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.
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