The structure and density of 𝒌-product-free setsin the free semigroup and group

Journal article

The free semigroup $\mathcal{F}$ on a finite alphabet $\mathcal{A}$ is the setof all finite words with letters from $\mathcal{A}$ equipped with theoperation 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 ele-ment of $S$ is a (non-trivial) concatenation of at most $k$words from $S$. We prove that a $k$-product-free subsetof $\mathcal{F}$ has upper Banach density at most $1/\rho(k)$, where$\rho(k) = \min\{\ell \colon \ell \nmid k - 1\}$. We also determine the struc-ture of the extremal $k$-product-free subsets for all $k \notin\{3, 5, 7, 13\}$; a special case of this proves a conjectureof Leader, Letzter, Narayanan, and Walters. We furtherdetermine the structure of all strongly $k$-product-freesets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banachdensity at most $1/\rho(k)$, which confirms a conjecture ofOrtega, 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