scholarly journals Explicit bounds for rational points near planar curves and metric Diophantine approximation

2010 ◽  
Vol 225 (6) ◽  
pp. 3064-3087 ◽  
Author(s):  
Victor Beresnevich ◽  
Evgeniy Zorin
Keyword(s):  
Diophantine Approximation ◽  
Rational Points ◽  
Planar Curves ◽  
Metric Diophantine Approximation ◽  
Explicit Bounds

scholarly journals Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan)

2007 ◽  
Vol 166 (2) ◽  
pp. 367-426 ◽  
Author(s):  
Victor Beresnevich ◽  
Detta Dickinson ◽  
Sanju Velani ◽  
Robert Vaughan
Keyword(s):  
Diophantine Approximation ◽  
Rational Points ◽  
Planar Curves

scholarly journals An Inhomogeneous Jarník type theorem for planar curves

2016 ◽  
Vol 163 (1) ◽  
pp. 47-70 ◽  
Author(s):  
DZMITRY BADZIAHIN ◽  
STEPHEN HARRAP ◽  
MUMTAZ HUSSAIN
Keyword(s):  
Diophantine Approximation ◽  
Type Theorem ◽  
Planar Curves ◽  
Metric Diophantine Approximation ◽  
Substantial Progress

AbstractIn metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental to each of them. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximation on manifolds for each of the classes above. In particular, both Khintchine and Jarník-type results have been established for approximation on planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves in the setting of dual approximation and in so doing complete the metric theory of Diophantine approximation on planar curves.

scholarly journals A Jarník type theorem for planar curves: everything about the parabola

2015 ◽  
Vol 159 (1) ◽  
pp. 47-60 ◽  
Author(s):  
MUMTAZ HUSSAIN
Keyword(s):  
Diophantine Approximation ◽  
Type Theorem ◽  
Rational Numbers ◽  
Real Numbers ◽  
Comprehensive Description ◽  
Planar Curves ◽  
Metric Diophantine Approximation ◽  
Comprehensive Study

AbstractThe well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.

scholarly journals Exponents of Diophantine Approximation in Dimension Two

2009 ◽  
Vol 61 (1) ◽  
pp. 165-189 ◽  
Author(s):  
Michel Laurent
Keyword(s):  
Diophantine Approximation ◽  
Necessary Conditions ◽  
Rational Points ◽  
Transference Principle ◽  
Linearly Independent ◽  
Quality Of Approximation

Abstract. Let Θ = (α, β) be a point in R2, with 1, α, β linearly independent over Q. We attach to Θ a quadruple Ω (Θ) of exponents that measure the quality of approximation to Θ both by rational points and by rational lines. The two “uniform” components of Ω (Θ) are related by an equation due to Jarník, and the four exponents satisfy two inequalities that refine Khintchine's transference principle. Conversely, we show that for any quadruple Ω fulfilling these necessary conditions, there exists a point Θ ∈ R2 for which Ω (Θ) = Ω.

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