SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS

2020 ◽  
Vol 102 (2) ◽  
pp. 186-195
Author(s):  
WEILIANG WANG ◽  
LU LI
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Continuous Functions ◽  
Lipschitz Functions ◽  
Image Position

Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$-transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set $$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$ where $\unicode[STIX]{x1D70F}_{1}$, $\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with $\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all $x,y\in [0,1]$.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

2014 ◽  
Vol 91 (1) ◽  
pp. 34-40 ◽  
Author(s):  
YUEHUA GE ◽  
FAN LÜ
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Real Numbers ◽  
Image Position

AbstractWe study the distribution of the orbits of real numbers under the beta-transformation$T_{{\it\beta}}$for any${\it\beta}>1$. More precisely, for any real number${\it\beta}>1$and a positive function${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

An integral which occurs in statistics

1933 ◽  
Vol 29 (2) ◽  
pp. 271-276 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Real Number ◽  
Positive Definite ◽  
Direct Evaluation ◽  
Independent Variables ◽  
Symmetrical Matrices ◽  
Image Position

1. In this note we give a direct evaluation of the integralwhose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation

scholarly journals Uniform Diophantine approximation related to -ary and -expansions

2014 ◽  
Vol 36 (1) ◽  
pp. 1-22 ◽  
Author(s):  
YANN BUGEAUD ◽  
LINGMIN LIAO
Keyword(s):  
Hausdorff Dimension ◽  
Real Number ◽  
Diophantine Approximation ◽  
Large Integer ◽  
Real Numbers

Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

On a Hardy-Littlewood problem of diophantine approximation

1939 ◽  
Vol 35 (4) ◽  
pp. 527-547 ◽  
Author(s):  
D. C. Spencer
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Diophantine Approximation ◽  
Irrational Numbers ◽  
Image Position

1. Let .When r is a positive integer, various writers have considered sums of the formwhere ω1 and ω2 are two positive numbers whose ratio θ = ω1/ω2 is irrational and ξ is a real number satisfying 0 ≤ ξ < ω1. In particular, Hardy and Littlewood (2,3,4), Ostrowski(9), Hecke(6), Behnke(1), and Khintchine(7) have given best possible approximations for sums of this type for various classes of irrational numbers. Most writers have confined themselves to the case r = 1, in which

Some metrical theorems in diophantine approximation

1951 ◽  
Vol 47 (1) ◽  
pp. 18-21 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Solutions ◽  
The Difference ◽  
Image Position

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequalityhas infinitely many integer solutions q > 0. In particular, if α is any real number, the inequalityhas infinitely many solutions.

Determination of the zeros of a cross-product Bessel function

1973 ◽  
Vol 74 (3) ◽  
pp. 477-483 ◽  
Author(s):  
L. Z. Salchev ◽  
V. B. Popov
Keyword(s):  
Critical Load ◽  
Real Number ◽  
Power Law ◽  
Bessel Functions ◽  
Cross Product ◽  
Positive Parameter ◽  
Inertia Moment ◽  
Simply Supported ◽  
Image Position

In many mechanical and other problems the following equationis reached, where Jν(α) and Yν(α) are Bessel functions of the first and second kind of any real order ν and β is a positive parameter.For example, equation (1) is reached in the case of determining the critical load Pcr, for a simply supported strut with a variable inertia moment by a power law, where the power m is any real number.

Diophantine approximation and Hausdorff dimension in Fuchsian groups

1993 ◽  
Vol 113 (2) ◽  
pp. 343-354 ◽  
Author(s):  
S. L. Velani
Keyword(s):  
Hausdorff Dimension ◽  
Unit Circle ◽  
Hyperbolic Space ◽  
Diophantine Approximation ◽  
Fuchsian Groups ◽  
Two Dimensional ◽  
Disc Model ◽  
Image Position ◽  
Straight Lines

The Poincaré disc modelof two-dimensional hyperbolic space supports a metric ρ derived from the differentialGeodesics for the metric ρ are arcs of circles orthogonal to the unit circle S, and straight lines through the origin.

scholarly journals Hausdorff dimension of level sets of generalized Takagi functions

2014 ◽  
Vol 157 (2) ◽  
pp. 253-278
Author(s):  
PIETER C. ALLAART
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continuous Functions ◽  
Zero Set ◽  
Arbitrary Integer ◽  
Open Problems ◽  
Formula Group ◽  
Nowhere Differentiable Function ◽  
Image Position ◽  
Class Of Functions

AbstractThis paper examines the Hausdorff dimension of the level sets f−1(y) of continuous functions of the form \begin{equation*} f(x)=\sum_{n=0}^\infty 2^{-n}\omega_n(x)\phi(2^n x), \quad 0\leq x\leq 1, \end{equation*} where φ(x) is the distance from x to the nearest integer, and for each n, ωn is a {−1,1}-valued function which is constant on each interval [j/2n,(j+1)/2n), j=0,1,. . .,2n − 1. This class of functions includes Takagi's continuous but nowhere differentiable function. It is shown that the largest possible Hausdorff dimension of f−1(y) is $\log ((9+\sqrt{105})/2)/\log 16\approx .8166$, but in case each ωn is constant, the largest possible dimension is 1/2. These results are extended to the intersection of the graph of f with lines of arbitrary integer slope. Furthermore, two natural models of choosing the signs ωn(x) at random are considered, and almost-sure results are obtained for the Hausdorff dimension of the zero set and the set of maximum points of f. The paper ends with a list of open problems.

Metric Diophantine approximation and Hausdorff dimension on manifolds

1989 ◽  
Vol 105 (3) ◽  
pp. 547-558 ◽  
Author(s):  
M. M. Dodson ◽  
B. P. Rynne ◽  
J. A. G. Vickers
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Smooth Manifolds ◽  
Metric Diophantine Approximation ◽  
Image Position

In this paper we discuss homogeneous Diophantine approximation of points on smooth manifolds M in ℝk. We begin with a brief survey of the notation and results. For any x,y ∈ℝk, let.

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