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

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 $t\mathrm{.}$ 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 $\text{ue}\mu$ has the cardinality of the continuum.