tilings and spectral sets

We consider functions ϕ ∈ L2(Rd) such that { | det(D) | 12ϕ(Dx−λ) : D ∈ D,λ ∈ T} forms an orthogonal basis for L2(Rd), where D ⊂Md(R) and T ⊂ Rd. Such a function ϕ 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 ⊂ Md(R) and a set T ⊂ Rd can be used as respectively the dilation set and the translation set of a wavelet? When restricted to wavelets of the form ϕ = χ̌Ω, this question has a surprising tie to spectral sets and their spectra. Key words and phrases. Wavelet, wavelet set, dilation, translation, spectral set, spec-trum, spectral pair, multiplicative tiling, complementing set.