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Let { A n } n = 1 ∞ be a sequence of expanding matrices with A n ∈ M 2 ⁢ (ℤ) , and let { D n } n = 1 ∞ be a sequence of three-element digit sets with { x ∈ (0 , 1) 2 : ∑ d ∈ D n e 2 ⁢ π ⁢ i ⁢ 〈 d , x 〉 = 0 } = { ± 1 3 ⁢ (1 , i) t } , i ∈ { 1 , 2 } . The associated Moran measure generated by the infinite convolution μ { A n } , { D n } = δ A 1 - 1 ⁢ D 1 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ D 2 * δ A 1 - 1 ⁢ A 2 - 1 ⁢ A 3 - 1 ⁢ D 3 * ⋯. In this paper, we give some necessary and sufficient conditions for μ { A n } , { D n } to be a spectral measure under some suitable conditions on A n and D n . [ABSTRACT FROM AUTHOR]

Let 0 < ρ < 1 and let { a j , b j , n j } j = 1 ∞ be a sequence of positive integers with an upper bound. Associated with them, there exists a unique Borel probability measure μ ρ , { 0 , a j , b j } , { n j } generated by the following infinite convolution of discrete measures: μ ρ , { 0 , a j , b j } , { n j } = δ ρ n 1 ⁢ { 0 , a 1 , b 1 } ∗ δ ρ n 1 + n 2 ⁢ { 0 , a 2 , b 2 } ∗ δ ρ n 1 + n 2 + n 3 ⁢ { 0 , a 3 , b 3 } ∗ ⋯ , where gcd ⁡ (a j , b j) = 1 for all j ∈ ℕ . In this paper, we show that L 2 ⁢ (μ ρ , { 0 , a j , b j } , { n j } ) admits an exponential orthonormal basis if and only if the following two conditions are satisfied: (i) { a j , b j } ≡ { ± 1 } ⁢ (mod ⁢ 3) for all j ≥ 1 ; (ii) there exists a natural number r such that ρ - r ∈ 3 ⁢ ℕ and n j ∈ r ⁢ ℕ for all j ≥ 2 . [ABSTRACT FROM AUTHOR]