Spectra of the Sierpiński type spectral measure and their Beurling dimensions

Abstract

In this paper, we study the structure of the spectra for the Sierpiński type spectral measure ${\mu }_{A,\mathcal{D}}$ on ${\mathbb{R}}^2$. We give a sufficient and necessary condition for the family of exponential functions $\{e^{-2\pi i⟨\lambda ,x⟩}:\lambda \in \Lambda \}$ to be a maximal orthogonal set in $L^2({\mu }_{A,\mathcal{D}})$. Based on this result, we obtain a class of regular spectra of ${\mu }_{A,\mathcal{D}}$. Moreover, we discuss the Beurling dimensions of the spectra and obtain the optimal upper bound of Beurling dimensions of all spectra, which is in stark contrast with the case of self-similar spectral measure. An intermediate property about the Beurling dimension of the spectra is obtained.