Abstract

AbstractThe free semigroup on a finite alphabet is the set of all finite words with letters from equipped with the operation of concatenation. A subset of is ‐product‐free if no element of can be obtained by concatenating words from , and strongly ‐product‐free if no element of is a (non‐trivial) concatenation of at most words from . We prove that a ‐product‐free subset of has upper Banach density at most , where . We also determine the structure of the extremal ‐product‐free subsets for all ; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly ‐product‐free sets with maximum density. Finally, we prove that ‐product‐free subsets of the free group have upper Banach density at most , which confirms a conjecture of Ortega, Rué, and Serra.

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