What is the importance of "small doubling" in the theory of approximate groups?

Question

One question I have is "why are approximate groups important?". If the small doubling constant is $1$ then it's definitely a group. If I read Green's note correctly. (1, 2)

To be more specific let's look at Freiman's theorem.

Thm Let $G$ be a group and $A \subset G$ be a finite subset such that $|A^2| < \frac{3}{2}|A|$. There exists a subgroup $H$ with $|H| = |A^2|$ such that for every $a \in A$, we have $A \subset aH = Ha$.

One motivation I could see for approximate groups is that the objects we are dealing with are not quite perfectly symmetric. Perhaps the object is not quite a perfect circle, so that when we rotate it doesn't quite map to itself $A \cap R_\theta A \subset A$. This might have a name in the literature already. Such a shape might appear in the Number Theory or Fourier Series or something.

So why are theorems like this important? Or why can we already look at this as objects of pure study? Also what's so special about the fraction $\frac{3}{2}$ that is making the proof easier?

The lemma in the textbook doesn't look any better. (Book)

Lemma Let $G$ be a group and let $A \subset G$ be a finite subset such that $|A^2| < \frac{3}{2}|A|$ then $H = A^{-1}A$ is a subgroup of $G$. Moreover $H = AA^{-1}$ and $|H| < 2|A|$.

So how "close" are we to proving the theorem here?

Answer 1

I would say that it is not the approximate groups per se, but rather the idea of set addition which is important. This idea is not an invention of one single person; it goes back to Fermat (1640(!), the two-square theorem), Lagrange (1770, the four-square theorem), Goldbach (1742, Goldbach's conjecture), Cauchy and Davenport (1813 / 1935, the Cauchy-Davenport theorem), Minkowski (around year 1890, Minkowski addition), Schnirelmann (around year 1930, Schnirelmann's density theorem) and many, many, many others.

Concerning "what's so special about the fraction $\frac32$ that is making the proof easier?" It is not that the proof gets easier; the structure breaks at $\frac32$, and the assertion becomes wrong if you replace the coefficient $\frac32$ with, say, $1,50001$. Maybe, the right way to understand this is to concentrate first on the commutative case and study Kneser's theorem first.