Sums of linear transformations in higher dimensions

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 $$\mathrm{\mid }q\cdot A+s\cdot A\mathrm{\mid }\ge (\mathrm{\mid }q\mathrm{\mid }+\mathrm{\mid }s\mathrm{\mid }+2d-2)\mathrm{\mid }A\mathrm{\mid }-O_{q,s,d}(\mathrm{\mid }A{\mathrm{\mid }}^{1-\delta })\mathrm{.}$$ The main term in this bound is sharp and improves upon an earlier result of Balog and Shakan. Secondly, let $\mathscr{L} \in $\text{GL}$_{2}( $\mathbb{R}$)$bealineartransformationsuchthat$\mathscr{L}$doesnothaveanyinvariantone-dimensionalsubspaceof$$\mathbb{R}$^2$\mathrm{.}Thenforallfinitesubsets$A$of$$\mathbb{R}$^2$,wehave$$\mathrm{\mid }A+\mathcal{L}(A)\mathrm{\mid }\ge 4\mathrm{\mid }A\mathrm{\mid }-O(\mathrm{\mid }A{\mathrm{\mid }}^{1-\delta }),$$ for some absolute constant $\delta >0$. The main term in this result is sharp as well.