THE CARDINALITY OF ORTHOGONAL EXPONENTIAL FUNCTIONS ON THE SPATIAL SIERPINSKI GASKET

Abstract

For the self-affine measures [Formula: see text] generated by a diagonal matrix [Formula: see text] with entries [Formula: see text] and the digit set [Formula: see text], Li showed that there exists an infinite orthogonal exponential functions set in [Formula: see text] if and only if at least two of the three numbers [Formula: see text] are even, and conjectured that there exist at most four mutually orthogonal exponential functions in [Formula: see text] for other cases [J.-L. Li, Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl. 432 (2015) 1005–1017]. This conjecture was disproved by Wang and Li through constructing a class of the eight-element orthogonal exponential functions [Q. Wang and J.-L. Li, There are eight-element orthogonal exponentials on the spatial Sierpinski gasket, Math. Nachr. 292 (2019) 211–226]. In this paper, we will show that there are any number of orthogonal exponential functions in [Formula: see text] if two of the three numbers [Formula: see text] are different odd and the other is even.