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

Abstract

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.

On weighted inhomogeneous Diophantine approximation on planar curves

“…This is Lemma 4 in [6]. For s = 1 the following is its generalization in terms of general Hausdorff measure and a consequence of Corollary 3 in [5].…”

“…As a consequence of this definition there is a series of measure theoretic results established in [5,6]. We will only be exploiting the above definition in the framework below.…”

“…It is a natural generalization of the homogeneous ubiquity statement established in [6] (Corollary 5).…”

“…In this section, for the sake of simplicity, we introduce a restricted form of ubiquity to deal with Diophantine approximation on planar curves. We refer to [6] for detailed acquisition of the ubiquity framework and only present a restricted version of it for our use. Note that so far the ubiquity framework has been used for one approximating function.…”

“…A combination of results established in [6] and [20] provides a complete analogue of the Khintchine-Jarník theorem for a set C f ∩ A(ψ). For weighted sets C f ∩ A(ψ 1 , ψ 2 ) within the homogeneous setup the results given in [2] and [10] provide only a Lebesgue measure analogue of the Khintchine-Jarník theorem.…”