Approximate group

In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of products of elements in the subset be not much bigger than the subset itself (while for a subgroup it is required that they be equal). The notion was introduced in the 2010s but can be traced to older sources in additive combinatorics.
Formal definition.
Let formula_1 be a group and formula_2; for two subsets formula_3 we denote by formula_4 the set of all products formula_5. A non-empty subset formula_6 is a "formula_7-approximate subgroup" of formula_1 if: 
It is immediately verified that a 1-approximate subgroup is the same thing as a genuine subgroup. Of course this definition is only interesting when formula_7 is small compared to formula_15 (in particular, any subset formula_16 is a formula_17-approximate subgroup). In applications it is often used with formula_18 being fixed and formula_15 going to infinity.
Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally arithmetic progressions in the integers. Indeed, for all formula_20 the subset formula_21 is a 2-approximate subgroup: the set formula_22 is contained in the union of the two translates formula_23 and formula_24 of formula_25. A "generalised arithmetic progression" in formula_26 is a subset in formula_26 of the form formula_28, and it is a formula_29-approximate subgroup.
A more general example is given by balls in the word metric in finitely generated nilpotent groups.
Classification of approximate subgroups.
Approximate subgroups of the integer group formula_26 were completely classified by Imre Z. Ruzsa and Freiman. The result is stated as follows: 
The constants formula_39 can be estimated sharply. In particular formula_40 is contained in at most formula_41translates of formula_35: this means that approximate subgroups of formula_26 are "almost" generalised arithmetic progressions.
The work of Breuillard–Green–Tao (the culmination of an effort started a few years earlier by various other people) is a vast generalisation of this result. In a very general form its statement is the following: 
The statement also gives some information on the characteristics (rank and step) of the nilpotent group formula_52.
In the case where formula_1 is a finite matrix group the results can be made more precise, for instance: 
The theorem applies for example to formula_71; the point is that the constant does not depend on the cardinality formula_72 of the field. In some sense this says that there are no interesting approximate subgroups (besides genuine subgroups) in finite simple linear groups (they are either "trivial", that is very small, or "not proper", that is almost equal to the whole group).
Applications.
The Breuillard–Green–Tao theorem on classification of approximate groups can be used to give a new proof of Gromov's theorem on groups of polynomial growth. The result obtained is actually a bit stronger since it establishes that there exists a "growth gap" between virtually nilpotent groups (of polynomial growth) and other groups; that is, there exists a (superpolynomial) function formula_73 such that any group with growth function bounded by a multiple of formula_73 is virtually nilpotent.
Other applications are to the construction of expander graphs from the Cayley graphs of finite simple groups, and to the related topic of superstrong approximation.