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Solving in elements of finite sets

Tom 163 / 2014

Vsevolod F. Lev, Rom Pinchasi Acta Arithmetica 163 (2014), 127-140 MSC: 11P70, 11B30. DOI: 10.4064/aa163-2-3

Streszczenie

We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c)\in A\times B\times (A\cup B) with a+b=2c is at most (0.15+o(1))(|A|+|B|)^2 as |A|+|B|\to \infty . As a corollary, if A is antisymmetric (that is, A\cap (-A)=\emptyset ), then there are at most (0.3+o(1))|A|^2 triples (a,b,c) with a,b,c\in A and a-b=2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c\in A and a-b=2c is at most (0.5+o(1))|A|^2. These estimates are sharp.

Autorzy

  • Vsevolod F. LevDepartment of Mathematics
    The University of Haifa at Oranim
    Tivon 36006, Israel
    e-mail
  • Rom PinchasiDepartment of Mathematics
    Technion – Israel Institute of Technology
    Haifa 32000, Israel
    e-mail

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