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 $$ |q\cdot A + s\cdot A| \geq (|q| +|s|+ 2d-2)|A| - O_{q,s,d}(|A|^{1-\delta}). $$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathscr{L} \in \textrm{GL}_{2}( \mathbb{R})$ be a linear transformation such that $\mathscr{L}$ does not have any invariant one-dimensional subspace of $\mathbb{R}^2$. Then for all finite subsets $A$ of $\mathbb{R}^2$, we have $$ |A + \mathscr{L}(A)| \geq 4|A| - O(|A|^{1-\delta}), $$ for some absolute constant $\delta > 0$. The main term in this result is sharp as well.