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

Let mu(M,D) be the self-affine measure corresponding to a diagonal matrix M with entries p(1),p(2),p(3) is an element of Z \ {0, +/- 1} and D = {0, e(1), e(2), e(3)} in the space R-3, where e(1), e(2), e(3) are the standard basis of unit column vectors in R-3. Such a measure is supported on the spatial Sierpinski gasket. In this paper, we prove the non-spectrality of mu(M,D). By characterizing the zero set Z((mu) over cap (M,D)) of the Fourier transform (mu) over cap (M,D), we obtain that if p(1) is an element of 2Z and p(2),p(3) is an element of 2Z + 1, then mu(M,D) is a non-spectral measure, and there are at most a finite number of orthogonal exponential functions in L-2(mu(M,D)). This completely solves the problem on the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L-2(mu(M,D)). (C) 2015 Elsevier Inc. All rights reserved.