Let $u,v\in\{0,1\}^n$ be $0-1$ vectors with $n$ components.
Let $I=\langle u,v \rangle$. Clearly $I$ can take values in $\{0,1,\dots,n-1,n\}$.
How many different values can $$I'=\frac{\langle u,v \rangle}{\sqrt{\langle u,u \rangle\langle v,v \rangle}}$$ take?
The question is essentially, how many different angles can one make?
A calculation is shown here https://math.stackexchange.com/questions/1058925/number-of-different-normalized-inner-products stating the number is around $cn^{2.75}$. Is there a proof for this?