The maximal cardinality of $$\mu _{M,D}$$-orthogonal exponentials on the spatial Sierpinski gasket

Abstract

The self-affine measure $$\mu _{M,D}$$ corresponding to an expanding integer matrix $$M= diag[p_{1},p_{2},p_{3}]$$ and the digit set $$D=\left\{ 0, e_{1}, e_{2}, e_{3} \right\} $$ in the space $$\mathbb {R}^{3}$$ is supported on the spatial Sierpinski gasket, where $$e_{1},e_{2}, e_{3}$$ are the standard basis of unit column vectors in $$\mathbb {R}^{3}$$ and $$p_{1}, p_{2}, p_{3}\in \mathbb {Z}{\setminus } \{0, \pm 1\}$$. In the case $$p_{1}\in 2\mathbb {Z}$$ and $$p_{2}, p_{3}\in 2\mathbb {Z}+1$$, it is conjectured that all the four-element orthogonal exponentials in the Hilbert space $$L^{2}(\mu _{M,D})$$ are maximal in the class of exponential functions. This conjecture has been proved to be false by giving a class of the five-element (and later, the eight-element) orthogonal exponentials in $$L^{2}(\mu _{M,D})$$. In the present paper, we completely determine the maximal cardinality of $$\mu _{M,D}$$-orthogonal exponentials on the spatial Sierpinski gasket. The main result shows that (i) if $$p_{3}\ne \pm p_{2}$$, then for any $$l\in \mathbb {N}$$, there exist $$(2l+6)$$-element orthogonal exponentials in the Hilbert space $$L^{2}(\mu _{M,D})$$, which is also maximal in the class of exponential functions; (ii) if $$p_{3} = -p_{2}$$, then there exist at most eight mutually orthogonal exponential functions in $$L^{2}(\mu _{M,D})$$, where the number eight is the best upper bound.