Spectrality of planar Moran–Sierpinski-type measures

Let Mn ∞ n=1 be a sequence of expanding positive integral matrices with Mn = pn 0 0 qn for each n ≥ 1, and let D = 0 0, 1 0, 0 1 be a finite digit set in Z 2. The associated Borel probability measure obtained by an infinite convolution of atomic measuresWe prove that, under certain conditions, µ Mn, D is a spectral measure if and only if 3 | pn and 3 | qn for each n ≥ 2.