We have a triangle $ T $ with vertices at the $ \mathbb{Z} \times \mathbb{Z} $ grid . Now, consider the surface $ 2T= \{x \in \mathbb{R}^2 : \frac{x}{2} \in T \} $ ( so, double $ T $ ). Is it possible to write every point of the grid inside $ 2T $ , as the sum of two points of the grid in $ T $ ? So, is the relation true:
$ \forall x \in 2T \cap \mathbb{Z}^2, \exists x_1,x_2 \in T \cap \mathbb{Z}^2 : x=x_1 + x_2 $
Can anyone help me with this problem?