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
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.
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