Diophantine approximation onplanar curves andthe distribution of rational points

By Victor Beresnevich, Detta Dickinson, Sanju Velani. 2004

76 citations recorded by the source corpus.

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6f783c6cc47e24eebb06b31542d05943e60b0423

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Citations in the corpus

Exponents of Diophantine approximation (2006, result): “We display an immediate consequence of results by Beresnevich, Dickinson, Vaughan and Velani [10, 79].”
Diophantine approximation on planar curves: the convergence theory (2006, methodology): “In [1], the following result is established.”
Metric Diophantine Approximation : aspects of recent work (2016, background): “5) for planar curves was established in [14].”