I am having difficulty proving that a unit ball can be approximated with a set of finite vectors. Specifically, I want to bound the error of the following approximation.
Let $D$ be a uniform distribution over an $d$-dimensional unit ball and $D'$ be a the uniform distribution over $\{-1,0,1\}^d$. Then show that, for any vector $a$ sampled from $D$, with high probability there exists a non-empty set of vectors $$ V = \left\{v \left| \frac{v\cdot a}{\|v\|\ \|a\|} \geq \frac{1}{\sqrt{d}}\right. \right\} $$ sampled from $D'$. Moreover, for $A = \sum_{v \in V} v$, show that $\frac{A\cdot a}{\|A\|\|a\|}$ is close to 1.
In high dimensions, any two vectors picked at random are orthogonal with high probability. So, even though each vector in $V$ has dot product at least $\frac{\|v\|\|a\|}{\sqrt{d}}$ with $a$, their average might be closer to $a$. I want to bound this error. Pointers to any existing results in the high dimensional probability for such approximations would be helpful too.