The Number of Orthogonal Exponential Functions of a Class of Self-Affine Measures in R3

Let ρi∈R such that 0<|ρi|<1, where i∈1,2,3. For an expanding real matrix M∈M3(R) and a three-element integer digit set D, let μM,D be the self-affine measures. In this paper, we prove that if there are two different numbers i,j∈1,2,3 such that ρi−1 is equal to ρj−1, and they belong to a rational number with an odd numerator and odd denominator, then there exist at most four mutually orthogonal exponential functions in L2(μM,D), and four is the best number.