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). 



Angles in an integral lattice
The proof that $\pi/3$ is not a lattice angle is due to Lucas, while the proof that an $n$-gon does not embed in $\mathbb Z^d$ is due to Schoenberg, and has a nice proof by Scherrer. There is a AMM article on the topic you ask about ["Triangles with vertices on lattice points"](http://www.jstor.org/pss/2325060), which has some history and references, too. Also, note that except for the Euclidean approach, there is another geometric approach to lattice angles related to continued fractions. Oleg Karpenkov has some nice [notes](http://www.geometrie.tugraz.at/karpenkov/cf2009w.html) on this.


