1.2. Approximate groups 17
Example 1.2.6 (Extensions). Let φ : G → H be a surjective group ho-
momorphism (thus G is a group extension of H by the kernel ker(φ) of φ).
If A is a K-approximate subgroup of H, then
is a K-approximate
subgroup of G. One can think of
as an extension of the approximate
group A by ker(φ).
The classification of approximate groups is of importance in additive
combinatorics, and has connections with number theory, geometric group
theory, and the theory of expander graphs. One can ask for a quantita-
tive classification, in which one has explicit dependence of constants on the
approximate group parameter K, or one can settle for a qualitative classifica-
tion in which one does not attempt to control this dependence of constants.
In this text we will focus on the latter question, as this allows us to bring
in qualitative tools such as the Gleason-Yamabe theorem to bear on the
In the abelian case when the ambient group G is additive, approximate
groups are classified by Freiman’s theorem for abelian
As before, we phrase this theorem in a slightly stilted fashion (and in a
qualitative, rather than quantitative, manner) in order to demonstrate its
alignment with the general principles stated in the introduction.
Theorem 1.2.7 (Freiman’s theorem in an abelian group). Let A be an
object with the following properties:
(1) (Group-like object) A is a subset of an additive group G.
(2) (Weak regularity) A is a K-approximate group.
(3) (Discreteness) A is finite.
Then there exists a finite subgroup H of G, and a subset P of G/H, with
the following properties:
(i) (P is close to A/H)
) is contained in 4A := A + A + A + A,
where π : G → G/H is the quotient map, and |P|
(ii) (Nilpotent type structure) P is a symmetric generalised arithmetic
progression of rank OK(1) (see Example 1.2.3).
Informally, this theorem asserts that in the abelian setting, discrete
approximate groups are essentially bounded rank symmetric generalised
arithmetic progressions, extended by finite groups (such extensions are also
known as coset progressions). The theorem has a simpler conclusion (and is
simpler to prove) in the case when G is a torsion-free abelian group (such
as Z), since in this case H is trivial.
3The original theorem of Freiman [Fr1973] obtained an analogous classification in the case
when G was a torsion-free abelian group, such as the integers Z.