Monatsh Math DOI 10.1007/s00605-017-1096-0

Spectrality of self-affine measures and generalized compatible pairs Jian-Lin Li1

Received: 7 March 2016 / Accepted: 31 August 2017 © Springer-Verlag GmbH Austria 2017

Abstract Let μ M,D be a self-affine measure associated with an expanding matrix M ∈ Mn (Z) and a finite digit set D ⊂ Zn . In this paper, we study the spectrality of μ M,D by introducing the generalized compatible pairs via Hadamard matrices. This is motivated by the problem of looking for conditions for μ M,D −orthogonal exponential system to be infinite. Based on the properties of Hadamard matrices, we first present some elementary properties concerning the generalized compatible pair. We then provide a method of getting μ M,D -orthogonal exponentials. Certain relationships between the generalized compatible pair and the spectrality of self-affine measure are established, which extend the known results in the appropriate manner. The research here is closely related to the spectral problem of self-affine measures. Keywords Self-affine measure · Spectral pair · Generalized compatible pair · Digit set Mathematics Subject Classification 28A80 · 42C05 · 46C05

1 Introduction and notations Hadamard matrices have wide applications to many areas in mathematics and in physics (see, e.g. [1,33] and references cited therein). We are motivated by the appear-

Communicated by A. Constantin.

B 1

Jian-Lin Li [email protected] School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, People’s Republic of China

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ance of Hadamard matrices in the context of Fourier analysis on fractals (see [15,16]). This is an interesting subject in adapting the traditional Fourier analysis to the fractal setting, where the Hadamard matrices play an important role (see [2,31,32]). In the aspect of fractals, a finite number of contractive affine mappings φd (x) = M −1 (x + d),

(x ∈ Rn , d ∈ D)

are considered, where M ∈ Mn (Z) is an expanding integer matrix, and D ⊂ Zn is a finite digit set of cardinality |D|. As an affine iterated function system (IFS) {φd (x)}d∈D , Hutchinson’s [13] famous theorem says that there is a unique nonempty compact subset T := T (M, D) ⊂ Rn satisfying T = d∈D φd (T ). Also there is a unique probability measure μ := μ M,D satisfying the self-affine identity with equal weight: 1  μ= μ ◦ φd−1 . (1.1) |D| d∈D

The set T (M, D) is called attractor or invariant set of the affine IFS {φd (x)}d∈D . It includes a complicated geometric structure and behaves like a fractal. The measure μ M,D is called self-affine measure, it includes the restriction of n−dimensional Lebesgue measure as its special case. Moreover, the support of μ M,D is T (M, D). From (1.1), the Fourier transform μˆ M,D (ξ ) of the measure μ M,D is given by  μˆ M,D (ξ ) =

e

2πi

dμ M,D (x) =

∞ 

m D (M ∗− j ξ ),

(ξ ∈ Rn )

(1.2)

j=1

where M ∗ denotes the transposed conjugate of M (in fact, M ∗ = M t ) and m D (x) =

1  2πid,x e , (x ∈ Rn ). |D| d∈D

The previous research on μ M,D and its Fourier transform (1.2) have revealed some surprising connections with a number of areas in mathematics such as harmonic analysis, number theory, dynamical system, and others (see, e.g. [12,14,30] and references cited therein). In the aspect of Fourier analysis, we know that the Lebesgue measure plays a central role in the theory of Fourier series and Fourier transforms. What happens to the known results if we replace the Lebesgue measure by a more general measure such as μ M,D ? There are a lot of common interests on the question. We mainly restrict our research on the spectrality of μ M,D . Definition 1.1 For a probability measure μ of compact support on Rn , we call μ a spectral measure if there exists a discrete set  ⊂ Rn such that E() := {e2πiλ,x : λ ∈ } forms an orthogonal basis (Fourier basis) for the Hilbert space L 2 (μ). The set  is then called a spectrum for μ; we also say that (μ, ) is a spectral pair.

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The spectral measures are the extension of spectral sets introduced by Fuglede [10] whose famous spectrum-tiling conjecture has received much attention in the past forty years. It should be pointed out that the disproof of such spectrum-tiling conjecture in the higher dimensions (n ≥ 3) by [9,29,34] depends upon a kind of complex Hadamard matrices (see [18]). There are many interesting conjectures that are related to the Fuglede’s conjecture, for example, the dual Fuglede’s spectral-set conjecture proposed by Jorgensen and Pedersen [17], the spectral-set duality conjecture or the weak spectral-set conjecture proposed by Lagarias et al. [20] (see also [21]), and the generalized Fuglede’s conjecture proposed by Gabardo and Lai [11]. For a given measure μ M,D , the most interesting problem in this respect is to determine the spectrality or non-spectrality of μ M,D . And in the case when it is a spectral measure, one needs to determine the Fourier bases in the Hilbert space L 2 (μ M,D ). All these are directly connected with the above (1.2) simply due to the fact that (μ M,D , ) is a spectral pair if and only if λ∈ |μˆ M,D (ξ + λ)|2 = 1 for all ξ ∈ Rn . Moreover, the appearance of Hadamard matrices in this subject leads the research to an approachable way. Definition 1.2 For two finite subsets G and P of Rn of the same cardinality q, we say (G, P) is a compatible pair if the q × q matrix HG,P := [e2πig, p ]g∈G, p∈P ∗ H −1/2 H satisfies HG,P G,P = q Iq , i.e. q G,P is unitary.

In some contexts, such a matrix HG,P is called a complex Hadamard matrix, which is orthogonal but not unitary. With the given M and D in the integer case, the key point with a Hadamard matrix is the existence of a set S ⊂ Zn such that (M −1 D, S) is a compatible pair. If so, i.e. there is a finite subset S ⊂ Zn of the cardinality |S| = |D| and 0 ∈ S such that (M −1 D, S) is a compatible pair, one can consider the dual IFS {ψs (x) = M ∗ x + s}s∈S , and use (M, S) to denote its invariant set. This is the expansive orbit of 0 under {ψs (x)}s∈S , that is, ⎧ ⎫ k ⎨ ⎬ (M, S) := M ∗ j s j : k ≥ 0 and all s j ∈ S . (1.3) ⎩ ⎭ j=0

Then the well-known result of Jorgensen and Pedersen [15,16] shows that E((M, S)) is an infinite orthogonal system in L 2 (μ M,D ). Furthermore, in some cases, E((M, S)) may be an orthogonal basis for the Hilbert space L 2 (μ M,D ). Łaba and Wang [19] extended this result in the dimension one (n = 1), and showed that μ M,D is always a spectral measure. In the higher dimensions (n ≥ 2), it often needs additional condition for E((M, S)) to be an orthogonal basis for L 2 (μ M,D ) (see [27]). Nevertheless, the recent result of Dutkay, Hausserman and Lai [3] (see Theorem A below) ensures the spectrality of μ M,D , and the remaining problem relating to this case is to determine all the spectra for such a measure μ M,D . If not so, i.e. there is no any set S ⊂ Zn of the cardinality |S| = |D| and 0 ∈ S such that (M −1 D, S) is a compatible pair, the above discussion fails and there are a

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few results on the spectrality of μ M,D . This arises a natural question of how to deal with the spectrality of μ M,D in this case. Note that, in this case one can still define the set (1.3) without the compatible pair (M −1 D, S). The set (M, S) depends upon a matrix M and a finite set S only. The research on the problem of looking for a suitable set S such that E((M, S)) is an infinite orthogonal system or an orthogonal basis for the Hilbert space L 2 (μ M,D ) is still an important subject. In the present paper, we find that the required condition |S| = |D| in a compatible pair (M −1 D, S) can be relaxed. By introducing a generalized compatible pair (M −1 D, S) in which the condition |S| = |D| may not be satisfied, we study the spectrality of μ M,D in a comparatively large range. We organize the paper as follows. In the next section (Sect. 2), we present some elementary properties on the generalized compatible pair. In Sect. 3, we provide a method of getting μ M,D −orthogonal exponentials. Certain relationships between the generalized compatible pair and the spectrality of self-affine measure are also established in the final section. The results here extend the corresponding known results in the appropriate manner.

2 Generalized compatible pairs We first introduce the following definition which generalizes Definition 1.2. Definition 2.1 For two finite subsets B and L given by B = {b1 , b2 , . . . , bk } ⊂ Rn and L = {l1 , l2 , . . . , ll } ⊂ Rn , where |B| = k > 1 and |L| = l > 1, we say (B, L) is a generalized compatible pair if the k × l matrix ⎡ 2πib ,l  2πib ,l  ⎤ 1 1 e 1 2 · · · e 2πib1 ,ll  e ⎢ e2πib2 ,l1  e2πib2 ,l2  · · · e2πib2 ,ll  ⎥ ⎢ ⎥ H B,L := ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . . e2πibk ,l1  e2πibk ,l2  · · · e2πibk ,ll 

∗ H satisfies H B,L B,L = k Il .

Note that if (B, L) is a generalized compatible pair, then we can translate either B or L and obtain another generalized compatible pair. So we may assume without essential loss of generality that 0 belongs to both B and L. In the case when k = l, the generalized compatible pair (B, L) is reduced to the compatible pair. It is the same as the compatible pair that for a given B such as B = {0, 1, 3}, there may not be any L such that (B, L) is a generalized compatible pair. Also there are many sets B such as B = {0, 1, 3, 4, 5, 7}, the compatible pair (B, L) does not exist, but the generalized compatible pair (B, L) (for example, take L = {0, 1/8}) exists. From Definition 2.1, one can check that the following result holds (see [22]). Proposition 2.1 Let B, L ⊂ Rn be two finite subsets. Then the following statements are equivalent:

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(i) (B, L) is a generalized compatible pair; (ii) (R B, R ∗−1 L) is a generalized compatible pair for any non-singular matrix R ∈ Mn (R); (iii) m B (l1 − l2 ) = 0, for any distinct l1 , l2 ∈ L. It should be pointed out that for a generalized compatible pair (B, L), if |B| ≤ |L|, then (L , B) is a generalized compatible pair; if |B| > |L|, then (L , B) is not necessarily a generalized compatible pair. Indeed, for any ξ ∈ Rn , we have 

|m L (ξ + b)|2 =

1    2πi<(ξ +b),(l1 −l2 )> e (|L|)2 b∈B l1 ∈L l2 ∈L

b∈B

|B|   2πi<ξ,(l1 −l2 )> = e m B (l1 − l2 ) (|L|)2 l1 ∈L l2 ∈L

=

|B| (by Proposition 2.1(iii)). |L|

(2.1)

 In the case when |B| ≤ |L|, we further have b∈B |m L (ξ + b)|2 ≤ 1(∀ξ ∈ Rn ) which implies m L (b1 − b2 ) = 0 for any distinct b1 , b2 ∈ B. That is, (L , B) is a generalized compatible pair. In the case when |B| > |L|, we have examples such as B = {0, 1/3, 2/3} and L = {0, 1} in R (see also Example 2.1 below) to illustrate that (B, L) is a generalized compatible pair but (L , B) is not. For the spectrality of self-affine measure μ M,D , we often need to look for a finite set S such that (M −1 D, S) is a compatible pair. This is due to the following result first conjectured by Dutkay and Jorgensen [5, Conjecture 2.5] [7, Conjecture 1.1] (see also [6, Problem 1]) and then settled by Dutkay, Hausserman and Lai [3]. Theorem A Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. If there exists a subset S ⊂ Zn , 0 ∈ S such that (M −1 D, S) is a compatible pair, then μ M,D is a spectral measure. Equivalently, Theorem A Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. If μ M,D is a non-spectral measure, then there exist no subset S ⊂ Zn , 0 ∈ S such that (M −1 D, S) is a compatible pair. Before the result of [3], Theorem A or A holds in the dimension n = 1 [19], in the case when |D| = |S| = | det(M)| [22] and in the case when | det(M)| is a prime number [23], it also holds in higher dimensions with additional conditions [4,5,7,8]. Applying Theorem A, one can obtain certain spectral self-affine measures. However there are many spectral self-affine measures whose spectrality cannot be obtained from the compatible pair (see [24, Section 2], [26, p. 308] and [35]). Also a spectral measure often has more than one spectrum (not translates of each other), and the question of determining all the spectra for such a measure μ M,D will need further research. For the pair (M, D) in the integer case, that is, M ∈ Mn (Z) is an expanding integer matrix and D ⊂ Zn is a finite digit set with 0 ∈ D, if there exists a subset

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S ⊂ Rn , 0 ∈ S such that (M −1 D, S) is a compatible pair, then there are examples to illustrate that μ M,D is a non-spectral measure (see [23, Example 1]). What we can say in this case is that E(S) is a finite orthogonal system in L 2 (μ M,D ). Indeed, from Proposition 2.1(iii) and (1.2), we have, for any distinct s1 , s2 ∈ S, m D (M ∗−1 (s1 − s2 )) = 0 and thus μˆ M,D (s1 − s2 ) = 0. Therefore, in the following discussion, we mainly consider the case that (M −1 D, S) is a generalized compatible pair, where M ∈ Mn (Z) is an expanding integer matrix and D, S ⊂ Zn are two finite set with 0 ∈ D ∩ S. We first give a method of looking for the set S. Note that HM −1 D,S = H D,M ∗−1 S , (M −1 D, S) is a generalized compatible pair if and only if (D, M ∗−1 S) is a generalized compatible pair. Let L = M ∗−1 S. From the definition, we can use the digit set D and ∗ H the condition H D,L D,L = |D|I|L| = |D|I|S| to get L, and then, the resulting set S follows by the formula S = M ∗ L. The detailed discussion of this process leads to the following conclusion. For the finite digit set D ⊂ Zn of cardinality |D| > 1, we denote by Z (m D ) := {x ∈ Rn : m D (x) = 0} and Z := {x ∈ [0, 1)n : m D (x) = 0}. Proposition 2.2 Let M ∈ Mn (Z) be a non-singular matrix, and D ⊂ Zn be a finite digit set of cardinality |D| > 1 and 0 ∈ D. Then there exists a finite set S ⊂ Zn of cardinality |S| = l > 1 and 0 ∈ S such that (M −1 D, S) is a generalized compatible pair if and only if there are l − 1 points a1 , a2 , . . . , al−1 ∈ Z such that (i) ai − a j ∈ Z (m D ) for i = j and i, j = 1, 2, . . . , l − 1; (ii) M ∗ a j ∈ Zn for each j = 1, 2, . . . , l − 1. Proof If there exist S ⊂ Zn , 0 ∈ S such that (M −1 D, S) is a generalized compatible pair, then, from Proposition 2.1(iii), m M −1 D (s1 − s2 ) = m D (M ∗−1 (s1 − s2 )) = 0 for any distinct s1 , s2 ∈ S.

(2.2)

By letting S = {0, s1 , s2 , . . . , sl−1 }, we have m D (M ∗−1 s j ) = 0 for any j = 1, 2, . . . , l − 1.

(2.3)

For any x ∈ Rn , the tiling property Rn = [0, 1)n + Zn implies that we can write x as x = xˆ + x˜ uniquely, where xˆ ∈ [0, 1)n and x˜ ∈ Zn . For simplicity, we denote the above xˆ by x mod [0, 1)n , and denote the above x˜ by [x]. Since m D (x) is Zn −periodic function, that is, m D (x + z) = m D (x) for any z ∈ Zn , we define the l − 1 points a1 , a2 , . . . , al−1 by a j = M ∗−1 s j

123

mod [0, 1)n and j = 1, 2, . . . , l − 1.

(2.4)

Spectrality of self-affine measures and generalized...

Then, (2.3) implies that a1 , a2 , . . . , al−1 ∈ Z , and (2.2) implies that ai − a j ∈ Z (m D ) for i = j and i, j = 1, 2, . . . , l − 1. Also, for each j = 1, 2, . . . , l − 1, we have      M ∗ a j = M ∗ M ∗−1 s j − M ∗−1 s j = s j − M ∗ M ∗−1 s j ∈ Zn .

(2.5)

This proves the necessary part. For the sufficient part, we only need to let S = {0, s1 , s2 , . . . , sl−1 } with s j = M ∗ a j for each j = 1, 2, . . . , l − 1. In fact, the condition (ii) implies that S ⊂ Zn , the other condition implies that (2.2) holds. Since the condition (2.2) is equivalent to the condition that (M −1 D, S) is a generalized compatible pair, we thus get the desired conclusion.   From Proposition 2.2, we observe that if Z = {x ∈ [0, 1)n : m D (x) = 0} is a finite set of cardinality 1 ≤ |Z | < ∞, then the cardinality of S in a generalized compatible pair (M −1 D, S) must satisfy 1 < |S| ≤ |Z | + 1. Moreover, if |D| > |Z |+1, then there exist no set S ⊂ Zn , 0 ∈ S such that (M −1 D, S) is a compatible pair. Next, we would like to point out that only a few properties on the compatible pair are remained for the generalized compatible pair. Let | det(M)| = m = p1b1 p2b2 · · · prbr be the standard prime factorization, where p1 < p2 < · · · < pr are prime numbers and b j > 0. Denote by W (m) the non-negative integer combination of p1 , p2 , . . . , pr , then the necessary condition |D| ∈ W (m) must be satisfied for the generalized compatible pair (M −1 D, S) in the integer case (see [23, Section 3]). That is, we have the following. Proposition 2.3 Let M ∈ Mn (Z) be an integer matrix with | det(M)| = m > 1 and / W (m), then let W (m) be defined as above. If D ⊂ Zn is a finite subset with |D| ∈ there is no finite subset S ⊂ Zn with |S| > 1 such that (M −1 D, S) is a generalized compatible pair. Also the following properties on the compatible pair are remained for the generalized compatible pair (see [22, Section 2]). Proposition 2.4 Let D, S ⊂ Zn and M ∈ Mn (Z) with | det(M)| > 1 such that (M −1 D, S) is a generalized compatible pair. Then the following statements hold: / M ∗ Zn for distinct si , s j (i) The elements in S are distinct modulo M ∗ (i.e., si − s j ∈ in S); ˆ is a general(ii) Suppose that Sˆ ⊂ Zn satisfies Sˆ ≡ S (mod M∗ ). Then (M −1 D, S) ized compatible pair; ˆ S) is a gener(iii) Suppose that Dˆ ⊂ Zn satisfies Dˆ ≡ D (mod M). Then (M −1 D, alized compatible pair; (iv) |S| ≤ | det(M)|.

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It is different from the compatible pair that in a generalized compatible pair (M −1 D, S), the elements in D are not necessarily distinct modulo M. There are examples to illustrate that in a generalized compatible pair (M −1 D, S) where D, S ⊂ Zn and M ∈ Mn (Z) with | det(M)| > 1, the elements in D are not distinct modulo M. In the end of this section, we give an example to illustrate the main difference between a compatible pair and a generalized compatible pair. Example 2.1 In the plane R2 , let M ∈ M2 (Z) and D, S ⊂ Z2 be given by 

         1 1 0 1 1 2 M= , D= , , , −1 1 0 0 −1 −1     0 k S= , , (k ∈ Z). 0 k+1

(2.6)

Then the following statements hold: (M −1 D, S) is a generalized compatible pair; The elements in D are not distinct modulo M; μ M,D is a non-spectral measure (see [25, Section 3.1]); There are infinite families of orthogonal exponentials in L 2 (μ M,D ); ˜ is a compatible pair; There is no subset S˜ ⊂ Z2 such that (M −1 D, S) 2 −1 ˆ is a compatible pair; ˆ There exists a subset S ⊂ R such that (M D, S) The above Theorem A or Theorem A is not true for the generalized compatible pair (M −1 D, S); (viii) (M ∗−1 S, D) is not a generalized compatible pair; (ix) μ M ∗ ,S is a spectral measure (see [28, p. 598]). (i) (ii) (iii) (iv) (v) (vi) (vii)

3 Connections with µ M, D −orthogonality The generalized compatible pairs have closely connection with μ M,D −orthogonal exponentials. We first extend the well-known result of Jorgensen and Pedersen [15,16] on the compatible pair to the generalized compatible pair. Theorem 3.1 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. If there exists a subset S ⊂ Zn , 0 ∈ S such that (M −1 D, S) is a generalized compatible pair, then E((M, S)) is an infinite orthogonal system in L 2 (μ M,D ). The proof of Theorem 3.1 is similar to the case of compatible pair, so we omit the proof here. For the given pair (M, D), the simplest case for the generalized compatible pair (M −1 D, S) is the case that |S| = 2 and S = {0, s1 }. In this case, we have (M −1 D, S) is a generalized compatible pair ⇐⇒ m D (M ∗−1 s1 ) = 0. This leads to the following conclusion.

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Corollary 3.2 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. If there exists a nonzero point s1 ∈ Zn such that m D (M ∗−1 s1 ) = 0, then E((M, S)) is an infinite orthogonal system in L 2 (μ M,D ), where S = {0, s1 }. Note that the condition m D (M ∗−1 s1 ) = 0 in Corollary 3.2 implies the condition that Z (μˆ M,D ) ∩ Zn = ∅ and the condition that there exist α ∈ Z such that M ∗ α ∈ Zn . Either of the above two conditions implies that there are infinite families of orthogonal ˜ ⊆ Zn . However the completeness of such ˜ in L 2 (μ M,D ) with  exponentials E() orthogonal exponentials remains open in a large degree. In this regard, the following theorem gives certain conditions on the spectral pair. Theorem 3.3 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. Suppose that S ⊂ Zn is a finite subset with 0 ∈ S and |S| > 1 such that the elements in S are distinct modulo M ∗ or M ∗ ((M, S)−(M, S))∩(S − S) = {0}. Then the following two conclusions hold: (i) If E((M, S)) is an orthogonal basis for L 2 (μ M,D ) or if (μ M,D , (M, S)) is a spectral pair, then (M −1 D, S) is a compatible pair; (ii) If (M ∗−1 S, D) or (S, M −1 D) is a generalized compatible pair with the cardinality |S| = |D|, then E((M, S)) is not an orthogonal basis for L 2 (μ M,D ), i.e. (μ M,D , (M, S)) is not a spectral pair. In particular, if (M ∗−1 S, D) or (S, M −1 D) is a generalized compatible pair with the cardinality |S| < |D|, then the infinite orthogonal system E((M, S)) is not complete in L 2 (μ M,D ). Proof From the definition of (M, S), we know that each of the following two conditions: (a) the elements in S are distinct modulo M ∗ ; (b) M ∗ ((M, S) − (M, S)) ∩ (S − S) = {0} implies that (M, S) = M ∗ (M, S) + S = M ∗ (M, S) ⊕ S(the direct sum).

(3.1)

For any ξ ∈ Rn , we define the function Q  (ξ ) :=



|μˆ M,D (ξ + λ)|2 .

λ∈(M,S)

It follows from (3.1) and the identity μˆ M,D (ξ ) = m D (M ∗−1 ξ )μˆ M,D (M ∗−1 ξ ) that 

Q  (ξ ) =

|μˆ M,D (ξ + λ)|2

λ∈M ∗ (M,S)+S

=





|μˆ M,D (ξ + M ∗ λ + s)|2

s∈S λ∈(M,S)

=





|m D (M ∗−1 (ξ + s) + λ)|2 |μˆ M,D (M ∗−1 (ξ + s) + λ)|2

s∈S λ∈(M,S)

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=



=

|μˆ M,D (M ∗−1 (ξ + s) + λ)|2

λ∈(M,S)

s∈S





|m D (M ∗−1 (ξ + s))|2 |m D (M

∗−1

(ξ + s))| Q  (M ∗−1 (ξ + s)), ∀ ξ ∈ Rn . 2

(3.2)

s∈S

In view of the fact that (μ M,D , (M, S)) is a spectral pair ⇐⇒ Q  (ξ ) = 1,

∀ ξ ∈ Rn ,

(3.3)

we know that if E((M, S)) is an orthogonal basis for L 2 (μ M,D ), then, by (3.2) and (3.3),  |m D (M ∗−1 (ξ + s))|2 = 1, ∀ ξ ∈ Rn , (3.4) s∈S

which implies m M −1 D (s2 − s1 ) = m D (M ∗−1 (s2 − s1 )) = 0 for any distinct s2 , s1 ∈ S. By Proposition 2.1(iii), we get that (M −1 D, S) is a generalized compatible pair. Furthermore, comparing (3.4) with (2.1), we obtain |D| = |S|, which proves (i). On the other hand, if (M ∗−1 S, D) is a generalized compatible pair, then m M ∗−1 S (d1 − d2 ) = m S (M −1 (d1 − d2 )) = 0 for any distinct d1 , d2 ∈ D, which implies  s∈S

=

|m D (M ∗−1 (ξ + s))|2 |S|   2πi e m S (M −1 (d1 − d2 )) (|D|)2 d1 ∈D d2 ∈D

|S| , ∀ ξ ∈ Rn . = |D|

(3.5)

In the case when |S| = |D|, it follows from (3.5) that (3.4) cannot be true, and hence Q  (ξ ) = 1 (∀ξ ∈ Rn ) cannot be true. This shows that E((M, S)) is not an   orthogonal basis for L 2 (μ M,D ). The proof of Theorem 3.3 is completed.

4 Applications to the spectral pairs Following the relationship between the generalized compatible pair and the μ M,D −orthogonal exponentials, we further consider the applications of generalized compatible pair to the spectral pairs. We first establish the following fundamental result. Theorem 4.1 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. Suppose that S ⊂ Zn is a finite subset with 0 ∈ S and |S| > 1 such

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that the elements in S are distinct modulo M ∗ or M ∗ ((M, S)−(M, S))∩(S − S) = {0}. If there exist s0 , s1 , . . . , s p−1 ∈ S for some p ≥ 1 such that (M ∗ p − I )−1 {s0 + M ∗ s1 + · · · + M ∗( p−1) s p−1 } ∈ Zn \{0},

(4.1)

then E((M, S)) is not an orthogonal basis for L 2 (μ M,D ), i.e. (μ M,D , (M, S)) is not a spectral pair. Proof With the same notation as above, let η0 := (M ∗ p − I )−1 {s0 + M ∗ s1 + · · · + M ∗( p−1) s p−1 },

(4.2)

and define the periodic sequence η1 , η2 , . . . , η p−1 , η p , . . . by ⎧ η1 ⎪ ⎪ ⎪ ⎪ ⎨ η2 ··· ⎪ ⎪ ⎪ η ⎪ ⎩ p−1 ηp

= (M ∗ p − I )−1 (s1 = (M ∗ p − I )−1 (s2 ··· = (M ∗ p − I )−1 (s p−1 = η0 , η p+1 = η1 , η p+2

+M ∗ s2 + · · · + M ∗( p−2) s p−1 +M ∗ s3 + · · · + M ∗( p−2) s0 ··· +M ∗ s0 + · · · + M ∗( p−2) s p−3 = η2 , . . . , η2 p = η p , η p+r = ηr ,

+M ∗( p−1) s0 ), +M ∗( p−1) s1 ), ··· +M ∗( p−1) s p−2 ), (r ∈ N). (4.3)

Then, from (4.2) and (4.3), we have ⎧ η1 ⎪ ⎪ ⎪ ⎪ ⎨ η2 ⎪ ⎪ η ⎪ ⎪ ⎩ p−1 ηp

= (M ∗ )−1 (η0 + s0 ) or M ∗ η1 = η0 + s0 , = (M ∗ )−1 (η1 + s1 ) or M ∗ η2 = η1 + s1 , ··· = (M ∗ )−1 (η p−2 + s p−2 ) or M ∗ η p−1 = η p−2 + s p−2 , = (M ∗ )−1 (η p−1 + s p−1 ) = η0 , or M ∗ η p = η p−1 + s p−1 .

(4.4)

Let s p = s0 and s p+r = sr for r ∈ N, we know, from (4.4) that η j = (M ∗ )−1 (η j−1 + s j−1 ) or η j = σs j−1 (η j−1 ) for all j ∈ N.

(4.5)

This shows that the set C := {η0 , η1 , . . . , η p−1 } is a cycle of length p for the IFS {σs (x) = (M ∗ )−1 (x + s)}s∈S (see Definition 4.1 below). From the hypothesis (4.1) we see that η0 = η p ∈ Zn \{0}, which combined with (4.4) or (4.5), yields that η j−1 = M ∗ η j − s j−1 ∈ Zn for all j ∈ N. That is C := {η0 , η1 , . . . , η p−1 } ⊂ Zn .

(4.6)

After analyzing the condition (4.1), we now prove that E((M, S)) is not an orthogonal basis for L 2 (μ M,D ), i.e. (μ M,D , (M, S)) is not a spectral pair. Suppose that it is not the case. Then (μ M,D , (M, S)) is a spectral pair, and hence (3.3), (3.4) hold. Let ξ = η j in (3.4), we have 

|m D (M ∗−1 (η j + s))|2 = 1,

∀ j ∈ N ∪ {0}.

(4.7)

s∈S

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From (4.2)–(4.6), we observe that m D (M ∗−1 (η j + s j )) = m D (η j+1 ) = 1. Combined with (4.7), we know that for any s ∈ S, m D (M ∗−1 (η j + s)) =



1, if s = s j , 0, if s = s j ,

∀ j ∈ N ∪ {0}.

(4.8)

This leads to the following. Claim 1 For any λ ∈ (M, S), we have μˆ M,D (η0 + λ) =

∞ 

m D (M ∗− j (η0 + λ)) = 0.

(4.9)

j=1

 It follows from Claim 1 that Q  (η0 ) = λ∈(M,S) |μˆ M,D (η0 + λ)|2 = 0, which shows that (μ M,D , (M, S)) is not a spectral pair, a contradiction. Therefore, the proof of Theorem 4.1 will be completed if the above Claim 1 is proved. Now, we prove Claim 1. For any λ ∈ (M, S), we write λ as λ=

∞ 

M ∗ j sˆ j , where sˆ j ∈ S and only finitely many sˆ j = 0.

(4.10)

j=0

From the first factor m D (M ∗−1 (η0 + λ)) of μˆ M,D (η0 + λ) in (4.9), we observe, from (4.8) that m D (M ∗−1 (η0 + λ)) = m D (M ∗−1 (η0 + sˆ0 ) + sˆ1 + M ∗ sˆ2 + · · · + M ∗( j−1) sˆ j + · · · ) = m D (M ∗−1 (η0 + sˆ0 ))  1, if sˆ0 = s0 , = 0, if sˆ0 = s0 .

(4.11)

If sˆ0 = s0 , then (4.9) holds. In the case when sˆ0 = s0 , we consider the second factor m D (M ∗−2 (η0 + λ)) of μˆ M,D (η0 + λ) in (4.9), and use (4.5) and (4.8) to conclude that m D (M ∗−2 (η0 + λ)) = m D (M ∗−2 (η0 + s0 + M ∗ sˆ1 ) + sˆ2 + M ∗ sˆ3 + · · · ) = m D (M ∗−1 (η1 + sˆ1 ))  1, if sˆ1 = s1 , = 0, if sˆ1 = s1 .

(4.12)

If sˆ1 = s1 , then (4.9) holds. By induction, we easily obtain that for each j = 1, 2, 3, . . ., m D (M ∗− j (η0 + λ)) = 0 ⇐⇒ sˆ j−1 = s j−1 .

(4.13)

Therefore, μˆ M,D (η0 +λ) = 0 only if sˆ j = s j for all j ≥ 0. This is impossible because {s0 , s1 , s2 , . . .} = {s0 , s1 , s2 , . . . , s p−1 } and sˆ j = 0 for all sufficiently large j. This proves Claim 1. The proof of Theorem 4.1 is completed.  

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Theorem 4.1 combined with Theorem 3.1 and Proposition 2.4(i) yields the following result which extends the corresponding result of [22, Theorem 3.4]. Corollary 4.2 Let M ∈ Mn (Z) be an expanding integer matrix, D and S be two finite subsets of Zn such that (M −1 D, S) is a generalized compatible pair and 0 ∈ D ∩ S. If there exist s0 , s1 , . . . , s p−1 ∈ S for some p ≥ 1 such that (4.1) holds, then the infinite orthogonal system E((M, S)) is not complete in L 2 (μ M,D ). Let S p = S + M ∗ S + · · · + M ∗( p−1) S. Theorem 4.1 can be stated in the following equivalent form, which is listed as Corollary 4.3. Corollary 4.3 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. Suppose that S ⊂ Zn is a finite subset with 0 ∈ S and |S| > 1 such that the elements in S are distinct modulo M ∗ or M ∗ ((M, S) − (M, S)) ∩ (S − S) = {0}. If (μ M,D , (M, S)) is a spectral pair, then (M ∗ p − I )−1 S p ∩ Zn = {0} for each p ∈ N.

(4.14)

For an expanding matrix M ∈ Mn (Z) and a finite subset S ⊂ Zn , the IFS {σs (x) = (M ∗ )−1 (x + s)}s∈S and its attractor or invariant set T (M ∗ , S) play an important role in the study of spectral pair (μ M,D , (M, S)). For example, from Theorem 4.1, we also have the following conclusion. Corollary 4.4 Let M ∈ Mn (Z) be an expanding integer matrix, and D ⊂ Zn be a finite digit set with 0 ∈ D. Suppose that S ⊂ Zn is a finite subset with 0 ∈ S and |S| > 1 such that the elements in S are distinct modulo M ∗ or M ∗ ((M, S) − (M, S)) ∩ (S − S) = {0}. If (μ M,D , (M, S)) is a spectral pair, then T (M ∗ , S) ∩ Zn = {0}.

(4.15)

The conditions (4.14) and (4.15) can be proved to be equivalent. Moreover, by introducing the following definition of m D −cycle and the two sets C(M, S) and C(M, S, D), we can state more results for the spectral pair. Definition 4.1 A finite set C := {x0 , x1 , . . . , x p−1 } is a cycle of length p for the IFS {σs (x)}s∈S if there exists s0 , s1 , s2 , . . . , s p−1 ∈ S such that σs j (x j ) = x j+1 for j ∈ {0, 1, 2, . . . , p − 1}, where x p = x0 . The cycle C is called a m D −cycle if |m D (x j )| = 1 for all j ∈ {0, 1, 2, . . . , p − 1}. For an expanding matrix M ∈ Mn (Z) and two finite subsets D, S ⊂ Zn with 0 ∈ D∩S, define the two sets C(M, S) and C(M, S, D) by  {C : C is a cycle of length p, p ≥ 1},  C(M, S, D) := {C : C is a m D − cycle of length p, p ≥ 1}. C(M, S) :=

Then the same method as [27] yields the following more general result, which is listed as Corollary 4.5.

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Corollary 4.5 Let M ∈ Mn (Z) be expanding, D and S be two finite subsets of Zn with |D| > 1, |S| > 1 and 0 ∈ D ∩ S. Consider the following four conditions: T (M ∗ , S) ∩ Zn = {0}; C(M, S) ∩ Zn = {0}; C(M, ∩ Zn = {0};  n ∞ S, D) ∗ p − I )−1 S Z = {0}. (M p p=1 If the elements in S are distinct modulo M ∗ , then the four conditions (i)– (iv) are equivalent, and any one of them is necessary for the spectral pair (μ M,D , (M, S)); (II) If (M −1 D, S) is a generalized compatible pair, then the four conditions (i)– (iv) are equivalent, and any one of them is necessary for the spectral pair (μ M,D , (M, S)). (III) If M ∗ ((M, S) − (M, S)) ∩ (S − S) = {0} and (μ M,D , (M, S)) is a spectral pair, then the four conditions (i)–(iv) hold. Moreover, the four conditions (i)–(iv) are equivalent. (i) (ii) (iii) (iv) (I)

Note that Corollary 4.5 extends the corresponding result of [27, Theorem 2.5]. Furthermore, in Corollary 4.5, there are examples to illustrate that the above necessary conditions for the spectral pair (μ M,D , (M, S)) are not sufficient. Acknowledgements The author thanks Prof. Deguang Han for a helpful discussion on the subject during his visit to Xi’an in the autumn of 2014. The author also thanks the anonymous referees for their valuable suggestions. This work is supported by the National Natural Science Foundation of China (No.11571214).

References 1. Bohnstengel, J.: Wavelet and Fourier Bases on Fractals. Ph.D. thesis, Universität Bremen (2011) 2. Dutkay, D.E., Han, D., Sun, Q.: Divergence of the mock and scrambled Fourier series on fractal measures. Trans. Am. Math. Soc. 366(4), 2191–2208 (2014) 3. Dutkay, D.E., Haussermann, J., Lai, C.-K.: Hadamard Triples Generate Self-Affine Spectral Measures, arXiv:1506.01503v1 (2015) 4. Dutkay, D.E., Jorgensen, P.E.T.: Iterated function systems, Ruelle operators, and invariant projective measures. Math. Comp. 75(256), 1931–1970 (2006) 5. Dutkay, D.E., Jorgensen, P.E.T.: Fourier frequencies in affine iterated function systems. J. Funct. Anal. 247, 110–137 (2007) 6. Dutkay, D.E., Jorgensen, P.E.T.: Fourier series on fractals: a parallel with wavelet theory. In: Olafsson, G., Grinberg, E.L., Larson, D., Jorgensen, P.E.T., Massopust, P.R., Quinto, E.T., Rubin, B (eds.) Radon Transform, Geometry, and Wavelets, Contemporary Mathematics, vol. 464, pp. 75–101. American Mathematical Society, Providence, RI (2008) 7. Dutkay, D.E., Jorgensen, P.E.T.: Probability and Fourier duality for affine iterated function systems. Acta Appl. Math. 107, 293–311 (2009) 8. Dutkay, D.E., Lai, C.-K.: Self-Affine Spectral Measures and Frame Spectral Measures on Rd , arXiv:1502.03209v1 (2015) 9. Farkas, B., Matolcsi, M., Móra, P.: On Fuglede’s conjecture and existence of universal spectra. J. Fourier Anal. Appl. 12(5), 483–494 (2006) 10. Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974) 11. Gabardo, J.-P., Lai, C.-K.: Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution. J. Fourier Anal. Appl. 20, 453–475 (2014) 12. Gröchenig, K., Madych, W.: Multiresolution analysis, Haar bases, and self-similar tilings. IEEE Trans. Inf. Theory 38, 556–568 (1992)

123

Spectrality of self-affine measures and generalized... 13. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981) 14. Jorgensen, P.E.T., Kornelson, K.A., Shuman, K.L.: Affine system: asymptotics at infinity for fractal measures. Acta Appl. Math. 98, 181–222 (2007) 15. Jorgensen, P.E.T., Pedersen, S.: Harmonic analysis of fractal measures. Constr. Approx. 12, 1–30 (1996) 16. Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal L 2 -spaces. J. Anal. Math. 75, 185–228 (1998) 17. Jorgensen, P.E.T., Pedersen, S.: Spectral pairs in Cartesian coordinates. J. Fourier Anal. Appl. 5, 285– 302 (1999) 18. Kolountzakis, M.N., Matolcsi, M.: Complex Hadamard matrices and the spectral set conjecture. Collect. Math. Extra 281–291 (2006) 19. Łaba, I., Wang, Y.: On spectral Cantor measures. J. Funct. Anal. 193, 409–420 (2002) 20. Lagarias, J.C., Reeds, J.A., Wang, Y.: Orthonormal bases of exponentials for the n-cube. Duke Math. J. 103, 25–37 (2000) 21. Li, J.-L.: On characterizations of spectra and tilings. J. Funct. Anal. 213, 31–44 (2004) 22. Li, J.-L.: Spectral self-affine measures in Rn . Proc. Edinb. Math. Soc. 50, 197–215 (2007) 23. Li, J.-L.: μ M,D -orthogonality and compatible pair. J. Funct. Anal. 244, 628–638 (2007) 24. Li, J.-L.: Spectra of a class of self-affine measures. J. Funct. Anal. 260, 1086–1095 (2011) 25. Li, J.-L.: Spectrality of a class of self-affine measures with decomposable digit sets. Sci. China Math. 55, 1229–1242 (2012) 26. Li, J.-L.: Analysis of μ M,D -orthogonal exponentials for the planar four-element digit sets. Math. Nachr. 287(2/3), 297–312 (2014) 27. Li, J.-L.: Extensions of Laba–Wang’s condition for spectral pairs. Math. Nachr. 288(4), 412–419 (2015) 28. Li, J.-L., Wen, Z.-Y.: Spectrality of planar self-affine measures with two-element digit sets. Sci. China Math. 55(3), 593–605 (2012) 29. Matolcsi, M.: Fuglede’s conjecture fails in dimension 4. Proc. Am. Math. Soc. 133, 3021–3026 (2005) 30. Strichartz, R.: Self-similarity in harmonic analysis. J. Fourier Anal. Appl. 1, 1–37 (1994) 31. Strichartz, R.: Mock Fourier series and transforms associated with certain Cantor measures. J. Anal. Math. 81, 209–238 (2000) 32. Strichartz, R.: Convergence of mock Fourier series. J. Anal. Math. 99, 333–353 (2006) 33. Szöll˝osi, F.: Construction, Classification and Parametrization of Complex Hadamard Matrices, PhD thesis, Central European University (2011) 34. Tao, T.: Fuglede’s conjecture is false in 5 and higher dimensions. Math. Res. Lett. 11, 251–258 (2004) 35. Yang, M.-S., Li, J.-L.: A class of spectral self-affine measures with four-element digit sets. J. Math. Anal. Appl. 423(1), 326–335 (2015)

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