On the intermediate value property of spectra for a class of Moran spectral measures

TL;DR

This paper demonstrates that for certain Moran spectral measures, the Beurling dimensions of their spectra can take any value between zero and the measure's upper entropy dimension, confirming a conjecture and revealing a rich spectrum structure.

Contribution

It establishes the intermediate value property of Beurling dimensions for spectra of Moran spectral measures, confirming a conjecture and analyzing the spectrum set cardinality.

Findings

Beurling dimensions of spectra are between 0 and the upper entropy dimension.

The intermediate value property holds for Beurling dimensions of spectra.

The set of spectra with a fixed Beurling dimension has continuum cardinality.

Abstract

We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are between $0$ and their upper entropy dimensions. Moreover, for such a Moran spectral measure $\mu$, we show that the Beurling dimension for the spectra of $\mu$ has the intermediate value property: let $t$ be any value between $0$ and the upper entropy dimension of $\mu$, then there exists a spectrum whose Beurling dimension is $t.$ In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution proposed by Fu, He and Wen in [J. Math. Pures Appl. 116 (2018), 105--131]. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value between $0$ and $\ue \mu$ has the cardinality of the continuum.