Brunn-Minkowski type estimates for certain discrete sumsets

Let $d,k$ be natural numbers and let ${\mathcal{L}}_1,\dots ,{\mathcal{L}}_k\in {\mathrm{G}\mathrm{L}}_d(\mathbb{Q})$ be linear transformations such that there are no non-trivial subspaces $U,V\subseteq {\mathbb{Q}}^d$ of the same dimension satisfying ${\mathcal{L}}_i(U)\subseteq V$ for every $1\le i\le k$. For every non-empty, finite set $A\subset {\mathbb{R}}^d$, we prove that $$\mathrm{\mid }{\mathcal{L}}_1(A)+\cdots +{\mathcal{L}}_k(A)\mathrm{\mid }\ge k^d\mathrm{\mid }A\mathrm{\mid }-O_{d,k}(\mathrm{\mid }A{\mathrm{\mid }}^{1-\delta }),$$ where $\delta >0$ is some absolute constant depending on $d,k$. Building on work of Conlon-Lim, we can show stronger lower bounds when $k$ is even and ${\mathcal{L}}_1,\dots ,{\mathcal{L}}_k$ satisfy some further incongruence conditions, consequently resolving various cases of a conjecture of Bukh. Moreover, given any $d,k\in \mathbb{N}$ and any finite, non-empty set $A\subset {\mathbb{R}}^d$ not contained in a translate of some hyperplane, we prove sharp lower bounds for the cardinality of the $k$-fold sumset $kA$ in terms of $d,k$ and $\mathrm{\mid }A\mathrm{\mid }$. This can be seen as a $k$-fold generalisation of Freiman’s lemma.