Product-free sets in the free group

Abstract

We prove that product-free subsets of the free group over a finite alpha- bet have maximum upper density 1/2 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 andWal- ters. In more general terms, we actually prove that strongly k-product- free sets have maximum upper density 1/k in terms of this measure. The bounds are tight.