Density of rational points on manifolds and Diophantine approximation on hypersurfaces

Abstract

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 $\psi$-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 $\{ (\mathbf{x}, f(\mathbf{x})) \in \mathbb{R}^{d+1}: \mathbf{x} \in B_{\nu}(\mathbf{x}_0) \}$ with non-zero Hessian matrix of $f$ at $\mathbf{x}_0$ and $\nu > 0$ sufficiently small.