Orthogonal exponentials on the generalized three-dimensional Sierpinski gasket

The self-affine measure μ M , D corresponding to the expanding integer matrix M = [ p 0 m 0 p 0 0 0 p ] and D = { ( 0 0 0 ) , ( 1 0 0 ) , ( 0 1 0 ) , ( 0 0 1 ) } is supported on the generalized three-dimensional Sierpinski gasket T ( M , D ) , where p is odd. In the present paper we show that there exist at most 7 mutually orthogonal exponential functions in L 2 ( μ M , 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 μ M , D corresponding to the expanding integer matrix M = [ p 0 0 p ] and D = { ( 0 0 ) , ( 1 0 ) , ( 0 1 ) , ( 1 1 ) } , where p is odd, there exist at most 5 mutually orthogonal exponential functions in L 2 ( μ M , D ) .