Sets with large additive energy and symmetric sets

  • Authors:

  • I. D. Shkredov;Sergey Yekhanin

  • Affiliations:

  • Moscow State University, Department of Number Theory, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, Russian Federation;Moscow State University, Department of Number Theory, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, Russian Federation

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

We show that for any set A in a finite Abelian group G that has at least c|A|^3 solutions to a"1+a"2=a"3+a"4, a"i@?A there exist sets A^'@?A and @L@?G, @L={@l"1,...,@l"t}, t@?c^-^1log|A| such that A^' is contained in {@?"j"="1^t@e"j@l"j|@e"j@?{0,-1,1}} and A^' has @?c|A|^3 solutions to a"1^'+a"2^'=a"3^'+a"4^', a"i^'@?A^'. We also study so-called symmetric sets or, in other words, sets of large values of convolution.