
Author(s):  
LUCA MARCHESE
Keyword(s):  
Series Expansion ◽  
Continued Fraction ◽  
Real Numbers ◽  
Continued Fraction Expansion

Abstract We consider the continued fraction expansion of real numbers under the action of a nonuniform lattice in $\text {PSL}(2,{\mathbb R})$ and prove metric relations between the convergents and a natural geometric notion of good approximations.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

scholarly journals On the metric theory of the optimal continued fraction expansion

1997 ◽  
Vol 56 (1) ◽  
pp. 69-79
Author(s):  
R. Nair
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Natural Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Special Case

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants byIn this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

scholarly journals COMPLEX NUMBERS WITH BOUNDED PARTIAL QUOTIENTS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 9-20 ◽  
Author(s):  
WIEB BOSMA ◽  
DAVID GRUENEWALD
Keyword(s):  
Continued Fraction ◽  
Strong Sense ◽  
Complex Numbers ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Bounded Complex ◽  
Partial Quotients

AbstractConjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree $d$, algebraic numbers of degree $d$ that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

scholarly journals Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

2019 ◽  
pp. 56-101
Author(s):  
Laura Capuano ◽  
Peter Jossen ◽  
Christina Karolus ◽  
Francesco Veneziano
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Hyperelliptic Curves ◽  
Pell Equation ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Algebraic Functions ◽  
Generalized Jacobians

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.

scholarly journals Beta-expansion and continued fraction expansion of real numbers

2019 ◽  
Vol 187 (3) ◽  
pp. 233-253 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu ◽  
Bing Li
Keyword(s):  
Continued Fraction ◽  
Real Numbers ◽  
Continued Fraction Expansion

Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

2011 ◽  
Vol 32 (3) ◽  
pp. 1073-1089 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Continued Fraction ◽  
Negative Integer ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Subexponential Growth ◽  
Complexity Function ◽  
Infinite Word ◽  
Zero Entropy

AbstractThe complexity function of an infinite wordwon a finite alphabetAis the sequence counting, for each non-negative integern, the number of words of lengthnon the alphabetAthat are factors of the infinite wordw. Letfbe a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whoseq-adic expansion has a complexity function bounded byfand the set of real numbers whose continued fraction expansion is bounded byqand has a complexity function bounded byf.

scholarly journals EXPANSIONS OF THE ORDERED ADDITIVE GROUP OF REAL NUMBERS BY TWO DISCRETE SUBGROUPS

2016 ◽  
Vol 81 (3) ◽  
pp. 1007-1027 ◽  
Author(s):  
PHILIPP HIERONYMI
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Additive Group ◽  
Second Order ◽  
Order Logic ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Discrete Subgroups ◽  
Second Order Logic ◽  
Monadic Second Order Logic

AbstractThe theory of (ℝ, <, +, ℤ, ℤa) is decidable if a is quadratic. If a is the golden ratio, (ℝ, <, +, ℤ, ℤa) defines multiplication by a. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of a to define the above structures in monadic second order logic of one successor. The converse that (ℝ, <, +, ℤ, ℤa) defines monadic second order logic of one successor, will also be established.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

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