Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups
-
Aleksandr A. Sapozhenko
and
Vahe G. Sargsyan
Abstract
A family (A1, …, Ak) of subsets of a group G is called k-solution-free family if the equation x1 + … + xk = 0 has no solution in (A1, …, Ak) such that x1 ∈ A1, …, xk ∈ Ak. We find the asymptotic behavior for the logarithm of the number of k-solution-free families in Abelian groups.
Originally published in Diskretnaya Matematika (2018) 30, №3, 117–126 (in Russian).
Funding: This research was carried out with the financial support of the Russian Fund for Basic Research (project 16-01-00593A).
References
[1] Cameron P. J., Erdös P., “On the number of sets of integers with various properties”, J. Number Theory, 1990, 61–79.10.1515/9783110848632-008Search in Google Scholar
[2] Calkin N. J., “On the number of sum-free sets”, Bull. London Math. Soc., 22 (1990), 140–144.10.1112/blms/22.2.141Search in Google Scholar
[3] Alon N., “Independent sets in regular graphs and sum-free subsets of Abelian groups”, Israel J. Math., 73 (1991), 247–256.10.1007/BF02772952Search in Google Scholar
[4] Sapozhenko A.A., “The Cameron.Erdős conjecture”, Dokl. Math., 68:3 (2003), 438–441.Search in Google Scholar
[5] Green B., “The Cameron.Erdös conjecture”, Bull. London Math. Soc., 36:6 (2004), 769–778.10.1112/S0024609304003650Search in Google Scholar
[6] Sapozhenko A.A., “On the number of sum-free sets in Abelian groups”, Vestn. Mosk. un-ta, ser. Matem., Mekh., 4 (2002), 14–17 (in Russian).Search in Google Scholar
[7] Lev V. F., Luczak T., Schoen T., “Sum-free sets in Abelian groups”, Israel J. Math., 125 (2001), 347–367.10.1007/BF02773386Search in Google Scholar
[8] Lev V. F., Schoen T., “Cameron–Erdös modulo a prime”, Finite Fields Appl., 8:1 (2002), 108–119.10.1006/ffta.2001.0330Search in Google Scholar
[9] Green, B., Ruzsa, I., “Sum-free sets in Abelian groups”, Israel J. Math., 147 (2005), 157–188.10.1007/BF02785363Search in Google Scholar
[10] Sapozhenko A.A., “Solution of the Cameron-Erdös problem for groups of prime order”, Comput. Math. and Math. Physics, 49:8 (2009), 1435–1441.10.1134/S0965542509080132Search in Google Scholar
[11] Calkin N. J., Taylor A. C., “Counting sets of integers, no k of which sum to another”, J. Number Theory, 57 (1996), 323–327.10.1006/jnth.1996.0051Search in Google Scholar
[12] Bilu, Yu., “Sum-free sets and related sets”, Combinatorica, 18:4 (1998), 449–459.10.1007/s004930050032Search in Google Scholar
[13] Calkin N. J., Thomson J. M., “Counting generalized sum-free sets”, J. Number Theory, 68 (1998), 151–160.10.1006/jnth.1997.2191Search in Google Scholar
[14] Schoen T., “A note on the number of (k, l)-sum-free sets”, Electr. J. Combinatorics, 17:1 (2000), 1–8.10.37236/1508Search in Google Scholar
[15] Lev V. F., “Sharp estimates for the number of sum-free sets”, J. fur die reine und angew. Math., 555 (2003), 1.25.10.1515/crll.2003.012Search in Google Scholar
[16] Sargsyan V. G., “Asymptotics of the logarithm of the number of (k, l )-sum-free sets in an Abelian group”, Discrete Math. Appl., 25:2 (2015), 93–99.10.1515/dma-2015-0009Search in Google Scholar
[17] Green B., “A Szemerédi-type regularity lemma in Abelian groups”, Geom. & Funct. Anal., 15:2 (2005), 340–376.10.1007/s00039-005-0509-8Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Periodic properties of pushdown automata
- Burnside-type problems in discrete geometry
- On affine classification of permutations on the space GF(2)3
- Limit distributions of the maximal distance to the nearest neighbour
- Elementary transformations of systems of equations over quasigroups and generalized identities
- Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups
- On the distribution of multiple power series regularly varying at the boundary point
- A letter to the Editor
Articles in the same Issue
- Frontmatter
- Periodic properties of pushdown automata
- Burnside-type problems in discrete geometry
- On affine classification of permutations on the space GF(2)3
- Limit distributions of the maximal distance to the nearest neighbour
- Elementary transformations of systems of equations over quasigroups and generalized identities
- Asymptotics for the logarithm of the number of k-solution-free sets in Abelian groups
- On the distribution of multiple power series regularly varying at the boundary point
- A letter to the Editor
