Real Irrational Number Research Articles

On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p / q such that | ξ − p / q | < q −2 . The correct generalization to the case of approximation by algebraic numbers of degree ⩽ n, n > 2 , is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n > 2 .

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

Nouvelle preuve d'un théorème de Yoccoz

If θ is an irrational real number such that ∑(ln q n+1 )/ q n =+∞ where p n / q n are the convergents of θ, then the quadratic polynomial P θ(z)= e i 2πθ z+z 2 is not linearizable at 0. This theorem has been proved in 1988 by J.C. Yoccoz, who first constructs a nonlinearizable germ by inverting a renormalisation procedure, and then proves universality of the quadratic family for that question. We give an alternative proof, based on the study of the explosion of parabolic cycles. To cite this article: A. Chéritat, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

On certain exact sequences for $\Gamma _0(m)$

We consider cohomology sets and exact sequences of groups with involutions. In particular, we study congruence subgroups of type $\Gamma_0(m)$ which are acted by the group generated either by the map $z \mapsto (-1/mz)$ of the upper half plane or by the map $x \mapsto (1/mx)$ of the set of irrational real numbers.

Transcendence of Sturmian or Morphic Continued Fractions

We prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values, and begins with arbitrary long blocks which are “almost squares,” then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence (1+(⌊nα⌋mod2))n⩾0, or are a “repetitive” fixed point of a binary morphism satisfying some technical conditions.

Good approximation and characterization of subgroups of R = Z

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.

On rational approximations with denominators from thin sets

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.

On Real Quadratic Number Fields and Simultaneous Diophantine Approximation

Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from \(\). Specifically, for all such pairs we have \(\).

On the Approximation of Irrationals by Rationals

Let ξ be a real irrational number, and a ≧ 0, b ≧ 0, s > 1 be integers. A theorem of S. UCHIYAMA states that there are infinitely many pairs of integers u and v ≠ 0 such that OVBARξ−u/vOVBAR ≤ s2/4v2 and u a, v b mod s, provided that it is not a 6 0 mod s. It is shown that this result is best-possible for all integers s > 1.

On a Problem of Schoenberg and Wills in Diophantine Approximation

Let n be a fixed natural number. Wills has shown that there exist irrational numbers α1,..., αn and real numbers β1,..., β1 with max1≤i≤n ‖qαi-βi‖ > 1/2 – 1/2n for all integers q (‖·‖ denotes the distance to the nearest integer). His example is αi = α and βi = i/n + δ, δ suitably chosen. Beyond that, he asked if αi can be found with pairwise different ‖αi‖. We prove that this does not hold for n ≤ 5, thereby revealing the close relation to Schoenberg's billiard ball problem for cubes and classifying its critical lines in these dimensions.

Computational classification of numbers and algebraic properties

In this paper, we propose a computational classification of finite characteristic numbers (Laurent series with coefficients in a finite field), and prove that some classes have good algebraic properties. This provides tools from the theories of computation, formal languages, and formal logic for finer study of transcendence and algebraic independence questions. Using them, we place some well-known transcendental numbers occurring in number theory in the computational hierarchy. Existence of or lack of patterns in the digit sequences of naturally occurring real numbers is a natural question. Rational numbers (and only rational numbers) have eventually periodic digit sequences. But the question has not been studied much for irrational real numbers, except for statistical studies on normality and randomness of digits for general numbers as well as special numbers such as π. Apart from the fact that no interesting patterns are found in general, the other reason for the lack of such studies is that irrationals usually are not naturally presented by their digit expansions (say decimals), at least in theoretical studies. Also, since it is hard to control carry-overs well, when we add or multiply, it is usually hard to manipulate the formulas to get good control on digit expansions of sums and products. The situation seems to be much better for finite characteristic numbers: There are well-known strong analogies between integers, rationals, and reals on one hand, and polynomials, rational functions, and Laurent series (all with coefficients in a finite field) on the other. Again rational functions correspond to eventually periodic Laurent series expansions. Also the Laurent series representation is widely used: There are no

On the distribution of αpk modulo 1

The fractional part of the sequence {αnk}, where α is an irrational real number and k is an integer, was first studied early this century, initiated by the work of Hardy, Littlewood and Weyl. It seems very natural to consider the subsequence {αpk}, where p denotes a prime variable. The pioneering work in this direction was conducted by Vinogradov [13,14]. Improvements have since been made by Vaughan [12], Ghosh [4], Harman [6,7,8] and Jia [11]. The best results to date have been obtained by Harman for k = 1 [9], by Baker and Harman for 2 ≤ k ≤ 12 [1], and by Harman for larger k [8]. In the following work, we shall adopt a sieve technique developed by Harman in [6] to show the following.

A generalization of Perron's theorem about Hurwitzian numbers

e = [2, 1, 2k + 2, 1]k=0 and e 1/q = [1, (2k + 1)q − 1, 1]k=0 when q is an integer ≥ 2 are well-known examples (see Euler [3], Perron [7], Davis [2], Matthews and Walters [6]). Other classical examples of Hurwitzian numbers are th(1/q) or tan(1/q) when q is a nonnegative integer, e2/q when q is odd and many other real numbers determined by means of Bessel functions (see Cabannes [1], Lehmer [4] and Stambul [10]). A recognizable Hurwitzian number whose quasi-period is determined by polynomials of degree ≥ 2 is still unknown. Let h : x 7→ (ax + b)/(cx + d) be a Mobius transformation where a, b, c, d are integers. If x is Hurwitzian, it follows from a theorem of O. Perron ([7], 127–131) that h(x) is also Hurwitzian. Moreover, the nonconstant polynomials in the quasi-periods of x and h(x) have the same degrees (see [10]). Denote by R the set of all irrational real numbers x whose continued fraction expansion has the form

A problem of Hooley in Diophantine approximation

In [5] Professor Hooley announced without proof the following result which is a variant of well-known work by Heilbronn [4]and Danicic [3] (see [1]).Let k≥2 be an integer, b a fixed non-zero integer, and a an irrational real number. Then, for any ɛ&gt; 0, there are infinitely many solutions to the inequalityHere

On dispersion and Markov constants

We proved some results on the dispersion of the real quadratic irrational numbers, and use LEO 386/25 to compute some numerical results for discriminant <200 (see the attached Table A).

Sums of Fractional Parts of Integer Multiples of an Irrational

Let α be a positive irrational real number, and let Cα(n) = ∑1 ≤ k ≤ n ({kα} − 12), n ≥ 1, where {x} denotes the fractional part of x. We give an explicit formula for Cα(n) in terms of the simple continued fraction for α, and use this formula to give simple proofs of several results of A. Ostrowski, G. H. Hardy and J.E. Littlewood, and V. T. Sós. We also show that there exist positive constants dA such that if α = [a0, a1, a2, ...] and (1/t) ∑1 ≤ j ≤ taj ≤ A holds for infinitely many t, then Cα(x) >dA log x and Cα(x) < −dA log x each hold for infinitely many x.

Diophantine approximation in 𝑅ⁿ

A modification of the Ford geometric approach to the problem of approximation of irrational real numbers by rational fractions is developed. It is applied to find an upper bound for the Hurwitz constant for a discrete group acting in a hyperbolic space. A generalized Khinchine’s approximation theorem is also given.

The Circular Dispersion Spectrum

Let ϑ be a real irrational number and N a positive integer. The one-dimensional torus T is divided by the N points nϑ mod 1, n = 1, 2, ..., N, into N arcs. Denote the length of the longest of these arcs by dN. Define the circular dispersion constant of ϑ, notation C(ϑ), by C(ϑ) ≔ lim supN → ∞NdN. The set of numbers C(ϑ) is called the circular dispersion spectrum. This paper determines the smallest accumulation point of this spectrum and all points below this accumulation point.

On Symmetrical and Asymmetric Diophantine Approximation by Continued Fractions

Let (Pn/Qn)n ≥ 0 be the sequence of regular continued fraction convergents of the real irrational number x. Define the approximation constantsdn, n ≥ 0 by dn = dn(x) ≔ Qn + 1|Qnx − Pn|, n ≥ 0. In this paper the distribution for almost all x of the sequences (dn − 1, dn)n ≥ 1 and (dn − 1, dn, dn + 1)n ≥ 1 is discussed. Furthermore, two recent theorems by Jingcheng Tong concerning arithmetical properties of the dn′s are generalized.

Continued Fractions and Linear Recurrences

We prove that the numerators and denominators of the convergents to a real irrational number $\theta$ satisfy a linear recurrence with constant coefficients if and only if $\theta$ is a quadratic irrational. The proof uses the Hadamard Quotient Theorem of A. van der Poorten.