A discrete set Λ⊆Rd\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda \subseteq {\mathbb {R}}^d$$\end{document} is called a spectrum for the probability measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} if the family of functions {e2πi⟨λ,x⟩:λ∈Λ}\documentclass[12pt]{minimal}
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\begin{document}$$\{e^{2 \pi i \langle \lambda ,\, x\rangle }: \lambda \in \Lambda \}$$\end{document} forms an orthonormal basis for the Hilbert space L2(μ).\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\mu ).$$\end{document} In this paper, we will give a characterization of the spectra of self-affine measures generated by compatible pairs in Rd.\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^d.$$\end{document} As an application, we show, for the Cantor measure μb,q\documentclass[12pt]{minimal}
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\begin{document}$$\mu _{b,~q}$$\end{document} on R\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}$$\end{document} with consecutive digit set and any integer p∈Z\documentclass[12pt]{minimal}
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\begin{document}$$p\in {\mathbb {Z}}$$\end{document} with gcd(p,q)=1,\documentclass[12pt]{minimal}
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\begin{document}$$\gcd (p,\,q)=1,$$\end{document} that the set {Λ⊆R:ΛandpΛare both spectra forμb,qand0∈Λ}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \{\Lambda \subseteq {\mathbb {R}}: \Lambda \ \hbox {and} \ p\Lambda \ \text {are both spectra for }\mu _{b,~q}\text { and }0 \in \Lambda \} \end{aligned}$$\end{document}has the cardinality of the continuum.
