论文标题
在高曲面上的歧管上有理点的密度和双磷酸近似值的密度
Density of rational points on manifolds and Diophantine approximation on hypersurfaces
论文作者
论文摘要
在本文中，我们建立了维度生长猜想的类似物，该模型涉及投射品种上的理性点的密度，用于$ \ mathbb {r}^n $的紧凑型子手机和非呈呈曲率。我们还为同时$ψ$ approximable的积分建立了融合理论，该点位于通用性超表面上，从而在同时设置通用的hypersurfaces中解决了广义的贝克 - 施密特问题。这些结果是作为最佳上限的后果而获得的。使用$ f $的非零Hessian矩阵，$ \ mathbf {x} _0 $和$ν> 0 $足够小。
In this article, we establish an analogue of the dimension growth conjecture, which is regarding the density of rational points on projective varieties, for compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also establish the convergence theory for the set of simultaneously $ψ$-approximable points lying on a generic hypersurface, thereby settling the generalized Baker-Schmidt problem in the simultaneous setting for generic hypersurfaces. These results are obtained as consequences of an optimal upper bound for the density of rational points near manifolds of the form $\{ (\mathbf{x}, f(\mathbf{x})) \in \mathbb{R}^{d+1}: \mathbf{x} \in B_ν(\mathbf{x}_0) \}$ with non-zero Hessian matrix of $f$ at $\mathbf{x}_0$ and $ν> 0$ sufficiently small.
