On the number of rational points close to a compact manifold under a less restrictive curvature condition

2022/05/12 by Florian Maximilian Munkelt, Munkelt, Florian
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.2205.06183

openalex publication_date 2022/05/12 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

Abstract

Let \mathscrM be a compact submanifold of ℝ^M. In this article we establish an asymptotic formula for the number of rational points within a given distance to \mathscrM and with bounded denominators under the assumption that \mathscrM fulfills a certain curvature condition. Our result generalizes earlier work from Schindler and Yamagishi, as our curvature condition is a relaxation of that used by them. We are able to recover a similar result concerning a conjecture by Huang and a slightly weaker analogue of Serre's dimension growth conjecture for compact submanifolds of ℝ^M.

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