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 ).
Let μ R,D be a self-affine measure associated with an expanding integer matrix R ∈ M n (Z) and a finite subset D ⊆ Z n . In the present paper we study the μ R,D -orthogonality and compatible pair conditions. We also show that any set of μ R,D -orthogonal exponentials contains at most 3 elements on the generalized plane Sierpinski gasket and the number 3 is the best.
The self-affine measure μ M,D corresponding to an expanding integer matrix M = a b c d and D = 0 0 , 1 0 , 0 1 is supported on the attractor (or invariant set) of the iterated function system {φ dIn the present paper we show that if (a + d) 2 = 4(ad − bc) and ad − bc is not a multiple of 3, then there exist at most 3 mutually orthogonal exponential functions in L 2 (μ M,D ), 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 transformμ M,D , and provides a way of dealing with the non-spectral problem.
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