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 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 METRICAL RESULTS ON SYSTEMS OF SMALL LINEAR FORMS

2013 ◽  
Vol 09 (03) ◽  
pp. 769-782 ◽  
Author(s):  
M. HUSSAIN ◽  
S. KRISTENSEN
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
Linear Forms ◽  
Approximation Function

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine–Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximation function.

scholarly journals Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves

2016 ◽  
Vol 12 (08) ◽  
pp. 2173-2187 ◽  
Author(s):  
Mohammad Sadek
Keyword(s):  
Hypergeometric Series ◽  
Hyperelliptic Curves ◽  
Character Sums ◽  
Rational Points ◽  
Quadratic Character ◽  
Jacobian Varieties ◽  
Gaussian Hypergeometric Series

We study the character sums [Formula: see text] [Formula: see text] where [Formula: see text] is the quadratic character defined over [Formula: see text]. These sums are expressed in terms of Gaussian hypergeometric series over [Formula: see text]. Then we use these expressions to exhibit the number of [Formula: see text]-rational points on families of hyperelliptic curves and their Jacobian varieties.

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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