Product-free sets in approximate subgroups of distal groups

Abstract: Recall that a subset $X$ of a group $G$ is 'product-free' if $X^2\cap X=\varnothing$, ie if $xy\notin X$ for all $x,y\in X$. Let $G$ be a group definable in a distal structure. We prove there are constants $c>0$ and $\delta\in(0,1)$ such that every finite subset $X\subseteq G$ distinct from ${1}$ contains a product-free subset of size at least $\delta|X|^{c+1}/|X^2|^c$. In particular, every finite $k$-approximate subgroup of $G$ distinct from ${1}$ contains a product-free subset of density at least $\delta/k^c$. The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma.