# Freiman's theorem

In mathematics, **Freiman's theorem** is a combinatorial result in number theory. In a sense it accounts for the approximate structure of sets of integers that 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

is small, in the sense that

for some constant . There exists an *n*-dimensional arithmetic progression of length

that contains *A*, and such that *c'* and *n* depend only on *c*.^{[1]}

A simple instructive case is the following. We always have

with equality precisely when *A* is an arithmetic progression.

This result is due to Gregory Freiman (1964,1966).^{[2]} Much interest in it, and applications, stemmed from a new proof by Imre Z. Ruzsa (1994).

## See also

## References

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*This article incorporates material from Freiman's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*