Let A∈M2(Z)\documentclass[12pt]{minimal}
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\begin{document}$$A\in M_2({\mathbb {Z}})$$\end{document} be an expanding integer matrix and D={d1=0,d2,d3}⊂Z2\documentclass[12pt]{minimal}
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\begin{document}$$D=\{d_1={\textbf{0}},d_2,d_3\}\subset {\mathbb {Z}}^2$$\end{document}. It follows from Hutchinson (Indiana Univ Math J 30:713–747, 1981) that the generalized Sierpinski self-affine set T(A,D)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D)$$\end{document} is the unique compact set determined by the pair (A, D) satisfing the set-valued equation AT(A,D)=⋃i=13(T(A,D)+di)\documentclass[12pt]{minimal}
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\begin{document}$$A{\textbf{T}}(A,D)=\bigcup _{i=1}^3({\textbf{T}}(A,D)+d_i)$$\end{document}. In this paper, we show that Fuglede’s conjecture holds onT(A,D)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D)$$\end{document}, which states that T(A,D)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D)$$\end{document} is a spectral set if and only if T(A,D)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D)$$\end{document} is a translational tile. For the classical Sierpinski self-affine set T(A,Dc)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D_{c})$$\end{document} with Dc={(0,0)t,(1,0)t,(0,1)t}\documentclass[12pt]{minimal}
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\begin{document}$$D_{\text {c}}=\{(0,0)^t,(1,0)^t, (0,1)^t\}$$\end{document}, a finer characterization of tiling set is given. As an application, we find that the classical Sierpinski self-affine tile T(A,Dc)\documentclass[12pt]{minimal}
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\begin{document}$${\textbf{T}}(A,D_{\text {c}})$$\end{document} is suitable for Kolountzakis’ conjecture on product domain. This enriches the results that are now known.
