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

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