Spectrality of Cantor–Moran measures with three-element digit sets.

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]