Sum of dilates in vector spaces

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

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