New bounds in Balog-Szemerédi-Gowers theorem


We prove, in particular, that every finite subset A of an abelian group with the additive energy κ|A|3 contains a set A′ such that |A′|≫κ|A| and |A′ − A′|≪κ −4|A′|.

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  1. [1]

    A. Balog: Many additive quadruples, Additive combinatorics, CRM Proc. Lecture Notes, 43, Amer. Math. Soc., Providence, RI, 2007, 39–49.

    Google Scholar 

  2. [2]

    A. Balog and E. Szemerédi: A statistical theorem of set addition, Combinatorica 14 (1994), 263–268.

    MATH  MathSciNet  Article  Google Scholar 

  3. [3]

    J. Bourgai, A. Glibichuk and S. Konyagin: Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. 73 (2006), 380–398.

    MathSciNet  Article  Google Scholar 

  4. [4]

    W. T. Gowers: A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.

    MATH  MathSciNet  Article  Google Scholar 

  5. [5]

    T. Sanders: Popular difference sets, Online J. Anal. Comb. 5 (2010), Art. 5, 4.

    Google Scholar 

  6. [6]

    B. Sudakov, E. Szemerédi and V. Vu: On a question of Erdős and Moser, Duke Math. J. 129 (2005), 129–155.

    MATH  MathSciNet  Article  Google Scholar 

  7. [7]

    T. Tao and V. Vu: Additive combinatorics, Cambridge University Press 2006.

    Google Scholar 

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Corresponding author

Correspondence to Tomasz Schoen.

Additional information

The author is supported by NCN grant 2012/07/B/ST1/03556.

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Schoen, T. New bounds in Balog-Szemerédi-Gowers theorem. Combinatorica 35, 695–701 (2015).

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Mathematics Subject Classication (2000)

  • 11B75
  • 11P99