1.2. Approximate groups 15

Figure 1. A schematic description of the steps needed to establish the

Gleason-Yamabe theorem. The annotation O, Q on an arrow indicates

that one has to pass to an open subgroup, and then quotient out a

compact normal subgroup, in order to obtain the additional structure

at the end of the arrow.

There are several ways to quantify what it means for a set A to be

“almost” closed under addition or multiplication. Here are some common

formulations of this idea (phrased in multiplicative notation, for the sake of

concreteness):

(1) (Statistical multiplicative structure) For a “large” proportion of

pairs (a, b) ∈ A × A, the product ab also lies in A.

(2) (Small product set) The “size” of the product set A · A := {ab :

a, b ∈ A} is “comparable” to the “size” of the original set A. (For

technical reasons, one sometimes uses the triple product A·A·A :=

{abc : a, b, c ∈ A} instead of the double product.)

(3) (Covering property) The product set A · A can be covered by a

“bounded” number of (left or right) translates of the original set

A.

Of course, to make these notions precise one would have to precisely quan-

tify the various terms in quotes. Fortunately, the basic theory of additive

combinatorics (and multiplicative combinatorics) can be used to show that

all these different notions of additive or multiplicative structure are “es-

sentially” equivalent; see [TaVu2006, Chapter 2] or [Ta2008b] for more

discussion.