Let $G$ be any group and $A,B\subset G$ are finite subsets of $G$. We define $AB:=\{ab |a \in A, b\in B\}$. In a similar fashion we can define $A^2, A^3, \dotsc, A^n,\dotsc$.
Now, we say $A$ is a $K$-approximate subset of $G$ if there is a $ X\subset G$ with $|X| \leq K$ such that $A^2 \subset XA$.
Another way to view this definition is to consider $|A^2|$ is at most $K|A|$, where $K$ is called the doubling constant.
Now my question: what is the significance of saying that $B\subset G$ is $O(K^9)$-approximate subset of $G$?
(cf. Proposition 2.5.5, Introduction to approximate groups by Matthew C.H. Tointon)