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

Abstract

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

Authors

Victor Beresnevich

Department of Mathematics
University of York
Heslington, York YO10 5DD
United Kingdom

Detta Dickinson

Mathematics Department
National University of Ireland
Maynooth, Co. Kildare
Ireland

Sanju Velani

Department of Mathematics
University of York
Heslington,York YO10 5DD
United Kingdom

R. C. Vaughan

Mathematics Department
Penn State University
State College, PA 16802
United States