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

Abstract

Suppose ${\bf b}=\{b_n\}_{n=1}^{\infty}$ is a sequence of integers bigger than 1 and ${\bf D}=\{{\mathcal D}_{n}\}_{n=1}^{\infty}$ is a sequence of consecutive digit sets. Let $\mu_{{\bf b},{\bf D}}$ be the Cantor-Moran measure defined by \begin{eqnarray*} \mu_{{\bf b},{\bf D}}&=& \delta_{\frac{1}{b_1}{\mathcal D}_{1}}\ast\delta_{\frac{1}{b_1b_2}{\mathcal D}_{2}}\ast \delta_{\frac{1}{b_1b_2b_3}{\mathcal D}_{3}}\ast\cdots. \end{eqnarray*} We prove that $L^2(\mu_{{\bf b},{\bf D}})$ possesses an exponential orthonormal basis if and only if $\mu_{{\bf b},{\bf D}}\ast\nu={\mathcal L}_{[0,N_1/b_1]}$ 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 ${\bf D}_n={\mathcal D}_{n}+b_n{\mathcal D}_{n-1}+b_2\cdots b_n{\mathcal D}_{1}$ for $n\geq1$.