Sums of linear transformations

Abstract

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 ${\mathbb{Z}}^d$, $$|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge {\left(|\det ({\mathcal{L}}_1){|}^{1/d}+|\det ({\mathcal{L}}_2){|}^{1/d}\right)}^d|A|-o(|A|).$$ 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 certain choices of ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\hat{I}»\cdot A|$ when $A$ is a finite set of real numbers and $\hat{I}»$ is an algebraic number. In particular, when $\hat{I}»$ 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 $$|A+\hat{I}»\cdot A|\ge (p^{1/d}+q^{1/d}{)}^d|A|-o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\hat{I}»=\sqrt{2}$.

24 pages