Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μM,DμM,D corresponding to the expanding integer matrixM=[p0m0p000p]andD={(000),(100),(010),(001)} is supported on the generalized three-dimensional Sierpinski gasket T(M,D)T(M,D), where p   is odd. In the present paper we show that there exist at most 7 mutually orthogonal exponential functions in L2(μM,D)L2(μM,D). This generalizes the result of Dutkay and Jorgensen [D.E. Dutkay, P.E.T. Jorgensen, Analysis of orthogonality and of orbits in affine iterated function systems, Math. Z. 256 (2007) 801–823] on the non-spectral self-affine measure problem. By using the same method, we also obtain that for self-affine measure μM,DμM,D corresponding to the expanding integer matrixM=[p00p]andD={(00),(10),(01),(11)}, where p   is odd, there exist at most 5 mutually orthogonal exponential functions in L2(μM,D)L2(μM,D).