### Abstract

Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x + y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2^{(μ(G)+o(1))n}, which is tight up to the o-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 of G. This extends a result of Lev, Łuczak and Schoen who found such a formula in the case n even.

Original language | English |
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Pages (from-to) | 157-188 |

Number of pages | 32 |

Journal | Israel Journal of Mathematics |

Volume | 147 |

Publication status | Published - 2005 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*147*, 157-188.

**Sum-free sets in abelian groups.** / Green, Ben; Ruzsa, I.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 147, pp. 157-188.

}

TY - JOUR

T1 - Sum-free sets in abelian groups

AU - Green, Ben

AU - Ruzsa, I.

PY - 2005

Y1 - 2005

N2 - Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x + y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2(μ(G)+o(1))n, which is tight up to the o-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 of G. This extends a result of Lev, Łuczak and Schoen who found such a formula in the case n even.

AB - Let A be a subset of an abelian group G with |G| = n. We say that A is sum-free if there do not exist x, y, z ∈ A with x + y = z. We determine, for any G, the maximal density μ(G) of a sum-free subset of G. This was previously known only for certain G. We prove that the number of sum-free subsets of G is 2(μ(G)+o(1))n, which is tight up to the o-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 of G. This extends a result of Lev, Łuczak and Schoen who found such a formula in the case n even.

UR - http://www.scopus.com/inward/record.url?scp=26244450787&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26244450787&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:26244450787

VL - 147

SP - 157

EP - 188

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -