Abstract
Let \(R=\varrho I_{n}\) and \(\mathcal {D}=\left\{ \textbf{0},\textbf{e}_{1},\ldots ,\textbf{e}_{n}\right\} \), where \(\varrho >1\) and \(\textbf{e}_{i}\) is the i-th coordinate vector in \(\mathbb {R}^n\). The spectral properties of the \(n-\)dimensional Sierpinski measure \(\mu _{R,\mathcal {D}}\) has been studied over two decades. In this paper, a special type of spectrum called a typical spectrum for \(\mu _{R,\mathcal {D}}\) is considered. We show that \(\varrho \in (n+1)\mathbb {N}\) is necessary and sufficient for \(\mu _{R,\mathcal {D}}\) to admit a typical spectrum if \(n+1\) is prime. And some necessary conditions for \(\mu _{R,\mathcal {D}}\) to admit a typical spectrum are provided when \(n+1\) is not a prime number. Furthermore, under the condition on real Hadamard matrix, we prove that \(\mu _{R,\mathcal {D}}\) admits a quasi-typical spectrum if and only if \(\varrho \in 2\mathbb {N}\). These results show that the spectral properties of the Sierpinski measure \(\mu _{R,\mathcal {D}}\) are really different between \(n+1\) is prime and non-prime. As a corollary, we prove that \(\varrho \in 2\mathbb {N}\) are the only integers such that \(\mu _{R,\mathcal {D}}\) becomes a spectral measure when \(n=3\).
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Acknowledgements
The authors would like to thank the anonymous referees for many valuable comments and suggestions which are helpful to improve the presentation of this manuscript. The research is supported by the NSFC of China (Nos.12271534, 12271194, 11922109), and Project funded by China Postdoctoral Science Foundation (2022M712821), Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University.
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Communicated by Jeff Geronimo.
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Dai, XR., Fu, XY. & Yan, ZH. Spectral Properties of Sierpinski Measures on \(\mathbb {R}^n\). Constr Approx 60, 165–196 (2024). https://doi.org/10.1007/s00365-023-09654-0
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DOI: https://doi.org/10.1007/s00365-023-09654-0