A Fourier Analytic Approach Towards Multiplicative Diophantine Approximation for Manifolds

How many rational points with a denominator of a given size lie inside a non-isotropic neighborhood of a manifold of arbitrary dimension? This talk presents recent progress toward answering this question, based on joint work with Sam Chow, Niclas Technau, and Han Yu. Our results resolve the convergence case of multiplicative Diophantine approximation for manifolds, both flat and nondegenerate. We shall focus on the latter case and describe a new approach to the problem, combining tools from harmonic analysis (oscillatory integrals) with methods from homogeneous dynamics (quantitative non-divergence estimates).