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Let the 2 × 2 expanding matrix Rk be an integer Jordan matrix, i.e., Rk = diag(rk, sk) or Rk = J(pk), and let Dk = {0, 1, . . ., qk - 1}v with v = (1, 1)T and 2 ≤ qk ≤ pk, rk, sk for each natural number k. We show that the sequence of Hadamard triples {(Rk, Dk, Ck)} admits a spectrum of the associated Moran measure provided that lim infk→∞ 2qk‖R-1 k ‖ < 1. [ABSTRACT FROM AUTHOR]

Let μ M , D {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix M ∈ M 3 ⁢ (ℝ) {M\in M_{3}(\mathbb{R})} with entries ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} and the digit set D = { (0 , 0 , 0) t , (1 , 0 , 0) t , (0 , 1 , 0) t , (0 , 0 , 1) t } {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}}. In this paper, we prove that for any ρ 1 , ρ 2 , ρ 3 ∈ (1 , ∞) {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)} , if ρ 1 , ρ 2 , ρ 3 ∈ { ± x 1 r : x ∈ ℚ + , r ∈ ℤ + } {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}} , then L 2 ⁢ (μ M , D) {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of ρ 1 , ρ 2 , ρ 3 {\rho_{1},\rho_{2},\rho_{3}} that are in the set { ± (p q) 1 r : p ∈ 2 ⁢ ℤ + , q ∈ 2 ⁢ ℤ + - 1 ⁢ and ⁢ r ∈ ℤ + } {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}}. In particular, if ρ 1 , ρ 2 , ρ 3 ∈ { p q : p , q ∈ 2 ⁢ ℤ + 1 } {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}} , then there exist at most 4 mutually orthogonal exponential functions in L 2 ⁢ (μ M , D) {L^{2}(\mu_{M,D})} , and the number 4 is the best possible. [ABSTRACT FROM AUTHOR]