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.
; for two subsets
the set of all products
A non-empty subset
if:[1] It is immediately verified that a finite 1-approximate subgroup is the same thing as a genuine subgroup.
Of course this definition is only interesting when
is small compared to
Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally arithmetic progressions in the integers.
is a 2-approximate subgroup: the set
is contained in the union of the two translates
A generalised arithmetic progression in
A more general example is given by balls in the word metric in finitely generated nilpotent groups.
Approximate subgroups of the integer group
were completely classified by Imre Z. Ruzsa and Freiman.
[2] The result is stated as follows: The constants
can be estimated sharply.
: this means that approximate subgroups of
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:[4] The statement also gives some information on the characteristics (rank and step) of the nilpotent group
is a finite matrix group the results can be made more precise, for instance:[5] The theorem applies for example to
; the point is that the constant does not depend on the cardinality
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).
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
such that any group with growth function bounded by a multiple of
is virtually nilpotent.
[6] 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.