The Main Conjecture on rational points near curved manifolds

Let $\mathscr{M}\subseteq\mathbb{R}^n$ be a compact manifold of dimension $m$ and co-dimension $c=n-m$. Let $\mathrm{N}_{\mathscr{M}}(\delta,Q)$ denote the relevant counting function for rational points within distance $\delta$ of $\mathscr{M}$ and of height at most $Q$. If $\mathscr{M}$ satisfies proper curvature conditions, then there exists a constant $c_{\mathscr{M}}>0$ such that

$$\mathrm{N}_{\mathscr{M}}(\delta,Q)\sim c_{\mathscr{M}}\delta^{c}Q^{m+1}\qquad(Q\rightarrow\infty)$$

for every fixed $\epsilon>0$, uniformly in the range

$$\delta\in(Q^{\epsilon-\frac{1}{c}},1/2).$$

Main Conjecture. The stated asymptotic holds under the proper curvature conditions above. This is presented as a basic open-ended conjecture motivated by the expected random distribution of rational points near manifolds; the precise curvature conditions remain to be determined.