Suppose $G$ is a finitely generated group. Suppose $A^1$ is a finite subset of $G$, such that $\langle A^1 \rangle = G$. Let’s define $A^n \subset G$ for $n \in \mathbb{N}$ using the recurrent relation $A^{n+1} = \{ab | a \in A^n, b \in A^1\}$. It happens so, that $A^3$ has many amazing non-trivial properties. The ones, that I know are:
Non-commutative Pluennecke inequality: $\forall k \in \mathbb{R} ((|A^3| \leq k|A^1|) \rightarrow \exists c_1, c2 \in \mathbb{R} \forall n \in \mathbb{N} (|A^n| \leq c_1 k^{c_2n}|A^1|)$.
Gowers-Nikolov-Pyber theorem: If $G = SL_n(\mathbb{F}_p)$, where $p$ is prime and $|A^1| > 2|G|^{1 - \frac{1}{3(n+1)}}$, then $A^3 = G$.
First Helfgott theorem: $\exists \epsilon > 0$ such, that if $G = SL_2(\mathbb{F}_p)$, where $p$ is prime, then either $A^3 = G$ or $|A^3| \geq |A^1|^{1 + \epsilon}$.
Second Helfgott theorem: If $G = SL_3(\mathbb{F}_p)$, where $p$ is prime, then $\forall \epsilon > 0 \exists \delta > 0 ((|A^1| < |G|^{1-\epsilon}) \rightarrow (|A^3| \geq |A^1|^{1 + \delta})$.
Shalev theorem: For every non-trivial group word $w$ $\exists n \in \mathbb{N} $ such that for all finite simple $G$, such that $|G| > n$ if $A^1$ is the set of all values of $w$ in $G$, then $A^3 = G$.
It would be interesting to know, what are other results like those.