scholarly journals Badly approximable numbers and Littlewood-type problems

2011 ◽  
Vol 150 (2) ◽  
pp. 215-226 ◽  
Author(s):  
YANN BUGEAUD ◽  
NIKOLAY MOSHCHEVITIN
Keyword(s):  
Hausdorff Dimension ◽  
Real Numbers ◽  
Image Position ◽  
Badly Approximable

AbstractWe establish that the set of pairs (α, β) of real numbers such that where ‖ · ‖ denotes the distance to the nearest integer, has full Hausdorff dimension in R2. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems.

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:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

scholarly journals Schmidt’s game, fractals, and orbits of toral endomorphisms

2010 ◽  
Vol 31 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
RYAN BRODERICK ◽  
LIOR FISHMAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Hausdorff Dimension ◽  
Recent Result ◽  
Lacunary Sequence ◽  
Alternative Proof ◽  
Integer Matrix ◽  
Image Position ◽  
Schmidt’S Game ◽  
Badly Approximable

AbstractGiven an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

The Hausdorff dimension of certain solenoids

1995 ◽  
Vol 15 (3) ◽  
pp. 449-474 ◽  
Author(s):  
H. G. Bothe
Keyword(s):  
Hausdorff Dimension ◽  
Dense Subset ◽  
Solid Torus ◽  
Cantor Set ◽  
Real Numbers ◽  
The Real ◽  
Mapping Degree ◽  
Image Position ◽  
Expanding Mapping

AbstractFor the solid torus V = S1 × and a C1 embedding f: V → V given by with dϕ/dt > 1, 0 < λi(t) < 1 the attractor Λ = ∩i = 0∞fi(V) is a solenoid, and for each disk D(t) = {t} × (t ∈ S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by where the real numbers pi are characterized by the condition that the pressure of the function log : S1 → ℝ with respect to the expanding mapping ϕ: S1 → S1 becomes zero. (There are two further characterizations of these numbers.)It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the C1 space of all embeddings f with sup λi > Θ−2 (Θ the mapping degree of ϕ: S1 → S1) all those f which have an intrinsically transverse attractor Λ form an open and dense subset.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

Inequalities related to Hardy's and Heinig's

1984 ◽  
Vol 96 (1) ◽  
pp. 1-7 ◽  
Author(s):  
James A. Cochran ◽  
Cheng-Shyong Lee
Keyword(s):  
Real Numbers ◽  
Image Position

In a 1975 paper [8], Heinig established the following three inequalities:where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;where is a sequence of nonnegative real numbers,and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

An Extremum Result

1962 ◽  
Vol 14 ◽  
pp. 597-601 ◽  
Author(s):  
J. Kiefer
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Ad Hoc ◽  
Continuous Functions ◽  
Orthogonality Condition ◽  
Technical Detail ◽  
Real Numbers ◽  
Discrete Probability ◽  
Linearly Independent ◽  
Image Position

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

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