A quantitative version of the idempotent theorem in harmonic analysis

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 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.

Locations

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