The Main Conjecture on rational points near curved submanifolds

Let $\mathcal{M}$ be a compact $m$-dimensional submanifold of $\mathbb{R}^n$ with codimension $k=n-m$ and proper curvature conditions. Let $N_{\mathcal{M}}(Q,\delta)$ count rational points near $\mathcal{M}$ with denominator at most $Q$. The Main Conjecture. For some constant $c_{\mathcal{M}}$, one should have

$$N_{\mathcal{M}}(Q,\delta)\sim c_{\mathcal{M}}\delta^kQ^{m+1}$$

when $\delta\ge Q^{-\frac{1}{k}+\varepsilon}$ for some $\varepsilon>0$ and $Q\to\infty$. The threshold is suggested by examples such as the parabola and hypersphere, where the heuristic asymptotic fails for smaller $\delta$; the conjecture is presented as plausible, but its general status is not resolved in the supplied text.