Counting rational points near smooth manifolds

Rajula Srivastava, University of Edinburgh

How many rational points with denominator of a given size lie within a certain distance from a compact, 'non-degenerate' manifold? This talk is about some recent progress towards answering this question. We shall describe how the geometric and analytic properties of the manifold play a key role in determining this count, and present a heuristic for the same. We shall then discuss how Fourier analytic techniques can be exploited to establish the desired asymptotic for rational points near manifolds satisfying certain strong geometric conditions. The key proof ingredient is a bootstrapping argument relying on Poisson summation, projective duality and oscillatory integral techniques.