A quantitative version of the idempotent theorem in harmonic analysis | Annals of Mathematics

Abstract 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 μˆ 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∈Gˆ:μˆ(γ)=1} belongs to the coset ring of Gˆ, that is to say we may write μˆ=∑j=1L±1γj+Γj where the Γj are open subgroups of Gˆ. In this paper we show that L can be bounded in terms of the norm ∥μ∥, and in fact one may take L≤expexp(C∥μ∥4). In particular our result is nontrivial even for finite groups.

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数学年刊(Annals of Mathematics)

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