Spectrality of generalized Sierpinski-type self-affine measures

In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μM,D on R2 generated by an expanding integer matrix M∈M2(Z) with det⁡(M)∈3Z and a non-collinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t} with α1β2−α2β1∈3Z. We give the sufficient and necessary conditions for μM,D to be a spectral measure, i.e., there exists a countable subset Λ⊂R2 such that E(Λ)={e2πi〈λ,x〉:λ∈Λ} forms an orthonormal basis for L2(μM,D). This completely settles the spectrality of the self-affine measure μM,D.