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A Proof of Lemma 3.3
Definition A.1. For any $\delta > 0$, $S \subseteq \mathbb{R}^n$, we say that $A \subseteq S$ is a $\delta$-net of $S$ if for each $\mathbf{v} \in S$, there is some $\mathbf{u} \in A$ such that $|\mathbf{u} - \mathbf{v}| \le \delta$.
Lemma A.2. For any $\delta > 0$, there exists a $\delta$-net of the unit sphere in $\mathbb{R}^n$ with at most $(1 + 2/\delta)^n$ points.
Proof. Let $N$ be maximal such that $N$ points can be placed on the unit sphere in such a way that no pair of points is within distance $\delta$ of each other. Clearly, there exists a $\delta$-net of size $N$.
So, it suffices to show that any collection of vectors $A$ in the unit sphere with $|A| > (1 + 2/\delta)^n$ must contain two points within distance $\delta$ of each other. Let
be the union of balls of radius $\delta/2$ centered at each point in $A$. Notice that $B \subseteq (1 + \delta/2)B_2^n$. If all of these balls were disjoint, then we would have
a contradiction. Therefore, two such balls must overlap. I.e., there must be two points within distance $\delta$ of each other, as needed. $\square$
We will need the following result from [Ver12, Lemma 5.4].
Lemma A.3. For a symmetric matrix $M \in \mathbb{R}^{n \times n}$ and a $\delta$-net of the unit sphere $A$ with $\delta \in (0, 1/2)$,