It is well-know that a $\Bbb R$-linear map $f : \Bbb R^n \to \Bbb R^n$ is injective iff it is surjective.
I am wondering, what happens if $f$ is only additive (i.e. $\Bbb Z$-linear, or equivalently, $\Bbb Q$-linear) ? Does injectivity imply surjectivity, or does the converse hold ? The only example of non-linear additive map I know is given by a projection $\Bbb R \cong \bigoplus_{x \in \Bbb R} \Bbb Q \to \Bbb Q \subset \Bbb R$, neither surjective nor injective...
Similarly, what happens if we assume that $f$ is homogeneous, i.e. $f(ax)=af(x)$ for any $a \in \Bbb R,x \in \Bbb R^n$ ? The only example I have is for $n=2$, and $f(x,y) = (x 1_{xy > 0}, 0)$, neither surjective nor injective...