Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 129 / 4 / b / Solution

Because $A\subseteq {\mathbb{Z}}^2$ is finite, choose $R$ with $A\subseteq [-R,R{]}^2$ and an integer $M>2sR$. DefineAn equality of $s$-term sums in ${\mathbb{Z}}^2$ plainly gives equality after applying this linear map. Conversely, equality of the images giveswhere $|{\Delta }_x|\le 2sR<M$. Therefore ${\Delta }_x=0$, and then ${\Delta }_y=0$. The same estimate with one term on each side shows that $\varphi$ is injective on $A$. Hence $\varphi$ is a Freiman s-isomorphism from $A$ to the subset $\varphi (A)\subseteq \mathbb{Z}$.