scholarly journals On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 127
Author(s):  
Qian Xiao ◽  
Chao Ma ◽  
Shuailing Wang
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Golden Ratio ◽  
Irreducible Polynomial ◽  
Negative Real Number ◽  
Lagrange's Theorem

In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

Markoff’s Theorem for Approximations

2018 ◽  
pp. 55-62
Author(s):  
Christophe Reutenauer
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Golden Ratio ◽  
Periodic Pattern ◽  
Square Root ◽  
Error Terms

This chapter provesMarkoff’s theorem for approximations: if x is an irrational real number such that its Lagrange number L(x) is <3, then the continued fraction of x is ultimately periodic and has as periodic pattern a Christoffel word written on the alphabet 11, 22. Moreover, the bound is attained: this means that there are indeed convergents whose error terms are correctly bounded. For this latter result, one needs a lot of technical results, which use the notion of good and bad approximation of a real number x satisfying L(x) <3: the ranks of the good and bad convergents are precisely given. These results are illustrated by the golden ratio and the number 1 + square root of 2.

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 A result for semi-regular continued fractions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 145-154 ◽  
Author(s):  
P. E. Blanksby
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Two Dimensional ◽  
Integer Part ◽  
Image Position

If Φ is a real number with |Φ| ≧ 1, then a semiregular continuet fraction development of Φ is denoted by where the ai are integers such that |ai| ≧ 2. The expansions arise geo-. metrically by considering the sequence of divided cells of two-dimensional grids (see [1]), and are described by the following algorithm: for all n ≧ 0, taking Φ = Φ.0 Hence where in this case the square brackets are used to signify the integer-part function. It follows that each irrational Φ has uncountably many such expansions, none of which has a constantly equal to 2 (or -2) for large n.

Cyclotomic approximation lattices

2015 ◽  
Vol 11 (02) ◽  
pp. 557-567
Author(s):  
Antonino Leonardis
Keyword(s):  
Continued Fractions ◽  
The Third ◽  
Lagrange's Theorem ◽  
Cyclotomic Integers

In this paper, we will consider the Approximation Lattices for a p-adic number, as defined in a work of de Weger, and construct a generalization called the Cyclotomic Approximation Lattices. In the latter case, we consider approximation by a pair of cyclotomic integers instead of rational ones. This can be useful for studying p-adic continued fractions with cyclotomic integral part. The first section will introduce this work and provide motivations. The second one will give some background theorems on number rings. In the third section, we will recall the work of de Weger with a new proof for Theorem 3.6, the analogue of classical Lagrange's theorem for continued fractions. In the fourth one, we will then see the cyclotomic variant and its analogous properties.

Numerical criteria for integral dependence

2011 ◽  
Vol 151 (1) ◽  
pp. 95-102 ◽  
Author(s):  
BERND ULRICH ◽  
JAVID VALIDASHTI
Keyword(s):  
Real Number ◽  
Direct Approach ◽  
Graded Algebras ◽  
Numerical Invariants ◽  
Negative Real Number

AbstractWe study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as ‘Rees criteria’. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

LINEAR CENTERS WITH PERTURBATIONS OF DEGREE 2d + 5

2012 ◽  
Vol 22 (01) ◽  
pp. 1250007 ◽  
Author(s):  
WENTAO HUANG ◽  
XINGWU CHEN ◽  
VALERY G. ROMANOVSKI
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Differential Systems ◽  
Isochronous Centers ◽  
Negative Real Number

We describe a method for studying the center and isochronicity problems for a class of differential systems in the form of linear center perturbed by homogeneous series of degree 2d + m, where d is a non-negative real number and m is a positive integer. As an application, we classify centers and isochronous centers for a particular case when m = 5.

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