Non-spectrality of self-affine measures on the spatial Sierpinski gasket

Let μM,DμM,D be the self-affine measure corresponding to a diagonal matrix M   with entries p1,p2,p3∈Z∖{0,±1}p1,p2,p3∈Z∖{0,±1} and D={0,e1,e2,e3}D={0,e1,e2,e3} in the space R3R3, where e1,e2,e3e1,e2,e3 are the standard basis of unit column vectors in R3R3. Such a measure is supported on the spatial Sierpinski gasket. In this paper, we prove the non-spectrality of μM,DμM,D. By characterizing the zero set Z(μˆM,D) of the Fourier transform μˆM,D, we obtain that if p1∈2Zp1∈2Z and p2,p3∈2Z+1p2,p3∈2Z+1, then μM,DμM,D is a non-spectral measure, and there are at most a finite number of orthogonal exponential functions in L2(μM,D)L2(μM,D). This completely solves the problem on the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2(μM,D)L2(μM,D).