Sum-free sets in abelian groups

@article{Green2003SumfreeSI,
  title={Sum-free sets in abelian groups},
  author={Ben Green and Imre Z. Ruzsa},
  journal={Israel Journal of Mathematics},
  year={2003},
  volume={147},
  pages={157-188}
}
  • B. Green, I. Ruzsa
  • Published 10 July 2003
  • Mathematics
  • Israel Journal of Mathematics
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… 

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References

SHOWING 1-10 OF 25 REFERENCES

Maximal sum-free sets in finite abelian groups

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) ∩ S = ø and ∣S∣ ≥ ∣T∣ for every sum-free set T in G. It is shown that if G is an elementary abelian p–group of order

Sum-free sets in abelian groups

AbstractWe show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is $$\left( {2^{\nu (G)} - 1} \right)2^{\left| G \right|/2} + O\left(

Counting sumsets and sum-free sets modulo a prime

The number of distinct sets of the form A of residues modulo p that are said to be sum-free if there are no solutions to a = a′ + a″ with a, a′, a″ ∈ A is counted.

On the Number of Sum-Free Sets

Cameron and Erdos have considered the question: how many sum-free sets are contained in the first n integers; they have shown that the number of sum-free sets contained within the integers {n3 , n 3

Additive Number Theory: Inverse Problems and the Geometry of Sumsets

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2(1/2+∈(k))n independent sets. This settles a

A Szemerédi-type regularity lemma in abelian groups, with applications

Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the

Cameron-Erdős Modulo a Prime

We prove that for p prime and sufficiently large, the number of subsets of Zp free of solutions of the equation x+y=z (that is, free of Schur triples) satisfies

Combinatorics: room squares, sum-free sets, Hadamard matrices

Now welcome, the most inspiring book today from a very professional writer in the world, combinatorics room squares sum free sets hadamard matrices. This is the book that many people in the world