Density of rational points near/on compact manifolds with certain curvature conditions

2021/03/09 by D. Schindler, Schindler, D., S. Yamagishi +1 · 3 citations
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.05281

openalex publication_date 2021/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold M of ℝ^M with a certain curvature condition. Our result generalises earlier work of Huang for hypersurfaces [J.-J. Huang, The density of rational points near hypersurfaces, Duke Math. J. 169 (2020), 2045--2077.], as our curvature condition reduces to Gaussian curvature being bounded away from 0 when M - dim M = 1. An interesting feature of our result is that the asymptotic formula holds beyond the conjectured range of the distance to M. Furthermore, we obtain an upper bound for the number of rational points on M with additional power saving to the bound in the analogue of Serre's dimension growth conjecture for compact submanifolds of ℝ^M when M - dim M > 1.

Cited by

• Counting rational points near manifolds: a refined estimate, a conjecture and a variant
• Lattice points in thickened parabolas and rational points near hypersurfaces
• Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

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