Density of rational points on manifolds and Diophantine approximation on hypersurfaces

TL;DR

This paper advances understanding of rational point density on manifolds and hypersurfaces, proving new bounds and convergence results that address longstanding conjectures in Diophantine approximation.

Contribution

It establishes an analogue of the dimension growth conjecture for submanifolds with curvature and solves the generalized Baker-Schmidt problem for generic hypersurfaces.

Findings

Optimal upper bounds for rational points near curved manifolds

Convergence results for $oldsymbol{\psi}$-approximable points on hypersurfaces

Resolution of the generalized Baker-Schmidt problem in the generic case

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.