Rational points near planar curves and Diophantine approximation

Huang, Jing–Jing

doi:10.1016/j.aim.2015.01.013

Cited by 34 publications

(41 citation statements)

References 22 publications

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“…The novel feature of this paper is that we adapt a counting result of the same type due to Huxley [25], which is essentially best possible, into an analytic form that is particularly suited for the application in Schmidt's method. This connection is established through the use of the dual curve, which appears, in one form or another, in the previous works [1,11,15,23,25,32].…”

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confidence: 99%

Hausdorff theory of dual approximation on planar curves

Huang

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Ten years ago, Beresnevich-Dickinson-Velani [10] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in R 2 , which represents the first complete theory of its kind for a general class of manifolds.

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confidence: 99%

Hausdorff theory of dual approximation on planar curves

Huang

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Indeed, it corresponds to the classical Jarník-Besicovitch theorem. For non-degenerate curves in R 2 , as a result of various works [3,13,19,30], we know that (1.5) holds for all τ ∈ [ 1 2 , 1). For completeness it is worth pointing out that non-degenerate planar curves are characterised by being C 2 and having non-zero curvature.…”

Section: Introductionmentioning

confidence: 98%

Diophantine Approximation on Manifolds and Lower Bounds for Hausdorff Dimension

et al. 2017

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Abstract. Given n ∈ N and τ > 1/n, let S n (τ ) denote the classical set of τ -approximable points in R n , which consists of x ∈ R n that lie within distance q −τ −1 from the lattice (1/q)Z n for infinitely many q ∈ N. In pioneering work, Kleinbock and Margulis showed that for any non-degenerate submanifold M of R n and any τ > 1/n almost all points on M are not τ -approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set M ∩ S n (τ ). In this paper we suggest a new approach based on the Mass Transference Principle of Beresnevich and Velani [A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971-992], which enables us to find a sharp lower bound for dim M ∩ S n (τ ) for any C 2 submanifold M of R n and any τ satisfying 1/n τ < 1/m. Here m is the codimension of M. We also show that the condition on τ is best possible and extend the result to general approximating functions. §1. Introduction. Throughout ψ : N → R + will denote a monotonic function that will be referred to as an approximating function, n ∈ N and S n (ψ) will be the set of ψ-approximable points y ∈ R n , that is points y = (y 1 , . . . , y n )

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“…We compare Theorem 1.1 with pointwise bounds in α 3 that can be obtained from work of Beresnevich, Dickinson and Velani [1] or alternatively Huang [6]. Indeed, the inequality (1.1) can be reformulated as a problem of finding rational points near the planar curve C ⊂ R 2 given by 1 − α 2 y k 2 − α 3 y k 3 = 0.…”

Section: Introductionmentioning

confidence: 99%