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

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 $t\in (0,{\dim }_H^w(\text{supp}(\mu ))]$ and $s\in [0,\mathrm{\infty })$, there exists a spectrum $\mathrm{\Lambda }:={\mathrm{\Lambda }}_{t,s}$ of $\mu$ satisfying $${\dim }_{Be}(\mathrm{\Lambda })=t\text{and}{D}_t^{+}(\mathrm{\Lambda })=s$$ 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.