Diophantine approximation on planar curves and the distribution of rational points

Details Diophantine approximation on planar curves and the distribution of rational points Diophantine approximation on planar curves and the distribution of rational points

2004-01-14

arxiv.org/abs/math/0401148v2

Mathematics

Number Theory

With an Appendix by Bob Vaughan: Sums of two squares near perfect squares

Scientific paper

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

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