The density of rational points near hypersurfaces

TL;DR

This paper derives a precise asymptotic count of rational points near hypersurfaces, introducing a novel bootstrap method combining advanced analytical techniques, and applies it to solve longstanding conjectures and approximation problems.

Contribution

It presents a new bootstrap approach that combines Poisson summation, projective duality, and stationary phase to count rational points near hypersurfaces, extending existing theories.

Findings

Established a sharp asymptotic formula for rational points near hypersurfaces.

Proved an analogue of Serre's Dimension Growth Conjecture in this context.

Derived an optimal Jarník type theorem for simultaneous approximation on hypersurfaces.

Abstract

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre's Dimension Growth Conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we obtain an optimal Jarn\'{i}k type theorem for simultaneous approximation on hypersurfaces.