LAUSR

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