Rational Points in Hyperbolic Neighborhoods of Manifolds

Are rational points distributed randomly near curved manifolds? After introducing a simple random model, we focus on this question in hyperbolic regions, which arise naturally in multiplicative Diophantine approximation. We then sketch how tools from homogeneous dynamics and harmonic analysis can be used to answer this question. Based on a forthcoming work with Sam Chow, Rajula Srivastava, and Han Yu.