Rational points near planar curves and Diophantine approximation

TL;DR

This paper derives precise asymptotic formulas for counting rational points near planar curves and advances the Lebesgue theory of Diophantine approximation on such curves, extending prior foundational results.

Contribution

It unifies and extends existing results on rational points near planar curves and completes the Lebesgue theory for Diophantine approximation on weakly non-degenerate curves.

Findings

Optimal error asymptotics for rational points near curves

Unified framework extending previous results

Complete Lebesgue theory for Diophantine approximation on curves

Abstract

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich-Zorin in the divergence case.