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

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