Let $K$ be a number field of degree $n$. Let also $\mathcal{O}(K)$ be the ring of integers of $K$ and let $B$ be a basis for $\mathcal{O}(K)$. Given $q\in K$ we write $\mathbf{q}$ for the column vector $\begin{bmatrix} q_0 & ... & q_{n-1} \end{bmatrix}^T$ associated to $q$ w.r.t. the basis $B$. We also write $A_q$ for the matrix corresponding to the linear transformation given by multiplication by $q$ on $K$ with respect to the basis $B$.
Fix an algebraic integer $q$. Is it true that, for each residue class, there is a unique representative $r$ of that class s.t. $(A_q^{-1})\mathbf{r} \in [0,1]^n$?