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Sum-free sets in abelian groups

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Abstract

LetA be a subset of an abelian groupG with |G|=n. We say thatA is sum-free if there do not existx, y, z εA withx+y=z. We determine, for anyG, the maximal densityμ(G) of a sum-free subset ofG. This was previously known only for certainG. We prove that the number of sum-free subsets ofG is 2(μ(G)+o(1))n, which is tight up to theo-term. For certain groups, those with a small prime factor of the form 3k+2, we are able to give an asymptotic formula for the number of sum-free subsets ofG. This extends a result of Lev, Luczak and Schoen who found such a formula in the casen even.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol University Walk, BS8 1TW, Bristol, England

    Ben Green

  2. Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, Pf. 127, H-1364, Budapest, Hungary

    Imre Z. Ruzsa

Authors
  1. Ben Green
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  2. Imre Z. Ruzsa
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Corresponding author

Correspondence to Ben Green.

Additional information

The first author is supported by a Fellowship of Trinity College Cambridge.

The second author is supported by Hungarian National Science Foundation (OTKA), Grants T38396, T43623, T42750.

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Green, B., Ruzsa, I.Z. Sum-free sets in abelian groups. Isr. J. Math. 147, 157–188 (2005). https://doi.org/10.1007/BF02785363

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  • DOI: https://doi.org/10.1007/BF02785363

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