We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are in 0 and their upper entropy dimensions. Moreover, for such a Moran spectral measure μ, we show that the Beurling dimension for the spectra of μ has the intermediate value property: let t be any value in 0 and the upper entropy dimension of μ, then there exists a spectrum whose Beurling dimension is t. In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution in Fu et al. (2018) [20]. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value in 0 and ${\bar{\dim }}_{\mathrm{e}}\mu$ has the cardinality of the continuum.

我们证明一类莫兰谱测度的谱的伯林维数为 0 及其上熵维数。此外，对于这样的莫兰谱测度μ，我们证明μ的谱的 Beurling 维数具有中间值性质：令t为 0 和μ的上熵维中的任意值，则存在一个谱，其 Beurling 维数是t 。特别是，这个结果肯定地解决了 Fu 等人涉及谱伯努利卷积的猜想。（2018）[20]。此外，我们证明了 Beurling 维数等于 0 和 0 中任意固定值的谱集${\overline{\mathrm{暗淡}}}_{\mathrm{e}}\mu$具有连续统的基数。