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

TL;DR

This paper introduces a new explicit method for estimating the number of rational points near non-degenerate curves, extending previous results and including inhomogeneous cases, with applications to divergence theorems in Diophantine approximation.

Contribution

It develops an explicit approach to count rational points near curves, generalizing prior analytic results and applying to divergence theorems in higher dimensions.

Findings

Optimal lower bounds on rational points near curves

Extension to inhomogeneous Diophantine approximation

Application to Khintchine-Jarník divergence theorem

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 $\mathbb{R}^n$.