Product-free sets in the free group

Abstract

We prove that product-free sets of the free group over a finite alphabet have maximum density 1/2 with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly k-product-free sets have maximum density 1/k in terms of the said measure.