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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]

For an expansive real matrix M=ρ−1C0ρ−1$M= \def\eqcellsep{&}\begin{bmatrix} \rho ^{-1} & \mathcal {C}\\ 0& \rho ^{-1} \end{bmatrix}$ and a noncollinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}$D=\lbrace (0,0)^t,(\alpha _1,\alpha _2)^t,(\beta _1,\beta _2)^t\rbrace$ with α2−2β2∉3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let μM,D$\mu _{M,D}$ be the self‐affine measure defined by μM,D(·)=13∑d∈DμM,D(M(·)−d)$\mu _{M,D}(\cdot)=\frac{1}{3}\sum _{d\in D}\mu _{M,D}(M(\cdot)-d)$. In this paper, some necessary and sufficient conditions for L2(μM,D)$L^2(\mu _{M,D})$ contains an infinite orthogonal exponential set or μM,D$\mu _{M,D}$ to be a spectral measure are given. [ABSTRACT FROM AUTHOR]

Let the iterated function systems { S i } i = 1 N be defined by S i ⁢ (x) = (- 1) i - 1 ⁢ ρ ⁢ (x + d i) , x ∈ R , d i ∈ D , where 0 < ρ < 1 , and 퐷 is a finite subset of ℤ. Let the measure μ ρ , P be generated by the IFS and the probability weight P = { p i } i = 1 N . In this paper, we introduce a new way to consider the spectrality of μ ρ , P , and obtain a sufficient and necessary condition for N = 2 , and construct two examples of spectral measures for N = 2 ⁢ L ≥ 4 and N = 2 ⁢ L + 1 , respectively. [ABSTRACT FROM AUTHOR]

Let { S d } d = 0 2 N − 1 be an iterated function system defined by S d (x) = (− 1) d ρ (x + d) , x ∈ R , where 0 < ρ < 1 and N is a positive integer. Let μ ρ , D 2 N be a self-similar measure defined by μ ρ , D 2 N (⋅) = 1 2 N ∑ d = 0 2 N − 1 μ ρ , D 2 N (S d − 1 (⋅)). We show that μ ρ , D 2 N is a spectral measure if and only if ρ = p − 1 for some nature number p and 2 N divides p. Moreover, we introduce a useful way to determine the spectrality of the convolution of two infinitely supported measures. [ABSTRACT FROM AUTHOR]

Let R k = diag (r k , 1 , r k , 2 , ... , r k , n) be the n × n expanding integer diagonal matrix, and let D k ⊂ Z n . We show that the sequence of Hadamard triples { (R k , D k , S k) } admits a spectrum of the associated Moran measure under specific situations. The results are new in all dimensions n greater than two. [ABSTRACT FROM AUTHOR]

We consider the self-similar measure μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} generated by an expanding real matrix M = ( ρ - 1 0 0 ρ - 1 ) ∈ M 2 ⁢ (ℝ) {M=\begin{pmatrix}\rho^{-1}&0\\ 0&\rho^{-1}\end{pmatrix}\in M_{2}({\mathbb{R}})} and a digit set 𝒟 = { ( 0 0 ) , ( a b ) , ( c d ) , ( a + c b + d ) } ⊆ ℤ 2. {{\mathcal{D}}=\Biggl{\{}\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}a\\ b\end{pmatrix},\begin{pmatrix}c\\ d\end{pmatrix},\begin{pmatrix}a+c\\ b+d\end{pmatrix}\Biggr{\}}\subseteq{\mathbb{Z}}^{2}}. In this paper, we study the spectral and non-spectral problems of μ M , 𝒟 {\mu_{M,{\mathcal{D}}}}. In this case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two independent vectors, we prove that if ρ - 1 ∈ ℤ {\rho^{-1}\in{\mathbb{Z}}} , then μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} is a spectral measure if and only if ρ - 1 ∈ 2 ⁢ ℤ {\rho^{-1}\in 2{\mathbb{Z}}}. For the case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two dependent vectors, we first give the sufficient and necessary condition for L 2 ⁢ (μ M , 𝒟) {L^{2}(\mu_{M,{\mathcal{D}}})} to contain an infinite orthogonal set of exponential functions. Based on this result, we can give the exact cardinality of orthogonal exponential functions in L 2 ⁢ (μ M , 𝒟) {L^{2}(\mu_{M,{\mathcal{D}}})} when L 2 ⁢ (μ M , 𝒟) {L^{2}(\mu_{M,{\mathcal{D}}})} does not admit any infinite orthogonal set of exponential functions by classifying the values of ρ. [ABSTRACT FROM AUTHOR]