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 the 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 )).