In
mathematics, an approximate group is a subset of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
which behaves like a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is 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
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A ...
.
Formal definition
Let
be a group and
; for two subsets
we denote by
the set of all products
. A non-empty subset
is a ''
-approximate subgroup'' of
if:
# It is symmetric, that is if
then
;
# There exists a subset
of cardinality
such that
.
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
is small compared to
(in particular, any subset
is a
-approximate subgroup). In applications it is often used with
being fixed and
going to infinity.
Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally
arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s in the integers. Indeed, for all
the subset
is a 2-approximate subgroup: the set