Freimanvariance

Freimanvariance is a concept in additive combinatorics that relates to the structure of sets whose sumsets have limited size. It is named after Gregory Freiman, a Russian mathematician. The core idea is that if a set of integers has a sumset of a certain size, then the set itself must exhibit a specific, structured form.

More formally, Freiman's theorem states that if a finite set of integers A has a sumset 2A = {a + b | a, b in A} with size |2A| <= c|A| for some constant c, then A must be contained within a generalized arithmetic progression of small rank. A generalized arithmetic progression is a set that can be expressed as the image of a standard arithmetic progression under a linear map. The "rank" quantifies how far the set deviates from being a simple arithmetic progression.

This theorem is a fundamental result in the study of additive structures. It implies that sets with small sumsets are not "random" but rather possess a high degree of order. The proof of Freiman's theorem is quite involved and relies on techniques from combinatorics, number theory, and sometimes model theory. Its applications extend to various areas, including theoretical computer science, cryptography, and other branches of mathematics. The theorem has been generalized and extended in many directions, leading to a rich body of research in additive combinatorics.