Sums of linear transformations in higher dimensions

Abstract

In this paper, we prove the following two results. Let $d$ be a natural number and $q,s$ be co-prime integers such that $1<qs$. Then there exists a constant $\delta >0$ depending only on $q,s$ and $d$ such that for any finite subset $A$ of ${\mathbb{R}}^d$ that is not contained in a translate of a hyperplane, we have $$|q\cdot A+s\cdot A|\ge (|q|+|s|+2d-2)|A|-O_{q,s,d}(|A{|}^{1-\delta }).$$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathcal{L}\in {\text{GL}}_2(\mathbb{R})$ be a linear transformation such that $\mathcal{L}$ does not have any invariant one-dimensional subspace of ${\mathbb{R}}^2$. Then for all finite subsets $A$ of ${\mathbb{R}}^2$, we have $$|A+\mathcal{L}(A)|\ge 4|A|-O(|A{|}^{1-\delta }),$$ for some absolute constant $\delta >0$. The main term in this result is sharp as well.

17 pages, minor correction in statement of Theorem 1.1