Corollary: Uniform limit of optimal lattice arrival times

For $j>\kappa$ and $b\in {\Lambda }_j$,

Consequently, for every $L>0$,

For rounded continuum endpoints $x_j={\mathsf{Q}}_{j}x$, the same conclusion holds uniformly for ${\Vert x\Vert }_2\le L$, after adding the endpoint allowance ${\epsilon }_j/c$.

Proof. The lower bound is Corollary cor:speed; the explicit route gives the upper bound. Lemma lem:routing supplies the uniform estimate. For rounded targets, $\left|{\Vert x_j\Vert }_2-{\Vert x\Vert }_2\right|\le {\epsilon }_j$.