The continuity of Beurling density and Beurling dimension of spectra of a class of self-affine spectral measures

Abstract

It is well-known that the Beurling dimension of the spectra of certain singularly continuous spectral measures possesses an intermediate property. In this paper, we establish that for a class of self-affine spectral measures $\mu$, both the Beurling dimension and Beurling density of their spectra attain full flexibility simultaneously. Specifically, for any $t\in (0,\dim_H^w(\supp(μ))]$ and $s\in [0,\infty )$, there exists a spectrum $\Lambda :={\Lambda }_{t,s}$ of $\mu$ satisfying \[\dim_{Be}(Λ)=t\quad\text{and}\quad D_{t}^+(Λ)=s\] where ${\dim }_H^w$ denotes the pseudo Hausdorff dimension, ${\dim }_{Be}$ denotes the Beurling dimension and $D_t^{+}$ denotes the $t$-Beurling density. These results provide new insights into the structure of the spectra for a singularly continuous spectral measure.