# Freiman’s theorem

Let $A$ be a finite set of integers such that the $2$-fold sumset $2A$ is “small”, i.e., $|2A| for some constant $c$. There exists an $n$-dimensional arithmetic progression (http://planetmath.org/MulidimensionalArithmeticProgression) of length $c^{\prime}|A|$ that contains $A$, and such that $c^{\prime}$ and $n$ are functions of $c$ only.

## References

• 1 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry of Sumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.
Title Freiman’s theorem FreimansTheorem 2013-03-22 13:39:05 2013-03-22 13:39:05 bbukh (348) bbukh (348) 7 bbukh (348) Theorem msc 11B25