A QUANTITATIVE VERSION OF THE NON-ABELIAN IDEMPOTENT THEOREM

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L-2(G) to L-2(G) by g -> f * g where f * g(z) := E-xy=z f(x)g(y) for all z is an element of G. This operator is a linear map L-2(G) -> L-2(G) between two finite dimensional Hilbert spaces, and so it has well-defined singular values; we write parallel to f parallel to(A(G)) for their sum. The quantity parallel to.parallel to(A(G)) is of particular interest because in the abelian setting it coincides with the l(1)-norm of the Fourier transform of f. Thus, in the abelian setting it is an algebra norm, and it turns out that this extends to the non-abelian setting as well when parallel to.parallel to(A(G)) is defined as above. It is relatively easy to see that if A := xH where H <= G and x is an element of G, then parallel to 1(A)parallel to(A(G)) = 1, so that indicator functions of cosets of subgroups have algebra norm 1. Since parallel to.parallel to(A(G)) is a norm we can easily construct other sets whose indicator functions have small algebra norm by taking small integer-valued sums of indicator functions of cosets (when these sums are themselves indicator functions of cosets); the object of this paper is to show the following converse. Suppose that A subset of G has parallel to 1(A)parallel to(A(G)) <= M. Then there is an integer L = L(M), subgroups H-1, ... , H-L <= G, elements x(1), ... , x(L) is an element of G and signs sigma(1), ... , sigma(L) {-1, 0, 1} such that 1(A) = Sigma(L)(i=1) sigma(i)1(xi)H(i), where L may be taken to be at most triply tower in O(M). This may be seen as a quantitative version of the non-abelian idempotent theorem.