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 $\epsilon \mathrm{/}Q$ of a fixed analytic nondegenerate manifold in ${\mathbb{R}}^n$, provided that $\epsilon \asymp Q^{-\tau }$ for some $\tau \le 3\mathrm{/}(2m+1)$, where $m$ is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range $\tau \le 1\mathrm{/}m$, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for $\tau \le 3\mathrm{/}(2n-1)$.