Rational points near planar curves and Diophantine approximation

2014/03/28 by Jing-Jing Huang, Huang, Jing-Jing · 1 citation
Mathematics · #11J83 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11J83

paper · pdf · doi:10.48550/arxiv.1403.7388

27 pages, corrected typos, to appear in Adv. Math

openalex publication_date 2014/03/28 · arxiv created 2015/02/09 · arxiv updated 2015/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• Rational points near manifolds and Khintchine theorem

Related

• Explicit bounds for rational points near planar curves and metric Diophantine approximation
• Diophantine approximation on curves and the distribution of rational points: divergence theory
• A note on rational points near planar curves
• Diophantine approximation on planar curves and the distribution of rational points
• Rational points near planar flat curves