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 mea-sure p E M(G) is said to be idempotent if mu * mu = mu, or alternatively if (mu) over cap takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure mu is idempotent if and only if the set {gamma is an element of (G) over cap : (mu) over cap(gamma) = 1} belongs to the coset ring of (G) over cap, that is to say we may write (mu) over cap = Sigma(L)(j=1) +/- 1(gamma j+Gamma j) where the Gamma(j) are open subgroups of (G) over cap. In this paper we show that L can be bounded in terms of the norm parallel to mu parallel to, and in fact one may take L <= exp exp(C parallel to mu parallel to(4)). In particular our result is nontrivial even for finite groups.