in my textbook the author uses the notation <x>, whereas x is some vector in $\displaystyle \mathbb{R}^n$ without any comment.

Is it the same as $\displaystyle \left||x\right||$, that is the norm of the vector?
The whole statement is:
If P is a polynomial of order m on $\displaystyle \mathbb{R}^n$. then for almost all $\displaystyle \alpha \in [0,1]^n $there are constants C,N s.t.
\left| P(k+\alpha)^{-1}\right| \leq C*<k>^N \ \forall k\in \mathbb{Z}^n