Diophantine approximation on curves and the distribution of rational points: divergence theory

Abstract

In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are also proved in the inhomogeneous setting. Applications of the main theorem include the Khintchine-Jarn\'ik type theorem for divergence for arbitrary non-degenerate curve in $\mathbb{R}^n$.