Product-free sets in the free group

Abstract

We prove that product-free subsets of the free group over a finite alphabet have maximum upper density $$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly $$-product-free sets have maximum upper density $$ in terms of this measure. The bounds are tight.