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

TL;DR

This paper demonstrates that for certain self-affine spectral measures, the Beurling dimension and density of their spectra can be independently varied across a full range, revealing new structural insights.

Contribution

It establishes the simultaneous full flexibility of Beurling dimension and density for spectra of a class of self-affine spectral measures, a novel result in spectral measure analysis.

Findings

Beurling dimension and density can be independently controlled for spectra.

Spectra of self-affine measures can achieve any prescribed Beurling dimension within a specific range.

Spectral measures exhibit a rich structure allowing diverse spectral configurations.

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 $t\in (0,\dim_H^w(\supp(\mu))]$ and $s\in [0,\infty)$, there exists a spectrum $\Lambda:=\Lambda_{t,s}$ of $\mu$ satisfying \[\dim_{Be}(\Lambda)=t\quad\text{and}\quad D_{t}^+(\Lambda)=s\] 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.