Sum-free set

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

a

+

b

=

c

{\displaystyle a+b=c}

has no solution with

a

,

b

,

c

∈

A

{\displaystyle a,b,c\in A}

.

For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N} forms a large sum-free subset of the set {1, ..., 2N}. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free set.

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 that the answer is

O

(

2

N

/

2

)

{\displaystyle O(2^{N/2})}

, as predicted by the Cameron–Erdős conjecture.

How many sum-free sets does an abelian group G contain?

What is the size of the largest sum-free set that an abelian group G contains?

A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.

Let

f

:

[

1

,

∞

)

→

[

1

,

∞

)

{\displaystyle f:[1,\infty )\to [1,\infty )}

be defined by

f

(

n

)

{\displaystyle f(n)}

is the largest number

k

{\displaystyle k}

such that any set of n nonzero integers has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma,

lim

n

f

(

n

)

n

{\displaystyle \lim _{n}{\frac {f(n)}{n}}}

exists.