Approximate subgroup

This article defines a property of subsets of groups
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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 product of subsets $A^{2}$ coincides with the product of subsets $SA$:

$A^{2}=SA$

where $A^{2}=\{a_{1}a_{2}\mid a_{1},a_{2}\in A\}$ 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.

External links

• Terence Tao's blog post on some results related to approximate subgroups