Let $d\geq 1$ be a fixed integer, and $\mathbb{Z}^d$ be the lattice of all integers; consider the set $$A_d:= \{ \alpha \mbox{ such that } \alpha=\frac{v \cdot w}{\|v\| \|w\|} , \mbox{ for some } \ v,w \in \mathbb{Z}^d \}$$ Of course $A_d$ is a set of quadratic irrationals and $\overline{A_d}=[-1,1]$.

Question: is it possible to caracterize all values that belong to $A_d$?

In other words, can one caracterize all angles which occour between two integer vectors? For instance: are there two integer vectors in the plane which form an angle of $\pi/3$?

PS: this problem comes from the question of a collegue, who wants to cook different versions of a linear algebra test for undergraduate students in such a way that computations are different, but equally simple (I guess the question has already been raised and solved - but I am not an expert).

Let $d\geq 1$ be a fixed integer, and $\mathbb{Z}^d$ be the lattice of all integers; consider the set $A_d \subset [-1,1]$ defined by $$ \alpha \in A_d \ \iff \ \alpha=\frac{v \cdot w}{\|v\| \|w\|} \mbox{ for some } \ v,w \in \mathbb{Z}^d $$ Of course $A_d$ is a set of quadratic irrationals and $\overline{A_d}=[-1,1]$.

Question: is it possible to caracterize all values that belong to $A_d$?

In other words, can one caracterize all angles which occour between two integer vectors? For instance: are there two integer vectors in the plane which form an angle of $\pi/3$?

PS: this problem comes from the question of a collegue, who wants to cook different versions of a linear algebra test for undergraduate students in such a way that computations are different, but equally simple (I guess the question has already been raised and solved - but I am not an expert).