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Let $$\mu $$ be a Borel probability measure on $${\mathbb {R}}^n$$ . We call $$\mu $$ a spectral measure if there exists a countable set $$\Lambda \subset {\mathbb {R}}^n$$ such that $$E_\Lambda :=\{e^{2\pi i }:\lambda \in \Lambda \}$$ forms an orthogonal basis for the Hilbert space $$L^2(\mu )$$ . Let the measure $$\mu _{\{M,{\mathcal {D}}_n\}}$$ be defined by the following expression $$\mu _{\{M,{\mathcal {D}}_n\}}=\delta _{M^{-1}{\mathcal {D}}_1}*\delta _{M^{-2}{\mathcal {D}}_2}*\cdots $$ , where $$M=\text {diag}(\rho ^{-1},\rho ^{-1})$$ with $$|\rho

Let the 2 × 2 {2\times 2} expanding matrix R k {R_{k}} be an integer Jordan matrix, i.e., R k = diag ⁡ ( r k , s k ) {R_{k}=\operatorname{diag}(r_{k},s_{k})} or R k = J ⁢ ( p k ) {R_{k}=J(p_{k})} , and let D k = { 0 , 1 , … , q k - 1 } ⁢ v {D_{k}=\{0,1,\ldots,q_{k}-1\}v} with v = ( 1 , 1 ) T {v=(1,1)^{T}} and 2 ≤ q k ≤ p k , r k , s k {2\leq q_{k}\leq p_{k},r_{k},s_{k}} for each natural number k. We show that the sequence of Hadamard triples { ( R k , D k , C k ) } {\{(R_{k},D_{k},C_{k})\}} admits a spectrum of the associated Moran measure provided that lim inf k → ∞ ⁡ 2 ⁢ q k ⁢ ∥ R k - 1 ∥ < 1 {\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert .

Suppose that $0 and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

For an integral self-affine spectral measure, if the zeros of its Fourier transform are all integral vectors, it is proven that any its spectrum has a tree structure. For any subset with such tree structure, a sufficient condition and a necessary condition for the subset to be a spectrum are given, respectively. Applications are given to some known results as special cases.

Let n,b≥2 be two positive integers. For D={0,1,⋯,b−1}, let the self‐similar measure μbn,D be defined by μbn,D=1b∑d∈Dμbn,D(bnx−d). It is known [18] that μbn,D is a spectral measure with a spectrum Λ(bn,C)=∑j=0finiteajbnj:aj∈C,where C=bn−1{0,1,⋯,b−1}. In this paper, we give some conditions on τ∈Z under which the scaling set τΛ(bn,C) is also a spectrum of μbn,D.

We study the spectral property of a class of Sierpinski-type self-affine measures μ M , D ( ⋅ ) = 1 3 ∑ d ∈ D μ M , D ( M ( ⋅ ) − d ) on R 2 , where M = [ ρ 1 − 1 a 0 ρ 2 − 1 ] is a real upper triangular expanding matrix and D = { ( 0 0 ) , ( d 1 0 ) , ( d 2 d 3 ) } is a three-element real digit set with d 1 d 3 ≠ 0 . A necessary and sufficient condition for μ M , D to be a spectral measure is established.

In this paper, we study infinite families of orthogonal exponentials of some self-affine measures. The digit set [Formula: see text] and any [Formula: see text] expanding integer matrix [Formula: see text] can generate a self-affine measure [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the transposed conjugate of [Formula: see text], where [Formula: see text] and the elements of [Formula: see text] come from [Formula: see text]. In this paper, we prove the following results. For [Formula: see text], [Formula: see text] is a spectral measure. For [Formula: see text], there are infinite families of orthogonal exponentials, but none of them forms an orthogonal basis in [Formula: see text].

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

In this paper, we consider the non-spectral problem for the planar self-affine measures $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^2$. Let $p\geq2$ be a positive integer, $E_p^2:=\frac{1}{p}\{(i,j)^t:0\leq i,j\leq p-1\}$ and $\mathcal{Z}_{D}^2:=\{x\in[0, 1)^2:\sum_{d\in D}{e^{2\pi i\langle d,x\rangle}}=0\}$. We show that if $\emptyset\neq\mathcal{Z}_{D}^2\subset E_p^2\setminus\{0\}$ and $\gcd(\det(M),p)=1$, then there exist at most $p^2$ mutually orthogonal exponential functions in $L^2(\mu_{M,D})$. In particular, if $p$ is a prime, then the number $p^2$ is the best., Comment: 14 pages

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].

For a positive integer [Formula: see text], let [Formula: see text]. Let the self-affine measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a finite digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] is the [Formula: see text]th column of the [Formula: see text] identical matrix [Formula: see text], [Formula: see text]. In this paper, we prove that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].