Error terms for sums of two linear transformations

Let $d$ be a positive integer, and let $\mathcal L_1,\mathcal L_2$ be irreducible and coprime linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$. In this paper, we prove that, for every finite nonempty set $A\subseteq\mathbb{Z}^d$, \[ |\mathcal L_1A+\mathcal L_2A| \ge \left(|\det\mathcal L_1|^{1/d}+|\det\mathcal L_2|^{1/d}\right)^d|A|-O\left(|A|^{1-1/d}\right). \] This confirms the conjecture of Conlon and Lim on the error term for two linear transformations.