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In [[additive combinatorics]] and [[additive number theory|number theory]], a subset ''A'' of an [[abelian group]] ''G'' is said to be '''sum-free''' if the [[sumset]] ''A⊕A'' is disjoint from ''A''. In other words, ''A'' is sum-free if the equation <math>a + b = c</math> has no solution with <math>a,b,c \in A</math>. | |||
For example, the set of [[Even and odd numbers|odd numbers]] is a sum-free subset of the integers, and the set ''{N/2+1, ..., N}'' forms a large sum-free subset of the set ''{1,...,N}'' (''N'' even). [[Fermat's Last Theorem]] is the statement that the set of all nonzero ''n''<sup>th</sup> powers is a sum-free subset of the integers for ''n'' > 2. | |||
Some basic questions that have been asked about sum-free sets are: | |||
* How many sum-free subsets of ''{1, ..., N}'' are there, for an integer ''N''? [[Ben J. Green|Ben Green]] has shown<ref>Ben Green, ''[http://www.arxiv.org/pdf/math.NT/0304058 The Cameron–Erdős conjecture]'', Bulletin of the [[London Mathematical Society]] '''36''' (2004) pp.769-778</ref> that the answer is <math>O(2^{N/2})</math>, as predicted by the [[Cameron–Erdős conjecture]]<ref>P.J. Cameron and P. Erdős, ''On the number of sets of integers with various properties'', Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79</ref> (see Sloane's {{OEIS2C|id=A007865}}). | |||
* How many sum-free sets does an abelian group ''G'' contain?<ref name="abelian">Ben Green and Imre Ruzsa, [http://www.arxiv.org/pdf/math.CO/0307142 Sum-free sets in abelian groups], 2005.</ref> | |||
* What is the size of the largest sum-free set that an abelian group ''G'' contains?<ref name="abelian" /> | |||
A sum-free set is said to be '''maximal''' if it is not a [[proper subset]] of another sum-free set. | |||
==References== | |||
{{reflist}} | |||
[[Category:Sumsets]] | |||
[[Category:Additive combinatorics]] | |||
Latest revision as of 20:37, 10 January 2014
In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A⊕A is disjoint from A. In other words, A is sum-free if the equation has no solution with .
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N/2+1, ..., N} forms a large sum-free subset of the set {1,...,N} (N even). Fermat's Last Theorem is the statement that the set of all nonzero nth powers is a sum-free subset of the integers for n > 2.
Some basic questions that have been asked about sum-free sets are:
- How many sum-free subsets of {1, ..., N} are there, for an integer N? Ben Green has shown[1] that the answer is , as predicted by the Cameron–Erdős conjecture[2] (see Sloane's Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.).
- How many sum-free sets does an abelian group G contain?[3]
- What is the size of the largest sum-free set that an abelian group G contains?[3]
A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.
References
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- ↑ Ben Green, The Cameron–Erdős conjecture, Bulletin of the London Mathematical Society 36 (2004) pp.769-778
- ↑ P.J. Cameron and P. Erdős, On the number of sets of integers with various properties, Number theory (Banff, 1988), de Gruyter, Berlin 1990, pp.61-79
- ↑ 3.0 3.1 Ben Green and Imre Ruzsa, Sum-free sets in abelian groups, 2005.