论文标题
在较少限制的曲率条件下，在接近紧凑型歧管的理性点数上
On the number of rational points close to a compact manifold under a less restrictive curvature condition
论文作者
论文摘要
令$ \ mathscr {m} $为$ \ mathbb {r}^{m} $的紧凑型submanifold。在本文中，我们建立了一个在给定距离至$ \ mathscr {m} $内的理性点数量的渐近公式，并在有界分母的假设下，假设$ \ mathscr {m} $符合一定的曲率条件。我们的结果概括了Schindler和Yamagishi的早期工作，因为我们的曲率条件是对它们使用的曲率的放松。我们能够恢复有关黄的猜想的类似结果，以及$ \ mathbb {r}^{m} $的紧凑型子策略的Serre尺寸生长猜想的类似物。
Let $\mathscr{M}$ be a compact submanifold of $\mathbb{R}^{M}$. In this article we establish an asymptotic formula for the number of rational points within a given distance to $\mathscr{M}$ and with bounded denominators under the assumption that $\mathscr{M}$ 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 $\mathbb{R}^{M}$.
