On a Nonabelian Balog–szemerédi-Type Lemma

Abstract

We show that if G is a group and A ⊂ G is a finite set with |A 2 | ≤ K |A|, then there is a symmetric neighbourhood of the identity S such that S k ⊂ A 2 A −2 and |S| ≥ exp(−K O(k) )|A|. Suppose that G is a group and A ⊂ G is a finite set with doubling K , that is,Clearly if A is a collection of free generators then K = |A|, but if K is much smaller then it tells us that there must be quite a lot of overlap in the products aa with a, a ∈ A. The extreme instance of this is when K = 1 and A is necessarily a coset of a subgroup of A. We are interested in the extent to which some sort of structure persists when K is slightly larger than 1, say O(1) as |A| → ∞.If G is abelian then the structure of A is comprehensively described by the GreenRuzsa-Freȋman theorem [GR07], but in the nonabelian case no analogue is known. A number of remarkable results have been established (see [BG09a, BG09b, FKP09, Hru09, Tao09] for details of these), but a clear description has not yet emerged. The interested reader may wish to consult [Gre09] for a discussion of the state of affairs.Freȋman-type theorems for abelian groups are applied to great effect throughout additive combinatorics, and many of these applications can make do with a considerably less detailed description of the set A. Moreover, additive combinatorics is now beginning to explore many nonabelian questions and so naturally a Freȋman-type theorem in this setting would be very useful. This is the motivation behind our present work: we wish to trade in some of the strength of the description of A in exchange for the increased generality of working in arbitrary groups. Tao proved a result in this direction in [Tao09] for which we require a short definition. A set S in a (discrete) group G is a symmetric neighbourhood of the identity if 1 G ∈ S and S = S −1 .

On Sumsets of Convex Sets

“…which has recently found many applications [8,10,11,12,13]. Roughly speaking, we show that if |A − A| is not much bigger than |A| 3/2 then the energy E(A, A − A) is very large.…”

The structure of approximate groups

“…As stated above, the basic idea of the proof is to first establish that A itself admits a locally compact local group as a good model. Here results of multiplicative combinatorics, and in particular a lemma of Sanders [46] (see also [12]), are critical. Once this is done, Theorem 3.10 follows relatively quickly from the deep results in the literature on Hilbert's fifth problem.…”

“…Proof. We use the argument from [46], generalised to the setting of multiplicative sets. For the convenience of the reader, we reproduce it here.…”

Small Doubling in Groups

“…We said very little about the proof of the classification of approximate groups in general [31]. An important ingredient in it (used to establish the correspondence between approximate groups and locally compact groups) was the following result of Croot-Sisask [12] and Sanders [65]. Theorem 6.5.…”