For $\Sigma\geq 0$ a $k\times k$ matrix and large $n$, fix $E:= \{(x_i)_{i=1}^n: \sum_i x_i^\dagger \Sigma x_i \leq n\}$. Fix $(z_m)_m$ as $M$ points iid uniform on $\mathbb{S}^{nk-1}\subset \mathbb{R}^{nk}.$ How large must $M$ be for $\cup_m (z_m+ E)$ to likely cover $\mathbb{S}^{nk-1}$?

Similarly, how small must $M$ be so that overlaps, $\cup_{(i,j)}(z_i+E)\cap (z_j+E)$, normalized to $\mathbb{S}^{nk-1}$'s area, likely have small volume?