The density of rational points near hypersurfaces

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.