On the intermediate value property of spectra for a class of Moran spectral measures. (English) Zbl 07796954
Summary: 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 \(\mu\), we show that the Beurling dimension for the spectra of \(\mu\) has the intermediate value property: let \(t\) be any value in 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 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 \(\overline{\dim}_{\mathrm{e}} \mu\) has the cardinality of the continuum.
MSC:
| 28A80 | Fractals |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
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