Abstract
Let \(\mu _{M,D}\) be the self-affine measure generated by an expansive matrix \(M\in M_n({\mathbb {Z}})\) and a finite digit set \(D\subset {\mathbb {Z}}^n\). If \(\{ x\in (0, 1)^n:\sum _{d\in D}{e^{2\pi i\langle d,x\rangle }}=0\}=q^{-1}{\mathbb {Z}}^n\cap (0, 1)^n\) for an integer \(q\ge 2\), we show that \(\mu _{M,D}\) is a spectral measure if and only if \(M\in M_n(q{\mathbb {Z}})\) and \(\#D=q^n\). As an application, we settle down the spectral eigenmatrix problem of the spectral self-affine measure \(\mu _{M,D}\).
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The authors would like to thank the anonymous referees for their valuable suggestions.
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Communicated by Rosihan M. Ali.
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The research is supported in part by the NNSF of China (Nos. 12071125, 11831007 and 12061010), the Science and Technology Research Project of Jiangxi Provincial Department of Education (No. GJJ2201244), the Doctoral Scientific Research Foundation of Gannan Normal University (No. BSJJ202241).
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Chen, ML., Liu, JC. On Spectra and Spectral Eigenmatrices of Self-Affine Measures on \({\mathbb {R}}^n\). Bull. Malays. Math. Sci. Soc. 46, 162 (2023). https://doi.org/10.1007/s40840-023-01559-2
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DOI: https://doi.org/10.1007/s40840-023-01559-2