# Approximate subgroup

Suppose $d$ is a positive integer. A finite subset $A$ of a group $G$ is termed a $d$-approximate subgroup if it is a symmetric subset containing the identity element such that there exists a subset $S$ of $G$ of size at most $d$ such that the product of subsets $A^2$ coincides with the product of subsets $SA$:
$A^2 = SA$
where $A^2 = \{ a_1a_2 \mid a_1,a_2 \in A \}$ and $SA = \{ sa \mid s \in S, a \in A \}$
Note that any finite symmetric subset $A$ containing the identity element is always a $|A|$-approximate subgroup, and $A$ is a 1-approximate subgroup if and only if it is a subgroup. The minimum value of $d$ for which $A$ is a $d$-approximate subgroup describes how close $A$ is to being a subgroup.