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 write μ = ∑L ±1γj+Γj j=1 where 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.