Beurling dimension of a class of spectra of the Sierpinski-type spectral measures

Abstract

In this paper, we study the structure of the spectra for the Sierpiński type spectral measure µA,D on R 2 . We give a sufficient and necessary condition for the family of exponential functions {e −2πi λ,x : λ ∈ Λ} to be a maximal orthogonal set in L 2 (µA,D). Based on this result, we obtain a class of regular spectra of µA,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.