Theorem: Local event approximation and cone convergence

The event grids become locally dense. If $a_n\to a=(t_a,x_a)$, then

locally, uniformly for apexes in bounded sets.

If $e_n=d_c(a_n,a)$, then the excess of $J_n^{-}(a_n)\cap K$ over $J_2^{-}(a)$ is at most $e_n$. For a bounded $K$, the reverse excess is at most

for a fixed sufficiently large $R_K$ and all large $n$.

Proof. The grid $\Delta t_n\mathbb{Z}\times h_n{\mathbb{Z}}^D$ approximates every point with error at most $\max ({\epsilon }_n/2,\sqrt{D}{h}_n/2)$ in $d_c$.

The exact upper word-ball containment gives $J_n^{-}(a_n)\subseteq J_2^{-}(a_n)$. Translate a point by $a-a_n$; it then lies in $J_2^{-}(a)$, at distance at most $e_n$.

Conversely, translate $p\in J_2^{-}(a)\cap K$ by $a_n-a$ to a point $p^{\prime }\in J_2^{-}(a_n)$. Round its time downward to the tick grid. The time error is less than $\Delta t_n$, and its spatial coordinate lies in the Euclidean ball of radius $k{\epsilon }_n=c(T_n\Delta t_n-s\Delta t_n)$. For bounded $K$ and bounded apexes, these radii are bounded by a fixed $R_K$. Theorem thm:shape gives a word-ball point with spatial error at most $(D+1){\epsilon }_n+2{\delta }_n{R}_K$. Together with the translation and time rounding this gives the bound.