Wavelets, tiling, and spectral sets

We consider a function phi is an element of L-2 (R-d) such that {\ det(D)\(1/2)phi(Dx - lambda) : D is an element of D, lambda is an element of F} forms an orthogonal basis for L-2(R-d), where D subset of M-d(R) and F subset of R-d. Such a function phi is called a wavelet with respect to the dilation set D and translation set T. We study the following question: Under what conditions can a D subset of M-d(R) and a T subset of R-d be used as, respectively; the dilation set and the translation set of a wavelet? When restricted to wavelets of the form phi = chiOmega, this question has a surprising tie to spectral sets and their spectra.