Spectrality of self-affine measures on the three-dimensional Sierpiński gasket (Q2891995)

The self-affine measure \(\mu_{M,D}\) corresponding to \(M\)=diag\([p_1,p_2,p_3]\) (\(p_j\in {\mathbb Z}\backslash \{0,\pm 1\},\;j=1,2,3)\) and \(D=\{0,e_1,e_2,e_3\}\) in the space \({\mathbb R}^3\) is supported on the three-dimensional Sierpinski gasket \(T(M,D)\), where \(e_1,e_2,e_3\) are the standard basis of unit column vectors in \({\mathbb R}^3\). The author considered how to determine the spectrality and non-spectrality of \(\mu_{M,D}\), and showed that if \(p_j\in 2{\mathbb Z}\backslash \{0,2\}\) for \(j=1,2,3\), then \(\mu_{M,D}\) is a spectral measure, and if \(p_j\in (2{\mathbb Z}+1)\backslash \{\pm 1\}\) for \(j=1,2,3\), then \(\mu_{M,D}\) is a non-spectral measure and exist at most 4 mutually orthogonal exponential functions in \(L^2(\mu_{M,D}\), where the number 4 is the best possible. This generalizes the known results on the spectrality of self-affine measure.