Li, Hai-Xiong; Li, Qian Spectral structure of planar self-similar measures with four-element digit set. (English) Zbl 1525.28005 J. Math. Anal. Appl. 513, No. 1, Article ID 126202, 23 p. (2022). Summary: Let \(R = \text{diag}(2 q, 2 q)\) with \(q \geq 2\) be an expanding positive integer matrix, and let \[ D = \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \begin{pmatrix} - 1 \\ - 1 \end{pmatrix} \right\} \] be a four-element digit set. Chen and Yan [6] in 2021 proved that the self-similar measure \(\mu_{R , D}\) is a spectral measure. In this paper, we give a description of the structure of spectra of \(\mu_{R , D}\). By extending the maximal mapping to plane, we characterize all the maximal orthogonal sets of exponential functions of \(\mu_{R , D}\) and we indeed obtain some sufficient conditions for the maximal orthonormal set to be or not to be a basis for the space \(L^2( \mu_{R , D})\). Cited in 1 Document MSC: 28A80 Fractals 42B05 Fourier series and coefficients in several variables 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) Keywords:self-similar measures; spectral measures; spectral structure; tree mapping PDFBibTeX XMLCite \textit{H.-X. Li} and \textit{Q. Li}, J. Math. Anal. Appl. 513, No. 1, Article ID 126202, 23 p. (2022; Zbl 1525.28005) Full Text: DOI References: [1] L.X. An, X.H. Dong, X.G. He, On spectra and spectral eigenmatrix problems of the planar Sierpinski measures, Indiana Univ. Math. J., preprint. · Zbl 1487.42060 [2] An, L. X.; Fu, X. Y.; Lai, C. K., On spectral Cantor-Moran measures and a variant of Bourgain’s sum of sine problem, Adv. Math., 349, 84-124 (2019) · Zbl 1416.42005 [3] An, L. X.; He, X. G., A class of spectral Moran measures, J. Funct. Anal., 266, 343-354 (2014) · Zbl 1303.28009 [4] An, L. X.; He, X. G.; Lau, K. S., Spectrality of a class of infinite convolutions, Adv. 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