A Quantitative Version of the Idempotent Theorem in Harmonic Analysis

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 μ̂ 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 ∈ Ĝ : μ̂(γ) = 1} belongs to the coset ring of Ĝ, that is to say we may write