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Abstract: The self-affine measure corresponding to an expanding integer matrix is supported on the attractor (or invariant set) of the iterated function system . In the present paper we show that if and is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in , and the number 3 is the best. This extends several known results on the non-spectral self-affine measure problem. The proof of such result depends on the characterization of the zero set of the Fourier transform , and provides a way of dealing with the non-spectral problem. [Copyright &y& Elsevier]

Abstract: The self-affine measure corresponding to the expanding integer matrix is supported on the generalized three-dimensional Sierpinski gasket , where p is odd. In the present paper we show that there exist at most 7 mutually orthogonal exponential functions in . 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 corresponding to the expanding integer matrix where p is odd, there exist at most 5 mutually orthogonal exponential functions in . [Copyright &y& Elsevier]