Spectrality of certain self-affine measures on the generalized spatial Sierpinski gasket (Q2808133)

The authors consider a class of self-affine measures \(\mu_{M,D}\) arising from affine iterated function systems of the form \(\{\phi_d:=M^{-1}(\cdot-d):d\in D\}\), where \(M\in M_n(\mathbb Z)\) is an expanding matrix with integer entries and \(D\subset\mathbb Z^n\) a finite digit set of cardinality \(|D|\). For the case that \(D:=\{0, e_1, e_2, e_3\}\), where \(\{e_1,e_2,e_3\}\) denotes the canonical basis of \(\mathbb R^3\), and upper or lower triangular matrices \(M\), the existence of an orthogonal basis of exponentials for the Hilbert space \(L^2(\mu_{M,D})\) is investigated. A method to decide upon the spectrality or non-spectrality of the measure \(\mu_{M,D}\) is presented. This method generalizes already known results.