Approximate subgroup

Definition
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 defining ingredient::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.