Lemma: Directional resolution

The normalized directions $p/\Vert p{\Vert }_2$, $p\in P_n$, have angular covering error tending to zero. One sufficient bound is

Proof. For unit $v$, round $(K_n-1)v$ to a nearest integer vector $p$. The rounding error has Euclidean norm at most $\sqrt{2}/2<1$. Thus $p\ne 0$, $\Vert p{\Vert }_2\le K_n$, and its component in direction $v$ is at least $K_n-2$. The transverse component has norm less than one. Its angular error is therefore at most $\arctan (1/(K_n-2))$, which is bounded by the displayed quantity.