A PROBLEM OF ERDÖS, SZÖSZ AND TURÁN CONCERNING DIOPHANTINE APPROXIMATIONS

2008 ◽  
Vol 04 (04) ◽  
pp. 691-708 ◽  
Author(s):  
FLORIN P. BOCA
Keyword(s):  
Lebesgue Measure ◽  
Complete Solution ◽  
Probabilistic Argument ◽  
Diophantine Approximations ◽  
Coprime Integers

For A > 0 and c > 1, let S(N, A, c) denote the set of those numbers θ ∈ ]0,1[ which satisfy [Formula: see text] for some coprime integers a and b with N < b ≤ cN. The problem of the existence and computation of the limit f(A, c) of the Lebesgue measure of S(N, A, c) as N → ∞ was raised by Erdös, Szüsz and Turán [3]. This limit has been shown to exist by Kesten and Sós [5] using a probabilistic argument and explicitly computed when Ac ≤ 1 by Kesten [4]. We give a complete solution proving directly the existence of this limit and identifying it in all cases.

scholarly journals Diophantine approximations with a constant right-hand side of inequalities on short intervals. 1

2021 ◽  
Vol 65 (5) ◽  
pp. 526-532
Author(s):  
V. I. Bernik ◽  
N. V. Budarina ◽  
E. V. Zasimovich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Short Intervals ◽  
Integer Polynomials ◽  
Right Hand ◽  
Diophantine Approximations

The problem of finding the Lebesgue measure 𝛍 of the set B1 of the coverings of the solutions of the inequality, ⎸Px⎹ <Q−w, w>n , Q ∈ N and Q >1, in integer polynomials P (x) of degree, which doesn’t exceed n and the height H (P) ≤ Q , is one of the main problems in the metric theory of the Diophantine approximation. We have obtained a new bound 𝛍B1 <c(n)Q−w+n, n<w<n+1, that is the most powerful to date. Even an ineffective version of this bound allowed V. G. Sprindzuk to solve Mahler’s famous problem.

scholarly journals Diophantine approximation with the constant right-hand side of inequalities on short intervals

2021 ◽  
Vol 65 (4) ◽  
pp. 397-403
Author(s):  
V. I. Bernik ◽  
D. V. Vasilyev ◽  
E. V. Zasimovich
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Short Intervals ◽  
Right Hand ◽  
Best Estimate ◽  
Diophantine Approximations

In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).

Perturbation of functions by the paths of a Lévy process

1989 ◽  
Vol 105 (2) ◽  
pp. 377-380 ◽  
Author(s):  
Steven N. Evans
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Stable Process ◽  
Levy Process ◽  
Probabilistic Argument ◽  
Scaling Properties ◽  
Measurable Set ◽  
Lebesgue Measurable Set ◽  
Image Position ◽  
Intermediate Value

In a recent paper Mountford [4] showed, using an ingenious probabilistic argument, that if X is a real-valued stable process with index α < 1 and f: [0, ∞) → ℝ is a non-constant continuous function, thenwhere we use the notation |A| for the Lebesgue measure of a Lebesgue measurable set A ⊂ ℝ. The argument in [4] appears to make strong use of the strict scaling properties of X and the ‘intermediate value’ property of f.

A Comparative Overview of the Fair Wear and Tear Exception: the Duty of Holders of Temporary Interests to Preserve Property

2002 ◽  
Vol 6 (1) ◽  
pp. 85-100
Author(s):  
Raffaele Caterina
Keyword(s):  
Common Core ◽  
Complete Solution ◽  
Roman Law ◽  
Private Ownership ◽  
Legal Systems ◽  
Personal Rights ◽  
Wear And Tear ◽  
Long Life ◽  
The Law ◽  
The Relationship

“A system of private ownership must provide for something more sophisticated than absolute ownership of the property by one person. A property owner needs to be able to do more than own it during his lifetime and pass it on to someone else on his death.”1 Those who own things with a long life quite naturally feel the urge to deal in segments of time. Most of the owner's ambitions in respect of time can be met by the law of contract. But contract does not offer a complete solution, since contracts create only personal rights. Certain of the owner's legitimate wishes can be achieved only if the law allows them to be given effect in rem—that is, as proprietary rights. Legal systems have responded differently to the need for proprietary rights limited in time. Roman law created usufruct and other iura in re aliena; English law created different legal estates. Every system has faced similar problems. One issue has been the extent to which the holder of a limited interest should be restricted in his or her use and enjoyment in order to protect the holders of other interests in the same thing. A common core of principles regulates the relationship between those who hold temporary interests and the reversioners. For instance, every system forbids holder of the possessory interest to damage the thing arbitrarily. But other rules are more controversial. This study focuses upon the rules which do not forbid, but compel, certain courses of action.

Diagnostic Fault Simulation for the Failure Analyst

2004 ◽  
Author(s):  
Dan Bodoh ◽  
Anthony Blakely ◽  
Terry Garyet
Keyword(s):  
Fault Diagnosis ◽  
Failure Analysis ◽  
Complete Solution ◽  
Fault Simulation ◽  
Building Blocks ◽  
Physical World ◽  
Design For Test ◽  
Fault Diagnosis System ◽  
Diagnosis Algorithm ◽  
Diagnosis Tool

Abstract Since failure analysis (FA) tools originated in the design-for-test (DFT) realm, most have abstractions that reflect a designer's viewpoint. These abstractions prevent easy application of diagnosis results in the physical world of the FA lab. This article presents a fault diagnosis system, DFS/FA, which bridges the DFT and FA worlds. First, it describes the motivation for building DFS/FA and how it is an improvement over off-the-shelf tools and explains the DFS/FA building blocks on which the diagnosis tool depends. The article then discusses the diagnosis algorithm in detail and provides an overview of some of the supporting tools that make DFS/FA a complete solution for FA. It also presents a FA example where DFS/FA has been applied. The example demonstrates how the consideration of physical proximity improves the accuracy without sacrificing precision.

On Ramsey Minimal Graphs

10.37236/1184 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz Łuczak
Keyword(s):  
Infinite Number ◽  
Probabilistic Argument ◽  
Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.

LEBESGUE MEASURE AND GAMBLING

1993 ◽  
Vol 19 (1) ◽  
pp. 40
Author(s):  
Kanovei ◽  
Linton
Keyword(s):  
Lebesgue Measure

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

Sign in / Sign up

Export Citation Format

Share Document