Product-free sets in the free group

EUROCOMB’23

Abstract
We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ 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 $k$-product-free sets have maximum density $1/k$ in terms of the said measure. The bounds are tight.

Pages:
748–753
References

László Babai and Vera T. Sós. Sidon sets in groups and induced subgraphs of Cayley graphs. European Journal of Combinatorics, 6(2):101-114, 1985.
https://doi.org/10.1016/S0195-6698(85)80001-9

W. T. Gowers. Quasirandom groups. Combinatorics, Probability and Computing, 17(3):363-387, 2008.
https://doi.org/10.1017/S0963548307008826

Kiran S. Kedlaya. Product-free subsets of groups, then and now. Contemporary Mathematics 479:169-177, 2009.
https://doi.org/10.1090/conm/479/09350

Imre Leader, Shoham Letzter, Bhargav Narayanan, and Mark Walters. Product-free sets in the free semigroup. European Journal of Combinatorics, 83:103003, 2020.
https://doi.org/10.1016/j.ejc.2019.103003

Tomasz Łuczak and Tomasz Schoen. On innite sum-free sets of natural numbers. Journal of Number Theory, 66(2):211-224, 1997.
https://doi.org/10.1006/jnth.1997.2169

Tomasz Łuczak and Tomász Schoen. Sum-free subsets of right cancellative semigroups.

European Journal of Combinatorics, 22(7):999-1002, 2001.
https://doi.org/10.1006/eujc.2001.0520

Miquel Ortega, Juanjo Rué, and Oriol Serra. Product-free sets in the free group. ArXiv 2302.03748, 2023.
https://doi.org/10.5817/CZ.MUNI.EUROCOMB23-103

Terence Tao and Van Vu. Sum-free sets in groups: a survey. J. Comb., 8(3):541-552, 2017.
https://doi.org/10.4310/JOC.2017.v8.n3.a7

Metrics

0

Views

0

PDF views