In mathematics, Freiman's theorem is a combinatorial result in number theory. It in a sense accounts for the approximate structure of sets of integers which contain a high proportion of their internal sums, taken two at a time.
The formal statement is.
Let A be a finite set of integers such that the sumset
A + A
is small, in the sense that
|A + A| < c|A|
for some constant c. There exists an n-dimensional arithmetic progression of length
that contains A, and such that c′ and n depend only on c.
This result is due to G. A. Freiman (1966). Much interest in it, and applications, stemmed from a new proof by Imre Ruzsa.
* Melvyn B. Nathanson, Additive Number Theory: Inverse Problems and Geometry of Sumsets volume 165 of GTM. Springer, 1996. Zbl 0859.11003.