Jeck Lim, Sums of linear transformations

We show that if $L_1$ and $L_2$ are linear transformations from ${\mathbb{Z}}^d$ to ${\mathbb{Z}}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of ${\mathbb{Z}}^d$, $$|L_{1}A+L_{2}A|\ge (|det(L_1){|}^{1/d}+|det(L_2){|}^{1/d}{)}^d|A|-o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for many choices of $L_1$ and $L_2$. As an application, we prove a lower bound for $|A+\lambda \cdot A|$ when $A$ is a finite set of real numbers and $\lambda$ is an algebraic number.
Joint work with David Conlon.