  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.
