Sum of dilates in vector spaces

Abstract

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 \(|q| \geq 2\). We show \[|A+ q \cdot A| \geq (|q|+d+1)|A| - O(1).\]