Sums of linear transformations

We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$,

$$\begin{equation*} |\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left( |\det (\mathcal{L}_1)|^{1/d}+|\det (\mathcal{L}_2)|^{1/d} \right)^d|A|- o(|A|). \end{equation*}$$

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 certain choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. 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. In particular, when $\lambda$ is of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as small as possible for such a representation, we show that

$$\begin{equation*} |A + \lambda \cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|). \end{equation*}$$

This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\lambda = \sqrt {2}$.