Caltech/UCLA Joint Analysis Seminar
UCLA MS 6627
We will mainly talk about problems related to counting rational points in a thin neighborhood of a manifold. Using Fourier analytic methods, we have recently solved the case of hypersurfaces. A key step of the proof utilizes a duality relation between the counting functions for the hypersurface and its dual surface to bootstrap the counting bound. There are also significant applications of our results to diophantine inequalities and metric diophantine approximation on manifolds.