scholarly journals Ubiquity and large intersections properties under digit frequencies constraints

2008 ◽  
Vol 145 (3) ◽  
pp. 527-548 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Hausdorff Dimension ◽  
Rational Numbers ◽  
Intersection Property ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Countable Intersection ◽  
The Real ◽  
Large Intersection ◽  
Digit Frequencies

AbstractWe are interested in two properties of real numbers: the first one is the property of being well-approximated by some dense family of real numbers {xn}n≥1, such as rational numbers and more generally algebraic numbers, and the second one is the property of having given digit frequencies in some b-adic expansion.We combine these two ways of classifying the real numbers, in order to provide a finer classification. We exhibit sets S of points x which are approximated at a given rate by some of the {xn}n, those xn being selected according to their digit frequencies. We compute the Hausdorff dimension of any countable intersection of such sets S, and prove that these sets enjoy the so-called large intersection property.

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers, II

2021 ◽  
pp. 1-21
Author(s):  
Dzmitry Badziahin ◽  
Yann Bugeaud ◽  
Johannes Schleischitz
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Simultaneous Rational Approximation

Abstract Let $p$ be a prime number. For a positive integer $n$ and a real number $\xi$ , let $\lambda _n (\xi )$ denote the supremum of the real numbers $\lambda$ for which there are infinitely many integer tuples $(x_0, x_1, \ldots , x_n)$ such that $| x_0 \xi - x_1|_p, \ldots , | x_0 \xi ^{n} - x_n|_p$ are all less than $X^{-\lambda - 1}$ , where $X$ is the maximum of $|x_0|, |x_1|, \ldots , |x_n|$ . We establish new results on the Hausdorff dimension of the set of real numbers $\xi$ for which $\lambda _n (\xi )$ is equal to (or greater than or equal to) a given value.

scholarly journals Minimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers

10.37236/749 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Avi Berman ◽  
Shmuel Friedland ◽  
Leslie Hogben ◽  
Uriel G. Rothblum ◽  
Bryan Shader
Keyword(s):  
Computational Complexity ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Real Numbers ◽  
Minimum Rank ◽  
The Real ◽  
Sign Patterns ◽  
Symmetric Patterns

We use a technique based on matroids to construct two nonzero patterns $Z_1$ and $Z_2$ such that the minimum rank of matrices described by $Z_1$ is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by $Z_2$ is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture by Arav, Hall, Koyucu, Li and Rao about rational realization of minimum rank of sign patterns. Using $Z_1$ and $Z_2$, we construct symmetric patterns, equivalent to graphs $G_1$ and $G_2$, with the analogous minimum rank properties. We also discuss issues of computational complexity related to minimum rank.

scholarly journals Counting algebraic numbers in short intervals with rational points

2019 ◽  
pp. 4-11
Author(s):  
Vasily I. Bernik ◽  
Friedrich Götze ◽  
Nikolai I. Kalosha
Keyword(s):  
Uniform Distribution ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Small Interval ◽  
Real Numbers ◽  
Algebraic Integers ◽  
Small Denominators ◽  
Short Intervals ◽  
Regular Distribution ◽  
Regularity Properties

In 2012 it was proved that real algebraic numbers follow a non­uniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.

Two Unusual Representations for the Set of Real Numbers

1970 ◽  
Vol 63 (8) ◽  
pp. 665
Author(s):  
Sanderson M. Smith
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
The Real ◽  
Binary Operations

The rational numbers and the real numbers are both fields under the binary operations of addition and multiplication.

scholarly journals Two constructions of the real numbers via alternating series

1989 ◽  
Vol 12 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Arnold Knopfmacher ◽  
John Knopfmacher
Keyword(s):  
Continued Fractions ◽  
Rational Numbers ◽  
Equivalence Classes ◽  
Real Numbers ◽  
The Real ◽  
New Methods ◽  
Arbitrary Choice ◽  
Ordered Field

Two further new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The methods are motivated by some little known results on the representation of real numbers via alternating series of rational numbers. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of "base" or equivalence classes or any similar constructs. The methods bear similarities to a method of construction due to Rieger, which utilises continued fractions.

scholarly journals On shrinking targets for piecewise expanding interval maps

2015 ◽  
Vol 37 (2) ◽  
pp. 646-663 ◽  
Author(s):  
TOMAS PERSSON ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Gibbs Measure ◽  
Intersection Property ◽  
Interval Maps ◽  
Large Intersection ◽  
Set Of Points

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

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.

What's Between Q and R

1967 ◽  
Vol 60 (4) ◽  
pp. 308-314
Author(s):  
James Fey
Keyword(s):  
Mathematics Instruction ◽  
Rational Numbers ◽  
Complex Numbers ◽  
Valuable Insight ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Number Systems ◽  
Insight Into ◽  
Structure Properties

Among the objectives of school mathematics instruction, one of the most important is to develop understanding of the structure, properties, and evolution of the number systems. The student who knows the need for, and the technique of, each extension from the natural numbers through the complex numbers has a valuable insight into mathematics. Of the steps in the development, that from the rational numbers to the real numbers is the trickiest.

Arithmetical definability of field elements

1951 ◽  
Vol 16 (2) ◽  
pp. 125-126 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Characteristic Zero ◽  
Rational Numbers ◽  
Free Variable ◽  
The Other ◽  
Necessary Condition ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Fourth Root ◽  
Arithmetical Condition ◽  
Image Position

If F is a field, and α is an element of F, we say that α is arithmetically definable in F if there is a formula containing one free variable and any number of bound variables, involving only the concepts of elementary logic and the operations of addition and multiplication, which is satisfied by α and by no other element of F. The range of the bound variables is understood to be F. Without changing the sense of the above definition, we can allow in our formulas symbols for specific integers, or even (if F has characteristic zero) symbols for specific rational numbers, since these are arithmetically definable.As an example, consider the field F = R(2¼), obtained by adjoining the positive fourth root of 2 to the field R of rationals. Notice that 2¼ is not defined arithmetically by the formula x2 = 2, since this equation has two roots in F.However, 2¼ may be defined arithmetically by the equivalencewhere we have used the logical symbols ↔ (if and only if), ∨ (there exists), and ∧ (and). For the equation y4 = 2 is satisfied by no elements of F except y = ±2¼, and in both cases y2 = 2¼. On the other hand, 2¼ is not arithmetically definable in F, since there is an automorphism of F which takes 2¼ into −2¼, so that every arithmetical condition satisfied by 2¼ is also satisfied by −2¼.In any field F, a necessary condition for the arithmetical definability of an element α is that α should be fixed for all automorphisms of F. That this condition is not always sufficient is shown by considering the field of real numbers. Here there is no automorphism but the identity, but there can of course be but a denumerable infinity of arithmetically definable real numbers. Tarski has shown that only the algebraic numbers are arithmetically definable.

On Computable Numbers, with an Application to the Entscheidungsproblem (1936)

2004 ◽  
Author(s):  
Alan Turing
Keyword(s):  
Bessel Functions ◽  
Real Variable ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Large Classes ◽  
The Real ◽  
The Subject ◽  
Correct Application ◽  
Computable Functions

The ‘‘computable’’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the relations of the computable numbers, functions, and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine. In §§ 9, 10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable. In particular, I show that certain large classes of numbers are computable. They include, for instance, the real parts of all algebraic numbers, the real parts of the zeros of the Bessel functions, the numbers π, e, etc. The computable numbers do not, however, include all definable numbers, and an example is given of a definable number which is not computable. Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable. In § 8 I examine certain arguments which would seem to prove the contrary. By the correct application of one of these arguments, conclusions are reached which are superficially similar to those of Gödel. These results have valuable applications. In particular, it is shown (§ 11) that the Hilbertian Entscheidungsproblem can have no solution. In a recent paper Alonzo Church has introduced an idea of ‘‘effective calculability’’, which is equivalent to my ‘‘computability’’, but is very differently defined. Church also reaches similar conclusions about the Entscheidungsproblem.

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