  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 $1/n$.
