Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

TL;DR

This paper investigates the Beurling densities of regular maximal orthogonal sets for certain self-similar spectral measures, establishing their exact upper bounds based on the Hausdorff dimension of the support.

Contribution

It provides the first precise upper bounds for the Beurling densities of these orthogonal sets in the context of self-similar spectral measures with consecutive digit sets.

Findings

Exact upper bounds for $s$-Beurling densities were derived.

The results connect Beurling densities with the Hausdorff dimension of the measure's support.

The study advances understanding of spectral measures and their orthogonal sets.

Abstract

Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the $s$-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where $s$ is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.