On weighted inhomogeneous Diophantine approximation on planar curves

Hussain, Mumtaz; Yusupova, T. N.

doi:10.1017/s0305004112000497

Cited by 9 publications

(7 citation statements)

References 19 publications

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“…A complete metric theory for this set has already been established some time ago. In particular, the Lebesgue measure of the set W (ϕ 1 , ϕ 2 ) has been established in [13], the Hausdorff measure for W (ϕ, ϕ) in [4] and the Hausdorff measure for W (ϕ 1 , ϕ 2 ) follows from [12]. However, hardly anything is known for the set E(T β , f, g).…”

Section: Introductionmentioning

confidence: 99%

Two-Dimensional Shrinking Target Problem in Beta-Dynamical Systems

Hussain

Wang

2017

Bull. Aust. Math. Soc.

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In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let β > 1 be a real number and define the β-transformation on [0, 1] by T β : x → βx mod 1. Let Ψ i (i = 1, 2) be two positive functions on N such that Ψ i → 0 when n → ∞. We determine the Lebesgue measure and Hausdorff dimension for the lim sup set2010 Mathematics subject classification: primary 11K55; secondary 11J83, 28A80, 37F35.

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Section: Introductionmentioning

confidence: 99%

Two-Dimensional Shrinking Target Problem in Beta-Dynamical Systems

Hussain

Wang

2017

Bull. Aust. Math. Soc.

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“…We remark that by adapting the arguments in this paper and the ideas developed in [1,8,16], it is possible to prove the divergent part (stated above). As a first step to solving the above problem Beresnevich and Velani in [7] has determined the Hausdorff dimension of the set C f ∩ A(ψ, θ).…”

Section: Discussionmentioning

confidence: 83%

“…Subsequently, Badziahin and Levesley ( [2]) extended this to the case of arbitrary C (3) non-degenerate curves and to Hausdorff measures. Recently, Hussain and Yusupova ( [16]) generalized previously existing results to Inhomogeneous framework. Theorem 3.…”

Section: Discussionmentioning

confidence: 96%

On weighted inhomogeneous Diophantine approximation on planar curves

Hussain,

Yusupova

2012

Preprint

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“…In another direction, one can also consider the inhomogeneous Diophantine approximation on manifolds, which is considered to be the generalization of the homogeneous theory discussed so far. As in the homogeneous setup the inhomogeneous theory is almost complete for the simultaneous setup for non-degenerate planar curves-see [1,15,27] and the references therein. But for the dual setup, the best available result is recently established by Badziahin, Beresnevich and Velani for the s-dimensional analogue of Theorem 4 under some mild convexity conditions on the approximating function-see [2, theorem 2] for further details.…”

Section: Final Comments and Further Researchmentioning

confidence: 99%

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

Hussain¹

2015

Math. Proc. Camb. Phil. Soc.

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The 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.1.1. Dual Diophantine approximation. To set the scene for the problems considered in this paper, we first recall the fundamental results in the theory of dual Diophantine approximation. Let ψ : R + → R + denote a real positive decreasing function. We refer to ψ as an approximating function. Define the set W (ψ) := x = (x 1 , . . . , x n ) ∈ R n : |a 0 + a 1 x 1 + · · · + a n x n | < ψ(|a|)where 'i.m.' stands for 'infinitely many' and |a| = max{|a 1 |, . . . , |a n |} is the standard supremum norm. A vector x ∈ W (ψ) will be called ψ−approximable. In the case ψ(r) = r −τ for some τ > 0 we also say that x is τ −approximable and denote W (ψ) by W (τ ). The first significant result in the theory is Dirichlet's theorem which tells us that W (n) = R n .The following fundamental result provides a beautiful criterion for the 'size' of the set W (ψ) in terms of n-dimensional Lebesgue measure | · | n .

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On weighted inhomogeneous Diophantine approximation on planar curves

Cited by 9 publications

References 19 publications

Two-Dimensional Shrinking Target Problem in Beta-Dynamical Systems

Two-Dimensional Shrinking Target Problem in Beta-Dynamical Systems

On weighted inhomogeneous Diophantine approximation on planar curves

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

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