Definition
An approximate group is a finite subset $A$ of a (typically multiplicative) group $G$ that is symmetric (i.e., $A = A^{-1}$ and contains the identity element) and satisfies a bounded doubling condition: there exists a constant $K \ge 1$ such that the product set $A \cdot A = {ab : a,b \in A}$ can be covered by at most $K$ left translates of $A$. Formally,
$$
\exists, X \subseteq G,\ |X| \le K \text{ such that } A \cdot A \subseteq X \cdot A .
$$
Overview
Approximate groups were introduced in additive combinatorics to capture the notion of a set that behaves like a genuine subgroup up to a bounded amount of “error.” They play a central role in the study of growth in groups, the classification of sets with small doubling, and have applications to number theory, geometry, and theoretical computer science. The concept was formalized in the early 2000s by Terrence Tao and Ben Green, building on earlier work by Freiman on sets of integers with small sumsets. Subsequent breakthroughs, such as the Breuillard–Green–Tao theorem (2012), provided a structural description of approximate groups, showing that they are essentially controlled by nilpotent Lie groups.
Etymology/Origin
The term combines the adjective “approximate,” indicating that the set is not an exact subgroup but approximates one, with “group,” the algebraic structure under consideration. The notion originates from additive combinatorial research on “small doubling” phenomena, where Freiman’s theorem (1966) classified subsets of the integers with small sumsets. The modern definition was codified by Tao (2008) and Green–Tao (2009) in the context of non‑abelian groups.
Characteristics
| Property | Description |
|---|---|
| Symmetry | $A = A^{-1}$ and $e \in A$ (where $e$ is the identity). |
| Bounded doubling | There exists a constant $K$ (the doubling constant) such that $ |
| Finite size | The definition is usually restricted to finite subsets, though infinite versions exist under additional measure‑theoretic conditions. |
| Control by a genuine subgroup | In many cases, an approximate group is efficiently covered by a coset progression or a nilpotent subgroup of bounded rank (see Breuillard–Green–Tao). |
| Stability under homomorphisms | The image of an approximate group under a group homomorphism is again an approximate group (with possibly larger doubling constant). |
| Examples |
|
Related Topics
- Additive combinatorics – the broader field studying combinatorial properties of sumsets and product sets.
- Freiman’s theorem – a classification of subsets of abelian groups with small doubling.
- Breuillard–Green–Tao theorem – a structural theorem giving a precise description of finite approximate groups.
- Small doubling – the phenomenon where $|A\cdot A|$ is comparable to $|A|$.
- Product set estimates – results such as the Plünnecke–Ruzsa inequalities that bound sizes of iterated product sets.
- Growth in groups – investigations of how subsets expand under repeated multiplication, closely related to approximate groups.
- Nilpotent Lie groups – many approximate groups are modeled on subsets of nilpotent Lie groups.
- Model theory of approximate subgroups – logical frameworks for studying approximate groups via definable sets.