A Characterization of Spectra for a Class of Planar Spectral Measures

In this paper we study the Beurling dimensions and the Beurling densities of spectra for a class of planar self-similar spectral measures \(\mu _{R,D}\) ${\mu }_{R,D}$ . We show an intermediate property about the Beurling dimensions of spectra of \(\mu _{R,D}\) ${\mu }_{R,D}$ , i.e., for any value t in zero and the Hausdorff dimension of the support of the measure \(\mu _{R,D}\) ${\mu }_{R,D}$ , there exists a spectrum whose Beurling dimension is t. Moreover, we obtain an upper bound of the Beurling densities of regular spectra of \(\mu _{R,D}\) ${\mu }_{R,D}$ , under the scale of the Hausdorff dimension of its support.