Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals

Let $\mathcal{M}\subset \mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\mathcal{M}$ is nowhere completely flat. Let $N_{\mathcal{M}}(\delta,Q)$ denote the number of rational vectors $\mathbf{a}/q$ within a distance of $\delta/q$ from $\mathcal{M}$ so that $q \in [Q,2Q)$. We develop a novel method to analyse $N_{\mathcal{M}}(\delta,Q)$. The salient feature of our technique is the combination of powerful quantitative non-divergence estimates, in a form due to Bernik, Kleinbock, and Margulis, with Fourier analytic tools. The second ingredient enables us to eschew the Dani correspondence and an explicit use of the geometry of numbers. We employ this new method to address in a strong sense a problem of Beresnevich regarding lower bounds on $N_{\mathcal{M}}(\delta,Q)$ for non-analytic manifolds. Additionally, we obtain asymptotic formulae which are the first of their kind for such a general class of manifolds. As a by-product, we improve upon upper bounds on $N_{\mathcal{M}}(\delta,Q)$ from a recent breakthrough of Beresnevich and Yang and recover their convergence Khintchine type theorem for arbitrary nondegenerate submanifolds. Moreover, we obtain new Hausdorff dimension and measure refinements for the set of well-approximable points for a range of Diophantine exponents close to $1/n$.