We prove that product‐free subsets of the free group over a finite alphabet have maximum upper density with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly ‐product‐free sets have maximum upper density in terms of this measure. The bounds are tight.
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Ortega et al. (2024) studied this question.
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