Coset decision trees and the Fourier algebra

We show that if G is a finite group and f is a {0,1}-valued function on G with Fourier algebra norm at most M then f may be computed by a coset decision tree (that is a decision tree in which at each vertex we query membership of a given coset) having at most \exp(\exp(\exp(O(M^2)))) leaves. A short calculation shows that any {0,1}-valued function which may be computed by a coset decision tree with m leaves has Fourier algebra norm at most \exp(O(m)).
Comment: 28pp; corrections and typos