Our main result is that if A is a finite subset of an abelian group with jA C Aj 6 KjAj, then 2A 2A contains an O.log O.1/ 2K/-dimensional coset progression M of size at least exp. O.log O

We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.Comment: 16 pp. Corrected the proof of the Croot-Sisask Lemma. Corrected typos. Updated reference

Abstract. In this article we survey some of the recent developments in the structure theory of set addition.

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ * µ = µ, or alternatively if µ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ G : µ(γ) = 1} belongs to the coset ring of G, that is to say we may writewhere the Γ j are open subgroups of G.In this paper we show that L can be bounded in terms of the norm µ , and in fact one may take L exp exp(C µ 4 ). In particular our result is nontrivial even for finite groups.

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.Comment: 19 pp. Corrected typos. Updated references

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+\epsilon}}.Comment: 28 pp. Corrected typos. Updated references

Abstract. Let f : F n 2 → {0, 1} be a boolean function, and suppose that the spectral normand each H j is a subgroup of F n 2 .This result may be regarded as a quantitative analogue of the Cohen-Helson-Rudin structure theorem for idempotent measures in locally compact abelian groups.

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 .