Additive energy, Freiman's lemma, and an edge-isoperimetric problem on the boolean cube

Abstract

Let $d \in \mathbb{N}$ and let $A \subseteq \mathbb{R}^d$ be a finite, non-empty set not contained in any translate of a hyperplane. We show that there exists a subset $S \subseteq A$ such that $d \ll |S| \ll d$ and \[E(A,S) := |\{(a_1,a_2,s_1,s_2)\in A\times A\times S\times S:a_1+s_1=a_2+s_2\}|\ll\lvert A\rvert\lvert S\rvert.\] This can be interpreted as a robust, analytic generalisation of the classical Freiman's lemma, and recovers Freiman's lemma up to a constant multiplicative factor. In the process of proving this result, we also show that for any integer $1 \leq m \leq d$ and any set $A \subset \mathbb{R}^d$ with $|A| \leq \binom{d}{m}$, one has \[ E(A, B) \leq 4(m + 1/m) |A||B|,\] where $B$ is the canonical basis of $\mathbb{Z}^d$. This gives new estimates for multiplicative energies between finite sets of primes and arbitrary subsets of positive integers, improving a previous result of Shkredov. Using related ideas, we also prove new bounds for an edge-isoperimetric problem for certain graphs on the boolean cube, improving an earlier result of Kirshner and Samorodnitsky. We give two different proofs of the first result described in this abstract and three different proofs of the second one.