Sum of dilates in vector spaces

Let $d\ge 2$, $A\subset {\mathbb{Z}}^d$ be finite and not contained in a translate of any hyperplane, and $q\in \mathbb{Z}$ such that $\mathrm{\mid }q\mathrm{\mid }>1$. We show $$\mathrm{\mid }A+q\cdot A\mathrm{\mid }\ge (\mathrm{\mid }q\mathrm{\mid }+d+1)\mathrm{\mid }A\mathrm{\mid }-O_{q,d}(1)\mathrm{.}$$