How close can rational points get to a manifold?

We prove the heuristically predicted lower bound for the number of rational points of height at most $Q$ lying within $ε/Q$ of a fixed analytic nondegenerate manifold in $\mathbb{R}^n$, provided that $ε\asymp Q^{-τ}$ for some $τ\leq 3/(2m+1)$, where $m$ is the codimension of the manifold. Our result