scholarly journals On the abc conjecture and diophantine approximation by rational points

2000 ◽  
Vol 122 (4) ◽  
pp. 843-872 ◽  
Author(s):  
Paul Vojta
Keyword(s):  
Diophantine Approximation ◽  
Rational Points ◽  
Abc Conjecture

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.

scholarly journals Diophantine approximation on the parabola with non-monotonic approximation functions

2019 ◽  
Vol 168 (3) ◽  
pp. 535-542
Author(s):  
JING–JING HUANG
Keyword(s):  
Diophantine Approximation ◽  
Character Sums ◽  
Rational Points ◽  
Approximation Function

AbstractWe show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

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 Distribution locale des points rationnels de hauteur bornée sur une surface de del Pezzo de degré 6

2017 ◽  
Vol 13 (07) ◽  
pp. 1895-1930
Author(s):  
Zhizhong Huang
Keyword(s):  
Asymptotic Distribution ◽  
Diophantine Approximation ◽  
Rational Point ◽  
Rational Points ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Del Pezzo

We establish a measure which describes precisely the local asymptotic distribution of rational points outside the locally accumulating subvarieties around a general rational point on a del Pezzo surface of degree 6 in the sense of diophantine approximation.

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