Brunn-Minkowski type estimates for certain discrete sumsets

TL;DR

This paper establishes lower bounds for the size of sumsets formed by linear transformations of finite sets in Euclidean space, extending classical results and resolving parts of a conjecture by Bukh.

Contribution

It provides new Brunn-Minkowski type estimates for discrete sumsets under linear transformations, including stronger bounds under certain conditions, and generalizes Freiman's lemma to multiple sums.

Findings

Proved lower bounds for sumsets involving linear transformations with no common invariant subspace.

Extended results to cases with additional incongruence conditions, confirming parts of Bukh's conjecture.

Established sharp bounds for the size of k-fold sumsets, generalizing Freiman's lemma.

Abstract

Let $d,k$ be natural numbers and let $\mathcal{L}_1, \dots, \mathcal{L}_k \in \mathrm{GL}_d(\mathbb{Q})$ 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 $1 \leq i \leq k$. For every non-empty, finite set $A \subset \mathbb{R}^d$, we prove that \[ |\mathcal{L}_1(A) + \dots + \mathcal{L}_k(A) | \geq k^d |A| - O_{d,k}(|A|^{1- \delta}), \] 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 $|A|$. This can be seen as a $k$-fold generalisation of Freiman's lemma.