# .

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

$$c' |A|\,$$

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

A simple instructive case is the following. We always have

$$|A + A|\, ≥ 2|A|-1\,$$

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).

Markov spectrum

References

Nathanson (1996) p.251

Nathanson (1996) p.252

Freiman, G.A. (1964). "Addition of finite sets". Sov. Math., Dokl. (in English. Russian original) 5: 1366–1370. Zbl 0163.29501.
Freiman, G. A. (1966). Foundations of a Structural Theory of Set Addition (in Russian). Kazan: Kazan Gos. Ped. Inst. p. 140. Zbl 0203.35305.
Freiman, G. A. (1999). "Structure theory of set addition". Astérisque 258: 1–33. Zbl 0958.11008.
Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics 165. Springer. ISBN 0-387-94655-1. Zbl 0859.11003.
Ruzsa, Imre Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica 65 (4): 379–388. doi:10.1007/bf01876039. Zbl 0816.11008.