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

In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For $n \geq 3$, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in $\mathbb{R}^n$.