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Journal of Applied and Industrial Mathematics

, Volume 13, Issue 2, pp 317–326 | Cite as

Asymptotics for the Logarithm of the Number of (k, l)-Solution-Free Collections in an Interval of Naturals

  • A. A. SapozhenkoEmail author
  • V. G. SargsyanEmail author
Article
  • 3 Downloads

Abstract

A collection (A1, … ,Ak+l) of subsets of an interval [1, n] of naturals is called (k, l)-solution-free if there is no set (a1, … , ak+l) ∈ A1 × ⋯ × Ak+l that is a solution to the equation x1 + ⋯ + xk = xk+1 + ⋯ + xk+l. We obtain the asymptotics for the logarithm of the number of sets (k, l)-free of solutions in an interval [1, n] of naturals.

Keywords

set group coset characteristic function progression 

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References

  1. 1.
    P. J. Cameron and P. Erdös, “On the Number of Sets of Integers with Various Properties,” in Number Theory (Proceedings of the 1st Conference. Canadian Number Theory Association, Banff, Canada, April 17–27, 1988) (de Gruyter, Berlin, 1990), pp. 61–79.Google Scholar
  2. 2.
    N. J. Calkin, “On the Number of Sum-Free Sets,” Bull. London Math. Soc. 22, 140–144 (1990).MathSciNetzbMATHGoogle Scholar
  3. 3.
    N. Alon, “Independent Sets in Regular Graphs and Sum-Free Subsets of Abelian Groups,” Israel J. Math. 73, 247–256 (1991).MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. A. Sapozhenko, “Abstract Erdös Conjecture,” Dokl. Akad. Nauk 393(6), 749–752 (2003).MathSciNetGoogle Scholar
  5. 5.
    B. Green, “Abstract Erdös conjecture,” Bull. Lond. Math. Soc. 36(6), 769–778 (2004).zbMATHGoogle Scholar
  6. 6.
    A. A. Sapozhenko, “On the Number of Sum-Free Sets in Abelian Groups,” Vestnik Moskov. Gos. Univ. Ser. 1 Mat. Mekh. 4, 14–17 (2002).MathSciNetzbMATHGoogle Scholar
  7. 7.
    V. F. Lev, T. Łuczak, and T. Schoen, “Sum-free sets in Abelian groups,” Israel J. Math. 125, 347–367 (2001).MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. F. Lev and T. Schoen, “Cameronab—Erdös Modulo a Prime,” Finite Fields Appl. 8(1), 108–119 (2002).MathSciNetzbMATHGoogle Scholar
  9. 9.
    B. Green and I. Z. Ruzsa, “Sum-Free Sets in Abelian Groups,” Israel J. Math. 147, 157–188 (2005).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. A. Sapozhenko, “Abstract Erdös Problem for Groups of Prime Order,” Zh. Vychisl. Mat. Mat. Fiz. 49(8), 1503–1509 (2009) [Comput. Math. Math. Phys. 49 (6), 1435–1441 (2009)].MathSciNetzbMATHGoogle Scholar
  11. 11.
    N. J. Calkin and A. C. Taylor, “Counting Sets of Integers, No k of Which Sum to Another,” J. Number Theory 57, 323–327 (1996).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yu. Bilu, “Sum-Free Sets and Related Sets,” Combinatorica 18(4), 449–459 (1998).MathSciNetzbMATHGoogle Scholar
  13. 13.
    N. J. Calkin and J. M. Thomson, “Counting Generalized Sum-Free Sets,” J. Number Theory 68, 151–160 (1998).MathSciNetzbMATHGoogle Scholar
  14. 14.
    T. Schoen, “A Note on the Number of (k, l)-Sum-Free Sets,” Electron. J. Comb. 17(1), 1–8 (2000).MathSciNetzbMATHGoogle Scholar
  15. 15.
    V. F. Lev, “Sharp Estimates forthe Number of Sum-Free Sets,” J. Reine Angew. Math. 555, 1–25 (2003).MathSciNetzbMATHGoogle Scholar
  16. 16.
    V. G. Sargsyan, “Asymptotics of the Logarithm of the Number of (k, l)-Sum-Free Sets inanAbelianGroup,” Diskretn. Mat. 26(1), 91–99 (2014) [Discrete Math. Appl. 25 (2), 93–99 (2014)].MathSciNetGoogle Scholar
  17. 17.
    B. Green, “A Szemerédi-Type Regularity Lemma in Abelian Groups,” Geom. Funct. Anal. 15(2), 340–376 (2005).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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