The structure and density of $k$-product-free sets in the free semigroup

Abstract

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/\ddot{I}(k)$, where $\ddot{I}(k)=\min \{\ell :\ell \nmid k-1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k\notin \{3,5,7,13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/\ddot{I}(k)$, which confirms a conjecture of Ortega, Rué, and Serra.

31 pages, added density results for the free group