Product-free sets in approximate subgroups of distal groups

TL;DR

This paper proves that in distal groups, finite subsets and approximate subgroups always contain large product-free subsets, using combinatorial and model-theoretic tools.

Contribution

It establishes the existence of large product-free subsets in finite subsets and approximate subgroups of distal groups, extending combinatorial group theory results.

Findings

Existence of large product-free subsets in finite subsets of distal groups

Product-free subsets in approximate subgroups have positive density

Short proof leveraging Ruzsa calculus and distal regularity lemma

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.