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.