# Freiman's theorem

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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

:$A + A,$

is small, in the sense that

:$|A + A| < c|A|,$

for some constant $c$. There exists an "n"-dimensional arithmetic progression of length

:$c\text{'} |A|,$

that contains "A", and such that $c\text{'}$ 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.

ee also

*Markov spectrum

References

*cite book| last=Nathanson | first=Melvyn B. | year=1996 | title=Additive Number Theory: Inverse Problems and Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | publisher=Springer | id=Zbl|0859.11003

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