16 1. Introduction

For the purposes of this text, it will be convenient to focus on the use of

covering to describe approximate multiplicative structure. More precisely:

Definition 1.2.1 (Approximate groups). Let G be a multiplicative group,

and let K ≥ 1 be a real number. A K-approximate subgroup of G, or K-

approximate group for short, is a subset A of G which contains the identity,

is symmetric (thus

A−1 := {a−1

: a ∈ A} is equal to A) and is such that A·A

can be covered by at most K left-translates (or equivalently by symmetry,

right translates) of A, thus there exists a subset X of G of cardinality at

most K such that A · A ⊂ X · A.

In most combinatorial applications, one only considers approximate

groups that are finite sets, but one could certainly also consider countably

or uncountably infinite approximate groups. We remark that this definition

is essentially from [Ta2008b] (although the definition in [Ta2008b] places

some additional minor constraints on the set X which have turned out not

to be terribly important in practice).

Example 1.2.2. A 1-approximate subgroup of G is the same thing as a

genuine subgroup of G.

Example 1.2.3. In the additive group of the integers Z, the symmet-

ric arithmetic progression {−N,...,N} is a 2-approximate group for any

N ≥ 1. More generally, in any additive group G, the symmetric generalised

arithmetic progression

{a1v1 + · · · + arvr : a1,...,ar ∈ Z, |ai| ≤ Ni∀i = 1,...,r}

with v1,...,vr ∈ G and N1,...,Nr 0, is a

2r-approximate

group.

Exercise 1.2.1. Let A be a convex symmetric subset of

Rd.

Show that A

is a

5d-approximate

group. (Hint: Greedily pack 2A with disjoint translates

of

1

2

A.)

Example 1.2.4. If A is an open

precompact2

symmetric neighbourhood of

the identity in a locally compact group G, then A is a K-approximate group

for some finite K. Thus we see some connection between locally compact

groups and approximate groups; we will see a deeper connection involving

ultraproducts in Chapter 7.

Example 1.2.5. Let G be a d-dimensional Lie group. Then G is a smooth

manifold, and can thus be (nonuniquely) given the structure of a Riemannian

manifold. If one does so, then for suﬃciently small radii r, the ball B(1,r)

around the identity 1 will be a Od(1)-approximate group.

2A

subset of a topological space is said to be precompact if its closure is compact.