Submission date: 22 August 2022

Abstract:

Recall that a subset X of a group G is 'product-free' if X^2 ∩ X=∅, ie if xy not in; X for all x,y in X. Let G be a group interpretable in a distal structure; for example, G may be any complex algebraic group. We prove there are constants c>0 and δ in (0,1) such that every finite subset X⊆ G distinct from {1} contains a product-free subset of size at least δ|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 δ/k^c. The proof is short, and follows quickly from Ruzsa calculus and an iterated application of Chernikov and Starchenko's distal regularity lemma.

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Full text arXiv 2208.10604: pdf, ps.