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.

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