Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?

Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the unit sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-\epsilon)^n\to0$.

So for any fixed subdivision of the unit sphere's surface into a finite number of little areas, the probability that each area has at least one of the vectors approaches $1$.

So for larger $n$, it doesn't matter which direction $\sum X_j$ points in, because you can find one of those vectors with a similar direction and do the dot product with that.

The only question is, what is the expectation of $||\sum X_j||_2$? Obviously it is $\sqrt{n}$.

Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?

Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the unit sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-\epsilon)^n\to0$.

So for any fixed subdivision of the unit sphere's surface into a finite number of little areas, the probability that each area has at least one of the vectors approaches $1$.

So for larger $n$, it doesn't matter which direction $\sum X_j$ points in, because you can find one of those vectors with a similar direction and do the dot product with that.

The only question is, what is the expectation of $||\sum X_j||_2$?

Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?

Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-\epsilon)^n\to0$.

So for any fixed subdivision of the sphere's surface into a finite number of little areas, the probability that each area has at least one of the vectors approaches $1$.

So for larger $n$, it doesn't matter which direction $\sum X_j$ points in, because you can find one of those vectors with a similar direction and do the dot product with that.

The only question is, what is the expectation of $||\sum X_j||_2$? Obviously it is $\sqrt{n}$.

Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?

Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the unit sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-\epsilon)^n\to0$.

So for any fixed subdivision of the unit sphere's surface into a finite number of little areas, the probability that each area has at least one of the vectors approaches $1$.

So for larger $n$, it doesn't matter which direction $\sum X_j$ points in, because you can find one of those vectors with a similar direction and do the dot product with that.

The only question is, what is the expectation of $||\sum X_j||_2$? Obviously it is $\sqrt{n}$.