Product-free sets in the free group

TL;DR

This paper proves that product-free sets in free groups have a maximum density of 1/2, confirming a previous conjecture, and generalizes the result to strongly k-product-free sets with maximum density 1/k.

Contribution

It establishes the maximum density bounds for product-free and strongly k-product-free sets in free groups, confirming a conjecture and extending the understanding of their measure-theoretic properties.

Findings

Maximum density of product-free sets is 1/2 in free groups.

Maximum density of strongly k-product-free sets is 1/k.

Confirms a conjecture by Leader, Letzter, Narayanan, and Walters.

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.