There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure μM,D corresponding to an expanding matrix M=diag[p1,p2,p3] and the digit set D=0,e1,e2,e3 in the space R3 is supported on the spatial Sierpinski gasket, where e1,e2,e3 are the standard basis of unit column vectors in R3 and p1,p2,p3∈Z∖{0,±1}. In the case p1∈2Z and p2,p3∈2Z+1, it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space L2(μM,D) is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in L2(μM,D). In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space L2(μM,D) to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.