scholarly journals Good approximation and characterization of subgroups of R = Z

2001 ◽  
Vol 38 (1-4) ◽  
pp. 97-113 ◽  
Author(s):  
A. Bíró ◽  
J. M. Deshouillers ◽  
Vera T. Sós
Keyword(s):  
Continued Fraction ◽  
Irrational Number ◽  
Explicit Construction ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Positive Integers

Let be a real irrational number and A =(xn) be a sequence of positive integers. We call A a characterizing sequence of or of the group Z mod 1 if lim n 2A n !1 k k =0 if and only if 2 Z mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of R = Z . Inthespecialcase Z mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems.

scholarly journals The continuous Diophantine approximation mapping of Szekeres

1995 ◽  
Vol 59 (2) ◽  
pp. 148-172 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Andrew D. Pollington
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Unit Square ◽  
Alternative Characterization ◽  
Continuous Analogue

AbstractSzekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

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.

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 Bernoulliness of when is an irrational rotation: towards an explicit isomorphism

2020 ◽  
pp. 1-26
Author(s):  
CHRISTOPHE LEURIDAN
Keyword(s):  
Continued Fraction ◽  
Bernoulli Shift ◽  
Irrational Number ◽  
Unit Interval ◽  
Positive Answer ◽  
Short Paper ◽  
Continued Fraction Expansion ◽  
Skew Products ◽  
Image Position ◽  
Open Question

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$ : the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$ where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$ . Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

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.

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.

On the Lower Range of Perron's Modular Function

1969 ◽  
Vol 21 ◽  
pp. 808-816 ◽  
Author(s):  
J. R. Kinney ◽  
T. S. Pitcher
Keyword(s):  
Finite Number ◽  
Continued Fraction ◽  
Modular Function ◽  
Finite Collection ◽  
Continued Fraction Expansion ◽  
Lower Range ◽  
Image Position ◽  
Positive Integers

The modular function Mwas introduced by Perron in (6). M(ξ) (for irrational ξ) is denned by the property that the inequalityis satisfied by an infinity of relatively prime pairs (p, q)for positive d,but by at most a finite number of such pairs for negative d.We will writefor the continued fraction expansion of ξ ∈ (0, 1) and for any finite collection y1,…, ykof positive integers we will writeIt is known (see 6) thatWhere

scholarly journals A theorem on approximation of irrational numbers by simple continued fractions

1988 ◽  
Vol 31 (2) ◽  
pp. 197-204 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Irrational Numbers

Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:Theorem 1.For n>1,

A NEW POINT IN LAGRANGE SPECTRUM

2012 ◽  
Vol 09 (02) ◽  
pp. 393-403
Author(s):  
K. C. PRASAD ◽  
HRISHIKESH MAHATO ◽  
SUDHIR MISHRA
Keyword(s):  
Continued Fraction ◽  
Infinite Sequence ◽  
Cluster Point ◽  
Continued Fraction Expansion ◽  
Irrational Numbers ◽  
Lagrange Spectrum ◽  
Positive Integers

Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a0, a1, …, an, …]. Then a0 is an integer and {an}n≥1 is an infinite sequence of positive integers. Let Mn(θ) = [0, an, an-1, …, a1] + [an+1, an+2, …]. Then the set of numbers { lim sup Mn(θ) ∣ θ ∈ I} is called the Lagrange Spectrum 𝔏. Notably 3 is the first cluster point of 𝔏. Essentially lim inf 𝔏 or [Formula: see text]. Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh.8 (1921) 12 pp.] has found that lim inf { lim sup Mn(θ) ∣ θ = [a0, a1, a2, …, an, …] and [Formula: see text]. This article forwards the value of lim inf{lim sup Mn(θ) ∣ θ = [a0, a1, …, an, …] and an ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

scholarly journals A Note on Continued Fractions

1960 ◽  
Vol 12 ◽  
pp. 303-308 ◽  
Author(s):  
A. Oppenheim
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Quadratic Irrational ◽  
Image Position ◽  
Positive Integers ◽  
Periodic Continued Fractions

Any real number y leads to a continued fraction of the type(1)where ai, bi are integers which satisfy the inequalities(2)by means of the algorithm(3)the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

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