SPECTRALITY OF PLANAR MORAN–SIERPINSKI-TYPE MEASURES

. Let { M n } ∞ n =1 be a sequence of expanding positive integral matrices with M n = (cid:18) p n

ces with Mn = pn 0 0 qn for each n ≥ 1, and let D = 0 0 , 1 0 , 0 1 be a finite digit set in Z 2 .The associated Borel probability measure obtained by an infinite convolution of atomic measures • is called a Moran-Sierpinski-type measure.We prove that, under certain conditions, µ {Mn},D is a spectral measure if and only if 3 | pn and 3 | qn for each n ≥ 2.

Introduction
Let µ be a Borel probability measure with compact support on R n .We say that µ is a spectral measure if there exists a countable discrete set Λ ⊂ R n such that E(Λ) := e −2πi λ,x : λ ∈ Λ forms an orthonormal basis for L 2 (µ).In this case, we call Λ a spectrum of µ.For the special case that a spectral measure is the restriction of the Lebesgue measure on a bounded Borel subset Ω of R n , we call Ω a spectral set.The existence of a spectrum is closely related to the famous Fuglede conjecture, which asserts that χ Ω dx is a spectral measure if and only if Ω is a translational tile [17].This conjecture has been proved to be false by Tao and others in both directions on R n for n ≥ 3 [24,23,29,30].But it is still open for n = 1 and n = 2.
Jorgensen and Pedersen initiated an investigation of spectral properties of fractal measures [22].They showed that the Cantor-typed measure µ 1/k , which is the invariant measure of the iterated function system {φ 0 (x) = x/k, φ 1 (x) = (x+1)/k}, with natural weight, is a spectral measure if k is even, but not a spectral one if k is odd.Since then, the study of the spectral properties of fractal measures became an active research topic, where, for example, self-similar measures, self-affine measures and Moran measures were considered and are still objects of study.The readers may see [4,5,1,3,2,7,9,6,8,12,11,10,13,16,18,20,25,28,27,32,26,19,14] and 66 QIAN LI AND MIN-MIN ZHANG the references therein for recent advances.In particular, Hu and Lau [20] showed a necessary and sufficient condition so that L 2 (µ ρ ) contains an infinite orthogonal set for the more general Bernoulli convolution µ ρ , 0 < ρ < 1.Recently, Dai [6] completely settled the problem that the only spectral Bernoulli convolution is µ 1/2k .The more general N -Bernoulli convolution was completely characterized by Dai et al. in [9].Let 0 < ρ < 1 and D = {0, 1, . . ., N − 1} with N > 1; they showed that µ ρ,D is a spectral measure if and only if N | ρ −1 .Unlike the one-dimensional situation, the study on the spectral properties of measures in higher dimensions is seldom addressed.See e.g.[14,11,10,8,12,26,28,27].We note that the most widely studied are the self-affine measures generated by an expanding matrix and a finite digit set.
A Sierpinski-type measure µ M,D is defined by where M = p 0 0 q is an expanding matrix and Sierpinski-type measure plays an important role in fractal geometry and in geometric measure theory [15,21].Deng and Lau [12], and Dai, Fu, and Yan [8] completely characterized the spectrality of the self-affine measure µ M,D .They proved that µ M,D is a spectral measure if and only if 3 | p and 3 | q.Motivated by the above results, in this paper we consider the spectral properties of a class of planar Moran-Sierpinski-type measures.Let {M n } ∞ n=1 be a sequence of expanding positive integral matrices (that is, all the eigenvalues of M n are strictly greater than 1 in module) with and let , where #D is the cardinality of D and δ d is the Dirac measure at point d.Then there exists a Borel probability measure with compact support generalized by the infinite convolution Here the sign * denotes the convolution of two measures, and the convergence is in the weak sense.The measure µ {Mn},D is called a Moran-Sierpinski-type measure, and its support is the Moran set Motivated by the above works, we extend the characterization of the spectrality of the Sierpinski-type measure to the Moran measure µ {Mn},D in (1.1).Note that in case M n = M = p 0 0 q , the measures µ {Mn},D and µ M,D coincide.The main result of this paper is as follows.
Theorem 1.1.Let µ {Mn},D be the Moran-Sierpinski-type measure defined as in (1.1) and p n ≡ ±q n (mod 3) for all n ≥ 2. Then µ {Mn},D is a spectral measure if and only if 3 | p n and 3 | q n for each n ≥ 2.
The most subtle part is proving the necessity.We note that convolution plays an important role in the study of the spectrality of the Moran measure µ {Mn},D .The following technical theorem gives a connection between two convolution measures, which will be used to prove the necessity.Theorem 1.2.Let B ⊂ Z 2 be a finite set and let ν be a Borel probability measure with compact support on R 2 .Suppose that µ := δ B * ν is a spectral measure.Further, suppose the following: Then both δ B and ν are spectral measures.Remark 1.3.Recently, An and Wang [5] proved the above theorem in dimension one, which is a special case of our conclusion.
We organize this paper as follows.In Section 2, we introduce some basic definitions and properties of spectral measures.In Section 3, we will give the proof of Theorem 1.2.We devote Sections 4 and 5 to prove Theorem 1.1.

Preliminaries
Let µ be a Borel probability measure with compact support on R 2 .The Fourier transform of µ is defined as usual, μ(ξ) = e −2πi ξ,x dµ(x) for any ξ ∈ R 2 .We will denote by Z(μ) = {ξ : μ(ξ) = 0} the zero set of μ.In what follows, e λ stands for the exponential function e −2πi λ,x .Then for a discrete (2.1) In this case, we say that Λ is a bi-zero set of µ.Since bi-zero sets (or spectra) are invariant under translation, without loss of generality we always assume that 0 ∈ Λ in this paper.
The following criterion is a universal test to decide whether a countable set Λ ⊂ R 2 is a bi-zero set (a spectrum) of µ or not.For ξ ∈ R 2 , we write Theorem 2.1 ([22]).Let µ be a Borel probability measure with compact support on R 2 , and let Λ ⊂ R 2 be a countable set.Then As a simple consequence of Theorem 2.1, the following useful theorem was proved in [9] and will be used to prove our main result.
Theorem 2.2.Let µ = µ 0 * µ 1 be the convolution of two probability measures µ i , i = 0, 1, which are not Dirac measures.Suppose that Λ is a bi-zero set of µ 0 .Then Λ is also a bi-zero set of µ, but it cannot be a spectrum of µ.
We define an equivalence relation Proof of Theorem 1.2.Let Λ be a spectrum of µ.Then We take 0 ∈ Λ ⊂ Λ as a maximal bi-zero set of δ B .Write for some t ∈ N.Then, for any λ ∈ Λ \ Λ , there is a λ i ∈ Λ such that λ − λ i ∈ Z(ν) \ Z( δB ).And we assert that the λ i is unique.Suppose, on the contrary, that The conditions of the theorem imply that Then the assertion follows.
where Λ i ∩ Λ j = ∅ for any i = j.Now we need the following two claims to complete the proof.
The first case follows directly from Lemma 3.1.And it is obvious that λ − λ ∈ Z(ν) \ Z( δB ) in the second case above.Hence the claim is proved.
Proof.For any λ ∈ Λ i , λ ∈ Λ j , we have For the case in which For the remaining three cases, it is easy to verify that λ − λ ∈ Z( δB ).Then the claim follows.
Due to Λ i \{0} ⊂ Λ\{0} ⊂ Z(μ) ⊂ A −1 Z 2 for some integral invertible matrix A, we know that Λ i /∼ is a finite set and Λ i /∼ is a partition of Λ i .And thus we write It follows that for any ξ ∈ (0, 1) 2 and 1 ≤ i ≤ t, there exists λ i,ξ(i) with ξ(i) ∈ {1, 2, . . ., n 1 } such that That is, for any ξ ∈ (0, 1) 2 , there exist {λ i,ξ(i) } t i=1 corresponding to it.As Λ i /∼ is a finite set for each 1 ≤ i ≤ t but there are infinitely many points in (0, 1) 2 , we can find a finite set { λ i } t i=1 in which λ i = λ i,ξ(i) for infinitely many ξ ∈ I ⊂ (0, 1) 2 .Then, for any ξ ∈ I, we have We know from Claim 3.2 that Λ i is an orthogonal set of ν for each 1 ≤ i ≤ t; then the second to last inequality in The property of entire function implies that, for any Hence { λ i } t i=1 is a spectrum of δ B and each Λ i is a spectrum of ν.

Sufficiency of Theorem 1.1
We will prove the sufficiency of Theorem 1.1 in this section.Wang and Dong [31] proved the sufficient case for more general 3-digit sets.In this section, we give another simple proof for it.This proof depends closely on the zero set of the Fourier transform μ{Mn},D .By the definition of Fourier transform of μ{Mn},D and (1.1), for any ξ ∈ R Then we have By calculation, we obtain where Then For any k ≥ 1, we define Then we have the following result.The sufficiency of Theorem 1.1 follows immediately from it.
Theorem 4.1.Let µ {Mn},D be the Moran-Sierpinski-type measure defined as in (1.1).If 3 | p n and 3 | q n for all n ≥ 2, then µ {Mn},D is a spectral measure with a spectrum where C is defined as in (4.3).
Proof.Firstly, we will show that Λ is a bi-zero set of µ {Mn},D .For any two distinct elements λ, λ ∈ Λ, we can write where m, l ≥ 1 and for some N 1 , N 2 ∈ Z.This together with (4.2) implies that Therefore Λ is a bi-zero set of µ {Mn},D .We now show the completeness of Λ.For any m ≥ 1, set where C is defined as in (4.3).Proceeding as in the proof above, we know that Λ m is a bi-zero set of µ m .Notice that #Λ m = 3 m = dim(L 2 (µ m )).Hence Λ m is a spectrum of µ m , and Theorem 2.1 implies that, for any ξ ∈ R 2 , we have Then, for any λ ∈ Λ, we have f (λ) = lim m→∞ f m (λ) and Moreover, We now claim that there exists a constant c > 0 such that for any m ≥ 1, where |ξ| < 1 3 and λ ∈ Λ m .Note that

QIAN LI AND MIN-MIN ZHANG
Combining with (4.5), we have If k ≥ m + 2, then we know from (4.6), (4.7), and (4.8) that These together with (4.5) imply that Thus the claim holds.Combining this claim with (4.4), we obtain By the dominated convergence theorem, we conclude that for any ξ ∈ R 2 with |ξ| < 1 3 .As Q Λ (ξ) is an entire function, we obtain that Q Λ (ξ) ≡ 1 for any ξ ∈ R 2 .By Theorem 2.1, we know that µ {Mn},D is a spectral measure.Now the proof is complete.

Necessity of Theorem 1.1
In this section, we will give the proof of the necessity of Theorem 1.1.For that purpose, we need the following technical theorem, which plays a crucial role in the proof.Moreover, the following theorem shows that if µ {Mn},D is a spectral measure, then any "truncation" of it is still a spectral measure.Theorem 5.1.Let µ {Mn},D be the Moran-Sierpinski-type measure defined by (1.1) and p n ≡ ±q n (mod 3) for all n ≥ 2. If µ {Mn},D is a spectral measure, then both µ k and µ >k are spectral measures for any k ≥ 1.
To prove Theorem 5.1, we need the following lemmas.Lemma 5.2.Let p n ≡ ±q n (mod 3) for all n ≥ 2. Suppose that µ {Mn},D is a spectral measure.If {λ 1 , λ 2 } is a bi-zero set of µ {Mn},D with λ 1 ∈ Z(μ k ) and λ 2 ∈ Z(μ >k ) \ Z(μ k ) for any k ≥ 1, then where j 1 ≤ k < j 2 and a 1 , a 2 ∈ A 1 ∪ A 2 .Suppose, on the contrary, that there exists j > k such that Then there exists Note that a i ∈ A 1 ∪ A 2 for all i = 1, 2, 3.This means that a i1 ≡ a i2 (mod 3) and a i1 , a i2 ∈ Z \ 3Z, i = 1, 2, 3. (5.2) Without loss of generality, we assume that j 2 ≤ j.If j 2 = j, then (5.1) implies Combining with (5.2), we obtain Furthermore, we have Let Λ denote a bi-zero set of µ {Mn},D .Then Λ is also a bi-zero set of ν.Hence it follows from Theorem 2.2 that Λ is not a spectrum of µ {Mn},D .Therefore, µ {Mn},D is not a spectral measure, a contradiction.Now we consider the case j 2 < j.Then (5.1) implies (5.3) This together with (5.2) implies that Moreover, applying the condition of p n ≡ ±q n (mod 3) for n ≥ 2, it follows that Applying (5.2) and (5.3) again, we get Proceeding as in the proof of the above case, we get a contradiction.Then we complete the proof of the lemma.
The conclusion follows from Theorem 2.1.
Proof of Theorem 5.1.For any k ≥ 1, we write Then we have Notice that (5.4) We know from (4.2) that Z μ{Mn},D ⊂ 1 3 Z 2 .This together with (5.4) implies that Z(μ {Mn},( is an integral invertible matrix.Then we know from Lemma 5.2 and Theorem 1.2 that both δ B and ν are spectral measures.Applying Lemma 5.3, we obtain that µ k and µ >k are all spectral measures.The proof is completed.
Recall that Proof.Suppose, on the contrary, that 3 p 2 or 3 q 2 .We just prove the case in which 3 p 2 .The proof of the remaining case is similar and we omit it here.Let Λ denote a bi-zero set of µ