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

Journal article

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

Authors: Illingworth, F; Michel, L; Scott, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Journal of the London Mathematical Society
• Volume: 111
• Issue: 1
• Article number: e70046
• Publication date: 2024-12-14
• Acceptance date: 2024-10-31
• DOI: 10.1112/jlms.70046
• EISSN: 1469-7750
• ISSN: 0024-6107
• Language: English