Fourier bases of a class of planar self-affine measures

Let $\mu_{M,D}$ be the planar self-affine measure generated by an expansive integer matrix $M\in M_2(\mathbb{Z})$ and a non-collinear integer digit set $D=\left\{\begin{pmatrix} 0\\0\end{pmatrix},\begin{pmatrix} \alpha_{1}\\ \alpha_{2} \end{pmatrix}, \begin{pmatrix} \beta_{1}\\ \beta_{2} \end{pmatrix}, \begin{pmatrix} -\alpha_{1}-\beta_{1}\\ -\alpha_{2}-\beta_{2} \end{pmatrix}\right\}$. In this paper, we show that $\mu_{M,D}$ is a spectral measure if and only if there exists a matrix $Q\in M_2(\mathbb{R})$ such that $(\tilde{M},\tilde{D})$ is admissible, where $\tilde{M}=QMQ^{-1}$ and $\tilde{D}=QD$. In particular, when $\alpha_1\beta_2-\alpha_2\beta_1\notin 2\Bbb Z$, $\mu_{M,D}$ is a spectral measure if and only if $M\in M_2(2\mathbb{Z})$.


Introduction
Let µ be a Borel probability measure with compact support on ‫ޒ‬ n , and let ⟨ • , • ⟩ denote the standard inner product on ‫ޒ‬ n .We say that µ is a spectral measure if there exists a countable set ⊂ ‫ޒ‬ n such that the exponential function system E := {e 2πi⟨λ,x⟩ : λ ∈ } forms an orthonormal basis for the Hilbert space L 2 (µ).In this case, we call a spectrum of µ and (µ, ) a spectral pair.In particular, if µ is the normalized Lebesgue measure supported on a Borel set , then is called a spectral set.
Spectral measure is a natural generalization of spectral set introduced by Fuglede [20], who proposed the famous conjecture that is a spectral set if and only if is a translational tile.It is known [22] that a spectral measure µ must be of pure type: µ is either discrete, or absolutely continuous or singularly continuous.The first singularly continuous spectral measure was constructed by Jorgensen and Pedersen in 1998 [24].They proved that the middle-fourth Cantor measure is a spectral measure with a spectrum Following this discovery, there is a considerable number of papers on the spectrality of self-affine measures and the construction of their spectra; see [2; 3; 5; 6; 7; 8; 12; 13; 16; 18; 29].These results are generalized further to some classes of Moran measures (see, e.g., [1; 9; 19]), and some surprising convergence properties of the associated Fourier series were discovered in [38; 39].These fractal measures also have very close connections with the theory of multiresolution analysis in wavelet analysis; see [11].
In [14], Dutkay and Jorgensen summarized some known results regarding iterated function systems (IFS); see [23] for details.Two approaches to harmonic analysis on IFS have been popular: one based on a discrete version of the more familiar and classical second-order Laplace differential operator of potential theory; see [27; 28; 30]; and the other is based on Fourier series.The first model in turn is motivated by infinite discrete network of resistors, and the harmonic functions are defined by minimizing a global measure of resistance, but this approach does not rely on Fourier series.In contrast, the second approach begins with Fourier series, and it has its classical origins in lacunary Fourier series [26].
For an expansive real matrix M ∈ M n ‫)ޒ(‬ and a finite digit set D ⊂ ‫ޒ‬ n with cardinality #D, the iterated function system (IFS) {φ d (x)} d∈D is defined by φ d (x) = M −1 (x + d) (x ∈ ‫ޒ‬ n , d ∈ D).By [23], there exists a unique probability measure µ M,D satisfying It is supported on the unique nonempty compact set T (M, D) = d∈D φ d (T (M, D)).Hence The measure µ M,D and the set T (M, D) are called self-affine measure and selfaffine set, respectively.It is known that a self-affine measure µ M,D can be expressed by the infinite convolution of discrete measures as where * is the convolution sign, δ E = 1 #E e∈E δ e for a finite set E and δ e is the Dirac measure at the point e.
Self-affine measures have the advantage that their Fourier transforms (see ) can be explicitly written down as an infinite product, which allows us to compute their zeros.The previous research on self-affine measures µ M,D and their Fourier transform have revealed some surprising connections with a number of areas in mathematics such as harmonic analysis, dynamical systems, number theory and others (see, e.g., [21; 25; 37]).
In the previous works, the spectral self-affine measures are usually generated by compatible pairs (known also as Hadamard triples).The appearance of compatible pairs stems from the terminology of [38].
is unitary, i.e., H * H = I , where I is a n×n identity matrix.
The well-known result of Jorgensen and Pedersen [24] shows that if (M, D) is admissible, then there are infinite families of orthogonal exponential functions in L 2 (µ M,D ).Dutkay and Jorgensen [13; 15] formulated the famous conjecture that if (M, D) is admissible, then µ M,D is a spectral measure.It was first proved in one dimension by Łaba and Wang [29].The conjecture is true in higher dimensions under some additional assumptions, introduced by Strichartz [38].There are many other papers that investigated it in higher dimensional cases; see [12; 32].In the end, Dutkay, Haussermann and Lai [16] proved that: Theorem 1.2.Let M ∈ M n ‫)ޚ(‬ be an expansive integer matrix, and let D ⊂ ‫ޚ‬ n be a finite digit set.If (M, D) is admissible, then µ M,D is a spectral measure.
In [18], Fu, He and Lau gave an example to illustrate that the sufficient condition in Theorem 1.2 is not necessary in one dimension.For an expansive integer matrix M ∈ M 2 ‫)ޚ(‬ and the classic digit set D = 0 0 , 1 0 , 0 1 , the spectrality and nonspectrality of the corresponding self-affine measure µ M,D has been widely investigated by many researchers; see [12; 31; 32].Eventually, An, He and Tao [2] completely settled the spectrality of µ M,D .More precisely, they showed that µ M,D is a spectral measure if and only if (M, D) is admissible.For a more general integer digit set D with 0 ∈ D and #D = 3, there is also a complete spectral characterization; see [4; 35; 36].In addition to these, another important integer digit set is where α 1 β 2 − α 2 β 1 ̸ = 0.The existence of infinitely many orthogonal exponentials in L 2 (µ M,D ) has been fully studied in [33; 40; 41].Recently, Fu and Tang [17] considered the special case where α 1 = 1, α 2 = 0, β 1 = 0 and β 2 = 1.They fully characterized the spectrality of the corresponding self-affine measures.However, to the best of our knowledge, the complete description of spectral properties of the general case (1-2) is not known yet.A natural subsequent question is: Question 1.For an expansive integer matrix M ∈ M 2 ‫)ޚ(‬ and the digit set D given by (1)(2), what is the sufficient and necessary condition for µ M,D to be a spectral measure?
In the study of the spectrality of self-affine measures µ M,D on ‫ޒ‬ n , the finiteness and rationality of the set Z n D := x ∈ [0, 1) n : d∈D e 2πi⟨d,x⟩ = 0 are pivotal.Many classic digit sets, such as {0, 1, . . ., N − 1}, {(0, 0) t , (1, 0) t , (0, 1) t } and the digit set D given by (1-2), exhibit the desired property.This has attracted a large number of researchers to study their spectrality of the corresponding self-affine measures.However, if Z n D is infinite or irrational, resolving the spectrality of the corresponding self-affine measure becomes a formidable challenge.For instance, consider This means that Z 2 D encompasses a submanifold characterized by the free variable a ∈ [0, 1).For the more general digit set D = {0, u, v, u + v} ⊂ ‫ޚ‬ 2 , the set Z 2 D is infinite and includes free variables.The spectral properties of these self-affine measures have not been resolved.
The cardinality #D of a digit set D significantly influences the properties of Z n D .In [3], An, He and Lai extensively classified four-element digit spectral self-similar measures on ‫.ޒ‬They showed that if #D = 4 and the corresponding self-similar measure is a spectral measure, then D is rational and Z 1 D is finite and rational.However, if D does not have any special structures and #D ≥ 5, the set Z n D is hard to calculate and may be irrational.For example, let D = {0, 1, 3, 5, 6}.Then [3,Example 5.2].This makes it very difficult to study the spectrality of the corresponding self-similar measure.
Inspired by the above researches and due to the finiteness and rationality of the set Z 2 D corresponding to the digit set D given by (1-2), we can give an answer to Question 1.Before presenting our results, a reasonable assumption for the digit set D is necessary.Without loss of generality, we can assume that gcd(α 1 , α 2 , β 1 , β 2 ) = 1 by Lemma 2.2.
Our first main result is as follows: Theorem 1.3.Let µ M,D be defined by (1-1), where M ∈ M 2 ‫)ޚ(‬ is an expansive integer matrix and D is given by (1-2).Then µ M,D is a spectral measure if and only if there exists a matrix Q ∈ M 2 ‫)ޒ(‬ such that ( M, D) is admissible, where We remark that Theorem 1.3 gives a complete answer to the spectral Question 1.We now outline the strategy of the proof of Theorem 1.3.The sufficiency of Theorem 1.3 follows directly from Theorem 1.2 and Lemma 2.2.The more challenging part of the proof is the necessity.The key point is to construct a self-affine measure µ M, D so that it has the same spectrality as the measure µ M,D , and then the necessity follows immediately from Theorems 1.5 and 1.6.What is exciting is that the proof method of the necessity is new and completely different from the previous work proving spectral self-affine measures.
Theorem 1.5.Let µ M, D and F2 p be defined by (1-1) and (1-3), respectively, where M and D are given by (1-4) and (1-5), respectively.If η = 0, then the following statements are equivalent: On the other hand, if η > 0 in D, the form of M is different from that in the case η = 0. | c.
We now give a brief explanation of the proofs of Theorems 1.5 and 1.6.The main technical difficulty in the proofs lies in "(i) ⇒ (ii)" of Theorem 1.5 and the necessity of Theorem 1.6.More precisely, the key point is to construct a Moran measure µ A, M, D (see (3-1)) so that it has the same spectrality as µ M, D .For the matrix A, we need to cleverly describe its complete residue system (Proposition 3.3).We carefully investigate the structure of the spectrum of µ A, M, D (see (3)(4)(5)(6)(7)(8)(9)(10)(11)).And then we get a property of decomposition on the spectrum of µ M, D under the assumption that µ A, M, D is a spectral measure (Lemma 3.5).With their help, the proof becomes within reach.
The paper is organized as follows.In Section 2, we introduce some basic definitions and lemmas.In Section 3, we focus on proving Theorems 1.5 and 1.6.Finally, we prove Theorems 1.3 and 1.4, and give some concluding remarks in Section 4.

Preliminaries
For the self-affine measure µ M,D defined by (1-1), the Fourier transform of µ M,D is defined by where M * denotes the transpose of M and m D ( • ⟩ is the mask polynomial of D. We denote the set of all the roots of f (x) by Z( f ), i.e., For a countable set If E forms an orthogonal family of L 2 (µ M,D ), then is called an orthogonal set of µ M,D .Note that the properties of spectra are invariant under a translation, so we can always assume that 0 ∈ .In a number of applications, one encounters a measure µ and a subset such that the functions e 2πi⟨λ,x⟩ indexed by are orthogonal in L 2 (µ), but a separate argument is needed in order to show that the family is complete.Let The following result is a basic criterion for the spectrality of µ.
Theorem 2.1 [24].Let µ be a Borel probability measure with compact support on ‫ޒ‬ n , and let ⊂ ‫ޒ‬ n be a countable set.Then: The following lemma indicates that the spectrality of µ M,D is invariant under a similarity transformation.Lemma 2.2 [12].Let D 1 , D 2 ⊂ ‫ޒ‬ n be two finite digit sets with the same cardinality, and let M 1 , M 2 ∈ M n ‫)ޒ(‬ be two expansive real matrices.If there exists a matrix The following result is a known fact, which was proved in [16] and will be used in the proof of Proposition 3.3.Lemma 2.3.Let M ∈ M n ‫)ޚ(‬ be an expansive integer matrix, and let D, S ⊂ ‫ޚ‬ n be two finite digit sets with the same cardinality.Then the following three statements are equivalent: Recalling that µ M,D is defined by (1-1), we let A be a nonsingular matrix and define the Moran measure The following lemma indicates the spectrality of µ M,D is independent of A. The proof is the same as that of [9, Lemma 3.1; 10, Lemma 2.6].For the convenience of readers, we include the proof here.
Proof.Applying (2-1) and (2-5), we have Hence the second assertion follows by Theorem 2.1.□ We conclude this section by recalling a useful lemma in our investigation, which was proved by Deng et al. in [9,Lemma 2.5].
Lemma 2.5.Let p i, j be positive numbers such that n j=1 p i, j = 1, and let q i, j be nonnegative numbers such that m i=1 max 1≤ j≤n q i, j ≤ 1.Then m i=1 n j=1 p i, j q i, j = 1 if and only if q i,1 = • • • = q i,n for 1 ≤ i ≤ m and m i=1 q i,1 = 1.
3. Proofs of Theorems 1.5 and 1.6 We focus on proving Theorems 1.5 and 1.6, that is, studying the spectrality of the measure µ M, D , where M and D are given by (1-4) and (1-5), respectively.For this purpose, we first give some properties of Z(m D ), and then investigate the structure of the spectrum of µ M, D under the assumption that µ A, M, D is a spectral measure, where µ A, M, D is defined by (2)(3)(4)(5).With these preparations, we will achieve our goal.By Lemma 2.4, without loss of generality, we assume in the rest of the paper that The matrix A will be pivotal in constructing the spectrum of µ M, D .Consequently, (3-1) We now make a detailed analysis on the zero set Z(m D ) of m D .
(ii) For any θ i ∈ i , it is easy to verify that Hence the assertion follows by using (3-3).
To investigate the spectrality of µ M, D , we need to construct a complete residue system of matrix A. In view of (3-1) and (3-3), one may easily get that Throughout this paper, we set h p = {0, 1, . . ., p − 1} for an integer p ≥ 1, and let where q is a nonnegative integer and Proposition 3.3.With the above notation, the following statements hold: (iii) S η ⊕ 2 η+1 αβT η is a complete residue system of matrix A in (ii).
Proof.According to the definitions of T η,i and i , (i) is obvious.We now prove (ii).
Proof.If is a nonempty set, we will complete the proof in the following two steps.
Step 1.We prove that is an orthogonal set of µ M, D .
) is a spectrum of µ A, M, D with 0 ∈ .Then we can conclude from (3-21) that for any s ∈ S η , one of the following two statements holds: (i) There exist some ℓ i s ∈ T η,i s such that s,ℓ is ̸ = ∅ for all 0 ≤ i s ≤ 3.
In order to prove Theorems 1.5 and 1.6 more conveniently, we define We have all ingredients for the proof of Theorem 1.5.
Proof of Theorem 1.5.We will prove this theorem by the circle (ii) Hence the assertion follows.
Proof of Theorem 1.6.We first prove the necessity.Suppose µ M, D is a spectral measure.In view of Lemma 3.
and D = Q D is given by .Since η > 0, it follows from Theorem 1.5 that µ M ′ , D is a spectral measure.Therefore, µ M, D is a spectral measure by Lemma 2.2.This completes the proof of Theorem 1.6.□ 4. Proofs of Theorems 1.3 and 1.4 We are committed to investigating the spectrality of the measure µ M,D , where M ∈ M 2 ‫)ޚ(‬ is an expansive integer matrix and D is given by (1-2).We first prove Theorem 1.3 by using Theorems 1.5 and 1.6, and then prove Theorem 1.4.Finally, we provide some concluding remarks.
Proof of Theorem 1.Therefore, the desired result now is obtained by appeal to Theorem 1.5.□ At the end of this paper, we give some further remarks and list an open question which is related to our main results.The following example is specifically used to display our results, which are convenient to judge whether the measure µ M,D in Question 1 is a spectral measure.(ii) µ M 1 ,D 2 is a nonspectral measure, while µ M 2 ,D 2 is a spectral measure.
Proof.By a simple calculation, this follows directly from Theorems 1.5 and 1.6.□ It is worth noting that if α 1 β 2 − α 2 β 1 ∈ ‫ޚ2‬ in Theorem 1.3, we cannot give the specific form of matrix M.However, if α 1 , α 2 , β 1 and β 2 are fixed, we can describe the specific form by applying Theorem 1.6.The following simple but interesting example is devoted to illustrating this fact.We remark here that the digit set D in (1-2) satisfies α 1 β 2 − α 2 β 1 ̸ = 0, and so it is of interest to consider the following question: Question 2. For an expansive matrix M ∈ M 2 ‫)ޚ(‬ and the digit set with α 1 β 2 − α 2 β 1 = 0, what is the sufficient and necessary condition for µ M,D to be a spectral measure? In fact, for the matrix M and the digit set D given in the above question, using the methods of [34], we can find an integer matrix Q such that M := Q M Q −1 and is independent of ξ 2 , where ξ = (ξ 1 , ξ 2 ) t .We have not yet discovered an effective method to address this situation.An answer to Question 2 may provide insights into the study of the spectrality of fractal measures.

Definition 1 . 1 .
Let M ∈ M n ‫)ޚ(‬ be an expansive integer matrix, and let D, S ⊂ ‫ޚ‬ n be two finite digit sets with #D = #S = N .We say that (M, D) is admissible (or (M −1 D, S) forms a compatible pair or (M, D, S) forms a Hadamard triple) if the matrix
is a spectral measure by Theorem 1.2.

3 .
The sufficiency follows directly from Theorem 1.2 and Lemma 2.2.Now we are devoted to proving the necessity.Suppose that µ M,D is a spectral measure.Let η = max{r : 2 r | (α 1 β 2 − α 2 β 1 )}, and let M and D be given by (1-4) and (1-5), respectively.That is, M = Q M Q −1 and D = Q D. In view of Lemma 2.2, µ M, D is a spectral measure.It suffices to prove that there exists a matrix Q ∈ M 2 ‫)ޒ(‬ such that ( M, D) is admissible, where M = Q M Q−1 and D = Q D. The proof will be divided into the following two cases.Case 1: η = 0. Since µ M, D is a spectral measure, it follows from η = 0 and Theorem 1.5 that ( M, D) is admissible.Thus the assertion follows by taking Q = diag(1, 1).Case 2: η > 0. Since µ M, D is a spectral measure, Theorem 1.6 implies that one may write D) is admissible by Theorem 1.5.This completes the proof of Theorem 1.3.□ Next, we focus on proving Theorem 1.4.Proof of Theorem 1.4.Let M and D be given by (1-4) and (1-5), respectively.That is,(4-1) M = Q M Q −1 and D = Q D,where the matrixQ ∈ M 2 ‫)ޚ(‬ satisfies det(Q) = 1.In view of Lemma 2.2, µ M,Dis a spectral measure if and only if µ M, D is a spectral measure.This implies that Theorem 1.4(i) is equivalent to Theorem 1.5(i).Note that det(Q) = 1; hence, by a simple calculation, one has thatM ∈ M 2 ‫)ޚ2(‬ ⇐⇒ M ∈ M 2 ‫.)ޚ2(‬ThusTheorem 1.4(ii) and (iii) are equivalent to Theorem 1.5(ii) and (iii), respectively.Finally, from the Definition 1.1 and (4-1), it is easy to see that ( M, D) is admissible ⇐⇒ there exists a set C ⊂ ‫ޚ‬ 2 such that ( M, D, C) is a Hadamard triple ⇐⇒ (M, D, Q * C) is a Hadamard triple ⇐⇒ (M, D) is admissible.Consequently, Theorem 1.4(iv) is equivalent to Theorem 1.5(iv).