# Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan) | Annals of Mathematics

Abstract Let  be a nondegenerate planar curve and for a real, positive decreasing function ψ let (ψ) denote the set of simultaneously ψ-approximable points lying on . We show that  is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on  of (ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that  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 (ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

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### 数学年刊（Annals of Mathematics）

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