Words and Quadratic Numbers

2018 ◽  
pp. 43-48
Author(s):  
Christophe Reutenauer
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Periodic Sequence ◽  
Periodic Pattern ◽  
Conjugation Class

This chapter offers an overview of words and quadratic numbers, and in particular ordering the conjugates of a Christoffel word. Within this topic the reader learns that the reversal of a Christoffel word is a conjugate, and that the lower and upper Christoffel words of the same slope are the smallest and the largest in their conjugation class. The chapter discusses computation in terms ofMarkoff numbers of the quadratic real number which has a periodic continued fraction with periodic pattern equal to a Christoffel word written on the alphabet 11, 22. It also reviews computation of theMarkoff supremum of a periodic biinfinite sequence, and of theLagrange number of a periodic sequence, both having a periodic pattern as above.

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.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

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.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

scholarly journals A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

2020 ◽  
Vol 15 (2) ◽  
pp. 1-8
Author(s):  
Francesco Amoroso ◽  
Moubinool Omarjee
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Partial Quotients

AbstractLet α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of \sqrt 2 /2 has bounded partial quotients, {S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus {S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough 1188.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (01) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilitiesp0,1= 1,pk,k–1=gk,pk,k+1= 1–gk(0 &lt;gk&lt; 1,k= 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

Linearly periodic continued fractions

2021 ◽  
Vol 105 (564) ◽  
pp. 442-449
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lalitphat Sukruan
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Integers ◽  
Periodic Continued Fractions

An infinite simple continued fraction representation of a real number α is in the form $$\eqalign{& {a_0} + {1 \over {{a_1} + {1 \over {{a_2} + {1 \over {{a_3} + {1 \over {}}}}}}}} \cr & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ddots \cr} $$ where $${a_0}$$ is an integer, and $${a_i}$$ are positive integers for $$i \ge 1$$. This is often written more compactly in one of the following ways: $${a_0} + {1 \over {{a_1} + }}{1 \over {{a_2} + }}{1 \over {{a_3} + }} \ldots \;{\rm{or}}\;\left[ {{a_0};\;{a_1},\;{a_2},\;{a_3} \ldots } \right]$$ .

scholarly journals Mixed periodic Jacobi continued fractions

1986 ◽  
Vol 104 ◽  
pp. 129-148 ◽  
Author(s):  
Yoshifumi Kato
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Jacobi Matrix ◽  
Padé Approximant ◽  
Positive Real ◽  
Pade Approximant

Let b0 be a positive real number andbe a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as followswhere An(z)/Bn(z) is the n-th Padé approximant of φ(z).

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