scholarly journals Rational points near planar curves and Diophantine approximation

2015 ◽  
Vol 274 ◽  
pp. 490-515 ◽  
Author(s):  
Jing-Jing Huang
Keyword(s):  
Diophantine Approximation ◽  
Rational Points ◽  
Planar Curves

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 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 A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

2007 ◽  
Vol 49 (2) ◽  
pp. 367-375 ◽  
Author(s):  
DZMITRY BADZIAHIN ◽  
JASON LEVESLEY
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
General Class ◽  
Simultaneous Approximation ◽  
Convergence Result ◽  
Planar Curves ◽  
Planar Curve ◽  
Induced Measure

AbstractLet $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

scholarly journals A note on rational points near planar curves

2017 ◽  
Vol 177 (4) ◽  
pp. 393-396
Author(s):  
Sam Chow
Keyword(s):  
Rational Points ◽  
Planar Curves
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