The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures

Suppose $\mathbf{b}=\{b_n{\}}_{n=1}^{\mathrm{\infty }}$ is a sequence of integers bigger than 1 and $\mathbf{D}=\{{\mathcal{D}}_n{\}}_{n=1}^{\mathrm{\infty }}$ is a sequence of consecutive digit sets. Let ${\mu }_{\mathbf{b},\mathbf{D}}$ be the Cantor-Moran measure defined by \begin{eqnarray*} μ_{{\bf b},{\bf D}}&=& δ_{\frac{1}{b_1}{\mathcal D}{1}}\astδ{\frac{1}{b_1b_2}{\mathcal D}{2}}\ast δ{\frac{1}{b_1b_2b_3}{\mathcal D}_{3}}\ast\cdots. \end{eqnarray*} We prove that $L^2({\mu }_{\mathbf{b},\mathbf{D}})$ possesses an exponential orthonormal basis if and only if ${\mu }_{\mathbf{b},\mathbf{D}}\ast \nu ={\mathcal{L}}_{[0,N_1\mathrm{/}{b}_1]}$ for some Borel probability measure $\nu$. This theorem shows that the generalized Fuglede’s conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of ${\mathbf{D}}_n={\mathcal{D}}_n+b_n{\mathcal{D}}_{n-1}+b_2\cdots b_n{\mathcal{D}}_1$ for $n\ge 1$.