Journal of Fourier Analysis and Applications https://doi.org/10.1007/s00041-018-9621-7
A Characterization on the Spectra of Self-Affine Measures Yan-Song Fu1 Received: 3 October 2017 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract A discrete set ⊆ Rd is called a spectrum for the probability measure μ if the family of functions {e2πiλ, x : λ ∈ } forms an orthonormal basis for the Hilbert space L 2 (μ). In this paper, we will give a characterization of the spectra of self-affine measures generated by compatible pairs in Rd . As an application, we show, for the Cantor measure μb, q on R with consecutive digit set and any integer p ∈ Z with gcd( p, q) = 1, that the set { ⊆ R : and p are both spectra for μb, q and 0 ∈ } has the cardinality of the continuum. Keywords Self-affine measures · Spectra · Spectral measures · Compatible pairs Mathematics Subject Classification 42B10 · 28A80 · 42C05
1 Introduction Given a Borel probability measure μ and a countable discrete set on the Euclidean space Rd , we call μ a spectral measure and a spectrum for μ if the family of complex exponential functions {e2πiλ, x : λ ∈ } forms an orthonormal basis for L 2 (μ), the space of all square-integrable functions with respect to μ. In this case, (μ, ) is often called a spectral pair. It is well known that the study of spectral measures dates back to the work of Fuglede [21], whose famous spectral-tiling conjecture and related problems have attracted much attention in the past 40 years, see [22,33] and the references cited therein.
Communicated by Dorin Ervin Dutkay.
B 1
Yan-Song Fu
[email protected] School of Science, China University of Mining and Technology, Beijing 100083, People’s Republic of China
Journal of Fourier Analysis and Applications
The purpose of this paper is to study the structure of the spectra for a class of self-affine measures μ R,D on Rd generated by the following iterated function systems (IFSs) {σd (x) = R −1 (x + d), d ∈ D}, where R ∈ Md (Z) is an expanding integer matrix (i.e., all eigenvalues of the matrix R have modulus greater than 1), and D is a finite subset of Zd . Moreover, the self-affine measure μ R,D considered here satisfies the following equation μ R,D =
1 μ R,D ◦ σd−1 , #D d∈D
and is supported on a fractal set T (R, D), called self-affine set, which has the following representation ⎧ ⎫ ∞ ⎨ ⎬ T (R, D) = R− j d j : d j ∈ D , ⎩ ⎭ j=1
where the symbol # D denotes the number of the set D, see [24]. It is well known that Jorgensen and Pedersen [25] first found that some non-atomic, singular self-affine measures can be spectral. More precisely, they proved that a fractal Cantor measure μ4,{0,2} generated by the IFS {4−1 (x + d) : d ∈ {0, 2}} is spectral and a spectrum is in the form: =
m
4 ci : ci ∈ {0, 1}, m ∈ N0 . i
i=0
Here and below, we use N to denote the set of positive integers and N0 = N ∪ {0}. Later on, Łaba and Wang [27] found that the spectrum for a given spectral measure is not unique. From then on, researchers began to develop Fourier analysis on fractal sets which involves extending the construction of [25] to a larger class of singular spectral measures (e.g., see [1–4,8,11,18,19,23,30–32], etc.), classifying the structure of spectra for a given spectral measure (e.g., see [5,6,9,10,12,17,20,26,28], etc.), and exploring the convergence and divergence of Mock Fourier series with respect to a given spectrum (e.g., see [13,32]). In all these researches, the concept of compatible pairs plays an important role. Definition 1.1 Let R ∈ Md (Z) be an expanding integer matrix, and let D and C be two finite subsets of Zd with the same cardinality q. We call (R −1 D, C) forms an (integral) compatible pair [or (R, D, C) forms a Hadamard triple] if the matrix −1 H R −1 D,C := q −1/2 e2πiR d,c d∈D,c∈C is unitary, that is, H R −1 D,C H R∗ −1 D,C = Iq .
Journal of Fourier Analysis and Applications
Recall that Łaba and Wang [27] showed, for R > 1 and D ∈ Z, that μ R,D on R is spectral if (R −1 D, C) forms a compatible pair for some C ⊆ Z. Over the past about 20 years, researchers tried their best to deal with the spectrality and non-spectrality of μ R,D on Rd when R ∈ Md (Z) is an expanding matrix and D ⊆ Zd is a finite set. Eventually, Dutkay et al. [14] proved that the compatible pair (R −1 D, C) actually implies that self-affine measures μ R,D on Rd are all spectral. Therefore, the remaining basic problem relating to self-affine measures μ R,D generated by compatible pairs is to determine all spectra for μ R,D . More recently, following the research of Strichartz [31] on the spectrality and non-spectrality of infinite convolutions, much more spectral property of random convolutions generated by compatible pairs have been obtained by An et al. [2,3], Dutkay and Lai [11], Fu et al. [17,19,20], etc. More specifically, let R ∈ Md (Z) be an expanding integer matrix, and let D1 , D2 , . . . , D N be a collection of finite subsets of Zd with the same cardinality such that (R −1 Di , C) forms a compatible pair for some C ⊆ Zd and all i = 1, 2, . . . , N . For any given infinite word w = w1 w2 w3 · · · ∈ {1, 2, . . . , N }N , the random convolution (see [11, p. 186]) to be studied is defined by μ R,w := δ R −1 Dw ∗ δ R −2 Dw ∗ · · · 1
2
in the weak∗ -topology, where δ E denotes the discrete measure δE =
1 δe , #E e∈E
and δe is the Dirac measure concentrated on the point e ∈ Rd . In particular, if N = 1, D := D1 and w = 111 · · · ∈ {−1, 1}N , then μ R,w is the self-affine measure μ R,D , see [29]. Motivated by the above study for random convolution μ R,w , the main question addressed in this paper is the dual problem in some sense. Question let R ∈ Md (Z) be an expanding integer matrix, and let D, C1 , C2 , . . . , C N be a collection of finite subsets of Zd with the same cardinality such that (R −1 D, Ci ) forms a compatible pair and 0 ∈ Ci for all i = 1, 2, · · · , N . For any given infinite word w = w0 w1 w2 · · · ∈ {1, 2, . . . , N }N0 , when is the set
2 N := Cw0 + R T Cw1 + R T Cw2 + · · · w R, {Ci }i=1
(1.1)
a spectrum for the self-affine measure μ R,D ? We remark here that when N = 1, C := C1 and w = 111 · · · ∈ {−1, 1}N0 , if
(R, C) := C + R C + R T
T 2
C + ··· =
m
R
T i
ci : ci ∈ C, m ∈ N0 ,
i=0
is a spectrum for μ R,D , then it is usually called a canonical spectrum for μ R,D , see, e.g., [10]. It is also pointed out that when d = 1, N = 2 and C1 = −C2 , some answers for the above question have been implicitly or explicitly given in [2,3,17,20].
Journal of Fourier Analysis and Applications N ) as in (1.1) random spectrum or For simplicity, one can call the set w (R, {Ci }i=1 mixed-type spectrum for μ R,D if it is actually a spectrum. The paper is organized as follows. In Sect. 2, we will give a condition [see (2.10)] to N ) to be a spectrum for the self-affine measure μ guarantee the set w (R, {Ci }i=1 R,D , see Theorem 2.3 for more detail. We have to say that the condition we give is inspired by Strichartz’s results on spectral theory of measures in [30] and [31, Theorem 2.8], and inspired by the construction of Moran sets by Feng et al. [15]. Furthermore, in terms of compatible pairs, Theorem 2.3 establishes a duality of Dutkay–Lai’s theorem [11, Theorem 1.5] to some extent. In Sect. 3, we investigate the structure of spectra for the Cantor measure μb, q with consecutive digit set on R, which is a particular self-affine measure. Our principal results Theorems 3.2 and 3.4 refine [17, Theorem 1.2] and provide a qualitative characterization on the spectra for μb, q . We would like to mention that the proof of Theorem 3.2 is an application of Theorem 2.3 and it also needs a careful construction of candidate spectrum as in (3.3) or Moran sets as in (3.4), which improves the techniques used in [17]. As a consequence of Theorem 3.4, one can settle a question for Bernoulli convolution proposed in [20, Sect. 5], see Proposition 3.5. At the end of Sect. 3, we provide some new spectra for Cantor measure μb, q , see Theorem 3.6.
2 A Duality of Dutkay–Lai’s Theorem The main result of this section is Theorem 2.3, which establishes a duality of Dutkay– Lai’s Theorem in [11, Theorem 1.5], (see Theorem A) to some extent. What’s more, Theorem 2.3 will be used to study the spectra for Cantor measures with consecutive digit sets in Sect. 3. Let R be a d × d expanding integer matrix and let S = R T be the transpose of the matrix R. Let C1 , C2 , . . . , C N be a finite collection of Zd with the same cardinality and 0 ∈ Ci for each i = 1, 2, . . . , N . Given an infinite word w = w0 w1 w2 · · · ∈ {1, 2, . . . , N }N0 , we define the Moran IFS
σ j,c j (x) = S −1 x + c j : c j ∈ Cw j , j ∈ N0 ,
(2.1)
and its (dual) Moran IFS
τ j,c j (x) = Sx + c j : c j ∈ Cw j , j ∈ N0 .
(2.2)
Our first result Theorem 2.1 shows that the Moran IFS {σ j,c j } Nj=1 as in (2.1) will generate a Moran measure μw (S, {C j } Nj=1 ) supported on the Moran set Tw (S, {C j } Nj=1 ) as in (2.3). Theorem 2.1 Given an infinite word w = w0 w1 w2 · · · ∈ {1, 2, . . . , N }N0 and the Moran IFS as in (2.1). Then there is a unique Borel probability measure μw (S, {C j } Nj=1 ), called the Moran measure, which is the weak∗ -limit of the discrete measures {μn } where
Journal of Fourier Analysis and Applications
μn := δ S −1 Cw
n−1
∗ δ S −2 Cw
n−2
∗ · · · ∗ δ S −n Cw0 (n ∈ N)
and is supported on the following compact set, called Moran set,
N Tw S, C j j=1 := Cl
⎧ n ⎨ ⎩
S − j cn− j : cn− j
j=1
⎫ ⎬ ∈ Cwn− j , n ∈ N . ⎭
(2.3)
Here, the symbol Cl(A) denotes the closure of the set A in the usual metric in Rd . We have to say that this result is well-known to many researchers, but we can not find it in this form in literatures. So we shall state and give a detailed proof. Moreover, the proof we provided here is elementary and essential to understand the dual relation between the compact set Tw (S, {C j } Nj=1 ) and the spectrum w (S, {C j } Nj=1 ) [see (2.9)] for the self-affine measure μ R,D in Theorem 2.3, which is our main result in this section. Proof We firstly show that for any complex-valued, compactly supported, continuous function σ ∈ Cc (Rd ), the sequence {μn (σ ) := Rd σ dμn }n∈N is a Cauchy sequence. Indeed, it easily follows from the fact that for any σ ∈ Cc (Rd ) and any p > q > 1, we have μ p (σ ) − μq (σ ) = 1 (# D) p 1 = (# D) p
p x∈ j=1 S − j C p− j
1 − (# D)q 1 = (# D) p
σ
p
c p− j ∈C p− j j=1,2,..., p
σ
q
cq− j ∈Cq− j j=1,2,...,q
σ
p
c p− j ∈C p− j j=1,2,..., p
(# D) p c p− j ∈C p− j j=1,2,..., p
c p− j
S
−j
cq− j
S
−j
c p− j
j=1
cq− j ∈Cq− j , j=q+1,..., p
1 = (# D) p
j=1
−
S
−j
q x∈ j=1 S − j Cq− j
σ (x)
j=1
1 σ (x) − (# D)q
⎛ ⎝σ
1
σ
q
cq− j ∈Cq− j j=1,2,...,q
p j=1
S − j c p− j
S
−j
cq− j
(2.4)
j=1
−σ
q j=1
⎞ S − j cq− j ⎠ .
Journal of Fourier Analysis and Applications
Since S is an expanding matrix, then there is a norm on Rd for which ρ := S −1 < 1 such that the quantity p p q p −j −j −j = ≤ S c − S c S c ρjM n− j n− j n− j j=1
j=1
j=q+1
(2.5)
j=q+1
will be very small if q is large enough, where M := {|c| : c ∈ C j for all j = 1, 2, . . . , N } is bounded. Using the facts that Tw (S, {C j } Nj=1 ) is compact (to be proved later) and n N −j j=1 S cn− j ∈ Tw (S, {C j } j=1 ), it follows from (2.4), (2.5) and the uniformly continuity of σ on compact sets that the sequence {μn (σ )}n∈N is a Cauchy sequence. Thus, the limit of {μn (σ )}n∈N exists for any σ ∈ Cc (Rd ) and we can define J (σ ) = lim
n→∞ Rd
σ (x)dμn (x)
∀σ ∈ Cc Rd .
Clearly, J (1) = 1 and J is positive, i.e., J (σ ) ≥ 0 if σ ≥ 0. Then, by Riesz representation theorem, there is a unique Borel probability measure, denoted by μw (S, {C j } Nj=1 ), satisfying that J (σ ) =
Rd
σ (x)dμw S, {C j } Nj=1 (x) ∀σ ∈ Rd .
This shows that μw (S, {C j } Nj=1 ) is the weak∗ -limit of the discrete measures {μn }. On the other hand, since S is an expanding integer matrix, then
σ j,c (x) − σ j,c (0) ≤ ρ |x| ∀ j ∈ N0 , c j ∈ Cw , j j j
(2.6)
where | · | denotes the usual metric in Rd . Let B(0, r ) be the closed ball in Rd centered at 0 with the radius r=
1 max σc (0) : c ∈ ∪ Nj=1 C j . 1−ρ
(2.7)
It is easy to see that σ j,c j (B(0, r )) ⊆ B(0, r ), cn− j ∈ Cwn− j , n ∈ N. This is true because for any x ∈ B(0, r ), we have, from (2.6), that σ j,c (x) ≤ ρ|x| + σ j,c (0) ≤ ρr + (1 − ρ)r = r . j j Now it is easy to check that
Tw S, {C j } Nj=1 =
σn,cn ◦σn−1,cn−1 ◦· · ·◦σ0,c0 (B(0, r )), (2.8)
n∈N0 {c j ∈Cw j , 0≤ j≤n}
where σn,cn ◦σn−1,cn−1 ◦· · · σ0,c0 is the composition of {σ j,c j : c j ∈ Cw j , 0 ≤ j ≤ n}.
Journal of Fourier Analysis and Applications
Hence Tw (S, {C j } Nj=1 ) is a compact set by the finite intersection property of compact sets. The following is a direct result of Theorem 2.1. Corollary 2.2 Let S be a d × d expanding integer matrix and let C be a finite digit set in Zd . Then there is a unique probability measure μ S,C , called self-affine measure, which is the weak∗ -limit of the measures μn := δ S −1 C ∗ δ S −2 C ∗ · · · ∗ δ S −n C (n ∈ N) and is supported on a compact set, called self-affine set, T (S, C) :=
⎧ ∞ ⎨ ⎩
S− j c j : c j ∈ C
j=1
⎫ ⎬ ⎭
= Cl
⎧ n ⎨ ⎩
j=1
⎫ ⎬ S − j c j : c j ∈ C, n ∈ N . ⎭
It is well known that, by Carathéodory–Kolmogorov’s extension theorem (see, e.g., [16, Theorem 1.14]), there is a unique Borel probability measure ν on the symbolic space N := {1, 2, . . . , N }N0 satisfying that
ν w0 w1 · · · wn−1 = N −n , where
w0 w1 · · · wn−1 = w = w1 w2 · · · ∈ N : wi = wi for i = 0, 1, . . . , n − 1
is a cylinder of N . In terms of the above measure ν, Dutkay and Lai provided a characterization on the spectrality of the measure μw (S, {C j } Nj=1 ) with some extra conditions, see [11, Theorem 1.5]. Theorem A Let (S −1 C j , D), j = 1, 2, . . . , N , be the compatible pairs for some D ⊆ Z. Assume that one of the following conditions holds: (i) the compatible pairs (S −1 C j , D) are on R1 , i.e., S is an integer; (ii) each C j is a complete set representative of S. Then there is a set such that is a spectrum for the measure μw (S, {C j } Nj=1 ), for ν-almost w ∈ N . Relating to (1.1), we know that
N w S, C j j=1 = Cw0 + SCw1 + S 2 Cw2 + · · ·
(2.9)
can be generated by the dual Moran IFS {τ j,c j } as in (2.2) via the following composition τ0,c0 ◦ τ1,c1 ◦ · · · ◦ τn,cn (0)
n ∈ N0 .
Journal of Fourier Analysis and Applications
The main Theorem 2.3 in the following will modify the condition of [31, Theorem 2.8], and complete a duality theorem of Theorem A, which shows, under some condition, that for all w ∈ N the sets w (S, {C j } Nj=1 ) are all spectra for some self-affine measure μ R,D . Theorem 2.3 Let R be an expanding integer matrix in Rd , and let D be a subset of Zd . Suppose that 0 ∈ C j and (R −1 D, C j ) forms a compatible pair for each j = 1, 2, . . . , N and for any infinite word w = w0 w1 w2 · · · ∈ {1, 2, . . . , N }N0 we define Tw (S, {C j } Nj=1 ) be as in (2.3). If
N Z μ R,D ∩ Tw S, C j j=1 = ∅,
(2.10)
then w (S, {C j } Nj=1 ) as in (2.9) are all spectra for the measure μ R,D . In order to prove Theorem 2.3, we shall need a lemma due to Jorgensen and Pedersen, see [25, Lemmas 3.3 and 4.2]. Lemma 2.4 Let μ be a compactly supported on Rd and let be a Borel measure d 2 μ(ξ + λ)| . Then discrete subset of R . Suppose Q (ξ ) = λ∈ | (i) is an orthogonal set for μ if and only if Q (ξ ) ≤ 1 for ξ ∈ Rd . Moreover, Q is an entire function on Cd if is an orthogonal set for μ. (ii) (μ, ) is a spectral pair if and only if Q (ξ ) ≡ 1 for all ξ ∈ B(0, r ), which is an open ball centered at 0 with radius r . Let us return to the proof of Theorem 2.3. Proof of Theorem 2.3 The proof is essentially identical to that of [31, Theorem 2.8], we write it down for completeness. We divide the proof into the following two steps. Step I. The orthogonality of w (S, {C j } Nj=1 ). For each n ∈ N we define νn = δ R −1 D ∗ δ R −2 D ∗ · · · δ R −n D ,
N nw S, C j j=1 = Cw0 + SCw1 + · · · + S n−1 Cwn−1 . By the property of compatible pairs (see e.g., [2, Lemma 2.2]), (νn , nw (S, {C j } Nj=1 )) is a spectral pair which is equivalent to say that
ν n (ξ + λ)2 = 1 ξ ∈ Rd .
(2.11)
λ∈nw (S, {C j } Nj=1 )
This implies that nw (S, {C j } Nj=1 ) is an orthogonal set for the measure νn , i.e.,
N
N
nw S, C j j=1 − nw S, C j j=1 ⊆ Z ν n ∪ {0}.
Journal of Fourier Analysis and Applications
Notice that ∞
N
N
∞ nw S, C j j=1 and Z μ w S, C j j=1 = R,D = ∪n=1 Z ν n . n=1
Therefore, w (S, {C j } Nj=1 ) is an orthogonal set for the measure μ R,D . Step II. The completeness of w (S, {C j } Nj=1 ). N d Since μ R,D is a continuous function on R and Z(μ R,D ) ∩ Tw (S, {C j } j=1 ) = ∅, then there exist positive numbers ε, δ > 0 such that
N dist Z μ R,D , Tw S, {C j } j=1 > δ, N 2 d and |μ R,D (ξ )| > ε holds for any ξ ∈ {x ∈ R : dist(Z(μ R,D ), Tw (S, {C j } j=1 )) < δ/2}. Fix ξ ∈ B(0, δ/2). By the orthogonality of w (S, {C j } Nj=1 ), Bessel’s inequality implies that
2 μ R,D (ξ + λ) ≤ 1 (ξ ∈ B(0, δ/2)).
(2.12)
λ∈w (S, {C j } Nj=1 )
Noting that for any λ ∈ nw (S, {C j } Nj=1 ) we have that
S −n λ ∈ S −n Cw0 + SCw1 + · · · + S n−1 Cwn−1 ⊆ Tw S, {C j } Nj=1 , and hence
μ R,D (ξ + λ)2 = νn (ξ + λ)2 μ R,D S −n (ξ + λ) 2 ≥ ενn (ξ + λ)2 .
(2.13)
By Lebesgue’s dominated convergence theorem, we obtain, from (2.11)–(2.13), that
2 μ R,D (ξ + λ) = 1 (ξ ∈ B(0, δ/2)).
λ∈w (R T , {C j } Nj=1 )
This finish the proof by Lemma 2.4.
Recall that the Bernoulli convolution μ R := μ R,D can be generated by the selfaffine IFS {R −1 (x + d) : d ∈ D}, where R > 1 and D = {−1, 1}. It has been shown in [4,25] that μ R,D is a spectral measure if and only if R ∈ 2Z. Applying Theorem 2.3 to the spectral Bernoulli convolution μ2k , k > 1, one can get some new mixed-type spectra that is not appeared in literatures, see [6,12,28] and references cited there. Example 2.5 Let k ∈ N \ {0, 1, 2}, D = {−1, 1} and let C1 , C2 , . . . , C N be a family of subsets of {−(2k − 1), . . . , −1, 0, 1, . . . , 2k − 1} satisfying that 0 ∈ C j and
Journal of Fourier Analysis and Applications
(R −1 D, C j ) forms a compatible pair for all j = 1, 2, . . . , N . Then for any infinite word w = w0 w1 w2 · · · ∈ {1, 2, . . . , N }N0 , the set
N w 2k, C j j=1 = Cw0 + 2kCw1 + (2k)2 Cw2 + · · · forms a spectrum for the Bernoulli convolution μ2k . Proof By Theorem 2.3, it suffices to show the condition (2.10) holds for which R = S = 2k. This is true because ∞
2Z + 1 (2k) j Z μ2k = , 4 j=1
and the elements in Tw (2k, {C j } Nj=1 ) are smaller than or equal to
(2k − 1) (2k)−1 + (2k)−2 + · · · = 1, and bigger or equal to −1, which implies that Tw (2k, {C j } Nj=1 ) ⊆ [−1, 1]. Then k > 2 implies that Z(μ2k ) ∩ Tw (2k, {C j } Nj=1 ) = ∅. The proof is complete. As another application of Theorem 2.3, we also get a new mixed-type of spectra for Jorgensen–Pedersen’s example μ4,{0,2} in [25]. Example 2.6 Let C1 = {0, 1}, C2 = {0, −1}, C3 = {0, 7} and C4 = {0, −7}. Then for any infinite word w = w0 w1 w2 · · · ∈ {1, 2, 3, 4}N0 , the set
4 w 4, C j j=1 := Cw0 + 4Cw1 + 42 Cw2 + · · · is a spectrum for the measure μ4,{0,2} . Proof It is easy to check that (4−1 {0, 2}, C j ) forms a compatible pair for each j = 1, 2, 3, 4. Thus, by Theorem 2.3, it suffices to show that
4
Z μ4,{0,2} ∩ Tw 4, C j j=1 = ∅. As in (2.7), we set σi (x) = 4−1 (x + i), i ∈ {0, ±1, ±7}, and set r=
1 max σi (0) : i ∈ {0, ±1, ±7} = 7/3. −1 1−4
(2.14)
Journal of Fourier Analysis and Applications
Then, from the construction of Tw (4, {C j }4j=1 ) as in (2.8), we get that
4 Tw 4, C j j=1 ⊆ σ0 (B(0, 7/3)) ∪ σ1 (B(0, 7/3)) = [−7/12, 7/12] ∪ [−1/3, 5/6], or
4 Tw 4, C j j=1 ⊆ σ0 (B(0, 7/3)) ∪ σ−1 (B(0, 7/3)) = [−7/12, 7/12] ∪ [5/6, −1/3], or
4 Tw 4, C j j=1 ⊆ σ0 (B(0, 7/3)) ∪ σ7 (B(0, 7/3)) = [−7/12, 7/12] ∪ [7/6, 7/3], or
4 Tw 4, C j j=1 ⊆ σ0 (B(0, 7/3)) ∪ σ−7 (B(0, 7/3)) = [−7/12, 7/12] ∪ [−7/3, −7/6]. j Combining with the fact that Z( μ4,{0,2} ) = ∪∞ j=0 4 (2Z + 1), we get the desired result (2.14), concluding the proof.
3 Spectra of Cantor Measure with Consecutive Digit Sets The main goal of this section is to investigate the structure of spectra for a Cantor measure μ R,D with consecutive digit set generated by the IFS {R −1 (x + d) : d ∈ D}, where D = {0, 1, 2, . . . , q − 1} and R > q, q|R. Some results on the structure of spectra of μ R,D have been obtained in [5,7,12,17], etc. Assumption Throughout this section, we set b := R and μb, q := μ R,D . In [7], the authors showed that μb, q is a spectral measure if and only if q|b. Recently, Fu et al. [17,20] borrowed the notation “eigenvalue” in linear algebra to characterize the real numbers that generate new spectra from old ones, which indicates a convenient way to find spectra as many as possible. Definition 3.1 Let μ be a Borel probability measure on R. A real number p is called a spectral eigenvalue of μ if there exists a discrete set such that both and p are spectra for μ. The set is called an eigen-spectrum for μ corresponding to the eigenvalue p. In particular, [17, Theorem 1.2], determine all spectral eigenvalues of the above spectral measure μb, q , where q|b. Theorem B Let p be a real number and q|b. Then p is a spectral eigenvalue of μb, q if and only if p = pp21 where p1 , p2 and q are pairwise co-prime. Following on Theorem B, we will investigate the structure of eigen-spectra corresponds to the same eigenvalue in this section. We shall use Card(A) to denote the cardinal number of A and c = Card(R) is the cardinality of the continuum. The following is our main qualitative structural result. Theorem 3.2 For any p ∈ Z with gcd( p, q) = 1, then the set A p := { ⊆ R : and p are both spectra for μb, q , and 0 ∈ }
Journal of Fourier Analysis and Applications
has the cardinality of the continuum, denoted by Card(A p ) = c. Proof We firstly show that Card(A p ) ≤ c. It is easy to check that the zero set of the Fourier transform μ b, q is ∞
= bk r (Z \ qZ), where b = qr . Z μ b, q k=0
Furthermore, if ∈ A p , then the orthogonality of implies that
−⊆Z μ b, q ∪ {0}. Now the condition 0 ∈ means that ⊆ Z( μb, q ) ∪ {0} ⊆ Z, which yields that ∈ P(Z), and hence A p ⊆ P(Z), where P(Z) denotes the family of all subsets of Z. Therefore,
Card A p ≤ Card(P(Z)) = c. Next, we show that Card(A p ) ≥ c. Without loss of generality, we only consider positive integers p ∈ qN + {1, . . . , q − 1}. Let C0 := r D ∪(−r D), where b = qr , and let T (b, ± pC0 ) be the attractor generated by the following self-similar IFS
σ (x) = b−1 (x + c) : c ∈ pC0 ∪ − pC0 . Notice that the compact set T (b, ± pC0 ) and the zero set Z( μb, q ) are both symmetric to the origin point. Thus, the cardinality of their intersection is finite and is an even number. Setting
A := T b, ± pC0 ∩ Z μ b, q = x 1 , x 2 , . . . , x m ,
(3.1)
where m ∈ 2N. Recall that [17, Lemma 3.5] says that each x ∈ T (b, ± pC0 )) has an unique, ultimate periodic expansion in base b and the expansion of xi ∈ A cannot be finite. Thus, for each point xi ∈ A, there exist m i ∈ N and i ∈ N such that xi = p
mi
ci, j b− j + pb−m i
j=1 ∞ !
×
" ci, m i +1 b−1 + ci, m i +2 b−2 + · · · + ci, m i + i b− i b− i k , (3.2)
k=0
where ci, j ∈ ±C0 and {ci, m i +1 , ci, m i +2 , . . . , ci, m i + i } are not zero.
Journal of Fourier Analysis and Applications
According to the minimal periodic section “ci, m i +1 ci, m i +2 · · · ci, m i + i ” of xi , the m can be divided into the following (at most) three classes: above points {xi }i=1 Class (a). for xi , 1 ≤ i ≤ s, s ∈ N, there are at least one term ci, m i + j > 0 and at least one term ci, m i + j < 0 for some 1 ≤ j, j ≤ i ; Class (b). for xi , s + 1 ≤ i ≤ s + t, t ∈ N, all terms ci, m i + j ≥ 0 for all 1 ≤ j ≤ i and there is at least one term ci, m i + j > 0 for some 1 ≤ j ≤ i ; Class (c). for xi , s + t + 1 ≤ i ≤ m, all terms ci, m i + j ≤ 0 for all 1 ≤ j ≤ i and there is at least one term ci, m i + j < 0 for some 1 ≤ j ≤ i . Letting M = max i : 1 ≤ i ≤ s + t and N = max i : s + t + 1 ≤ i ≤ m . Given any infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 . One define the sets
G M,N := − C0 + bC0 + · · · + b M−1 C0 + b M C0 + · · · + b M+N −1 C0 , and G M,N ,i := wi C0 + bG M,N . Next we define the set M,N ,w =
∞
p G M,N ,0 + b M+N +1 G M,N ,1 + · · · + b(M+N +1)i G M,N ,i i=0
= p
∞
ι(k)b ck : ck ∈ C0 , k ∈ N , k
(3.3)
k=0
where ι is a {1, −1}-valued function on N0 satisfying that ⎧ ⎨ wi , ι(k) = −1, ⎩ 1,
if k ∈ (M + N + 1)i, i ∈ N0 , if k ∈ {1, 2, . . . , M} + (M + N + 1)N0 , if k ∈ M + {1, 2, . . . , N } + (M + N + 1)N0 .
It is easy to check that pC0 ≡ C0 (mod b) if gcd( p, q) = 1, which yields that (b−1 D, pC0 ) forms a compatible pair. As in the proof of Theorem 2.3, we know that M,N ,w forms an orthogonal set for the measure μb, q . In the following, we will show that Claim 1 For any w ∈ {−1, 1}N0 , the set M,N ,w as in (3.3) is a spectrum for the measure μb, q . Proof Notice that for any w ∈ {−1, 1}N0 , one has # G M,N ,i =
C0 + bG M,N , −C0 + bG M,N ,
if wi = 1, if wi = −1.
Journal of Fourier Analysis and Applications
So one can apply Theorem 2.1 to the sets
C1 := p C0 + bG M,N and C2 := p − C0 + bG M,N . More precisely, as in the proof of Theorem 2.1, one can generate the Moran set T (b M+N +1 , pG M,N ,i ) by the Moran IFS {σi,gi (x) := b−(M+N +1) (x + pgi ) : gi ∈ G M,N ,i } via the following composition
σn, gn ◦ σn−1, gn−1 ◦ · · · ◦ σ0, g0 : n ∈ N0
as in (2.8). By Theorem 2.1 and the definition of G M,N ,i , we get that n
M+N +1 −(M+N +1)i T b , pG M,N ,i = Cl p b gn−i : gn−i ∈ G M,N ,n−i , n ∈ N = Cl p
i=1 n
τ (k)b−k ck : ck ∈ C0 , n ∈ (M + N + 1)N ,
k=1
(3.4) where Cl(A) denotes the closure of A in the usual topology of R and τ (k) is a {1, −1}valued function on N0 satisfying that ⎧ if k ∈ {1, 2, . . . , N } + (M + N + 1)N0 , ⎨ 1, if k ∈ N + {1, 2, . . . , M} + (M + N + 1)N0 , (3.5) τ (k) = −1, ⎩ ∈ {−1, 1}, if k ∈ (M + N + 1)(i + 1), i ∈ N0 . Here, we need to explain why one can not determine the exact value of τ at the positions (M + N + 1)N. This is because if we fix n = (M + N + 1)m for some m ∈ N in the right bracket of (3.4), then for all i = 0, 1, . . . , m − 1, we will have τ ((M + N + 1)(i + 1)) = wm−i , which means that the values of τ at these positions depend heavily on the length of the expansion of x ∈ T (b M+N +1 , pG M,N ,i ). In order to finish the proof of Claim 1, we need the following Claim 2. Claim 2 A ∩ T (b M+N +1 , pG M,N ,i ) = ∅. Proof First, since T (b M+N +1 , pG M,N ,i ) ⊆ T (b, ±C) and each element of T (b, ±C) has an infinite, unique and ultimate periodic expansion in base b, then (3.4) and (3.5) imply that any x ∈ T (b M+N +1 , pG M,N ,i ) has the following unique infinite expansion in base b x= p
n
c j b− j + pb−n
j=1
+cn+M+N +1 b
∞ ! cn+1 b−1 + cn+2 b−2 + · · · k=0
−(M+N +1)
"
b−(M+N +1)k ,
(3.6)
Journal of Fourier Analysis and Applications
where n ∈ (M + N + 1)N0 and cn+ j ∈ ±C0 , 1 ≤ j ≤ M + N + 1, satisfies that
cn+ j
⎧ ⎨ C0 , ∈ −C0 , ⎩ ±C0 ,
if j ∈ {1, 2, . . . , N }, if j ∈ N + {1, 2, . . . , M}, if j = M + N + 1.
(3.7)
Next, we will derive x ∈ / A by comparing the periodic section “cn+1 cn+2 · · · cn+M+N +1 ” in the expansion in the base b of x in (3.6) with the periodic section “ci, m i +1 ci, m i +2 · · · ci, m i + i ” in the expansion in the base b of x in (3.2). Recall that each element in A must belong to one of the Classes (a)–(c). We shall prove it in the following three cases. (i) x ∈ / Class (a). Otherwise, by the definition of class (a) and the choice of M, any section of length M in the (ultimate) periodic sequence
cn+1 · · · cn+M+N +1 cn+1 · · · cn+M+N +1 · · ·
corresponding to x in (3.6) must contain at least one term cn+ j > 0 and at least one term cn+ j < 0 for some j, j ∈ {1, 2, . . . , M + N + 1}. However, (3.7) implies that cn+ j ≤ 0 holds for all j ∈ N + {1, 2, . . . , M}, a contradiction. (ii) x ∈ / Class (b). The proof is similar to that of (i). Otherwise, by the definition of class (b) and the choice of M, any section of length M in the periodic sequence
cn+1 · · · cn+M+N +1 cn+1 · · · cn+M+N +1 · · ·
corresponding to x in (3.6) must satisfy that cn+ j ≥ 0 for all j ∈ {1, 2, . . . , M + N + 1} and at least one term cn+ j > 0. However, (3.7) implies that cn+ j ≤ 0 holds for all j ∈ N + {1, 2, . . . , M}, a contradiction. (iii) x ∈ / Class (c). The proof is similar to that of (i) and (ii). Otherwise, by the definition of class (c) and the choice of N , any section of length N in the periodic sequence
cn+1 · · · cn+M+N +1 cn+1 · · · cn+M+N +1 · · ·
corresponding to x in (3.6) must satisfy that cn+ j ≤ 0 for all j ∈ {1, 2, . . . , M + N + 1} and at least one term cn+ j < 0. However, (3.7) implies that cn+ j ≥ 0 holds for all j ∈ N + {1, 2, . . . , N }, a contradiction. Now the definition of A and the condition T (b M+N +1 , pG M,N ,i ) ⊆ T (b, ±C) imply that A ∩ T (b M+N +1 , pG M,N ,i ) = ∅, as desired. This finishes the proof of Claim 2. Returning to the proof of Claim 1 Combining Claim 2 with that T (b M+N +1 , pG M,N ,i ) ⊆ T (b, ± pC0 ), we can get, from (3.1), that
M+N +1
, pG M,N ,i = ∅. Z μ b, q ∩ T b By Theorem 2.3, the set M,N ,w is a spectrum for the measure μb, q . The Claim 1 is true.
Journal of Fourier Analysis and Applications
By [17, Lemma 3.4], we know that for any infinite word u = u 0 u 1 · · · ∈ {−1, 1}N0 , ⎧ ⎫ m ⎨ ⎬
u j b j−1 c j : c j ∈ C0 , u j ∈ {−1, 1}, m ∈ N , (3.8) u b, C0 := ⎩ ⎭ j=1
is a spectrum for the measure μb, q . In particular, for any w ∈ {−1, 1}N0 , the set p −1 M,N ,w [see (3.3)] forms a spectrum for the measure μb, q . Notice that B p := p −1 M,N ,w : w ∈ {−1, 1}N0 ⊆ A p and the set B p has the cardinality of the continuum c. Thus, Card(A p ) ≥ Card(B p ) = c. The proof of Theorem 3.2 is complete. Repeating the arguments of the proof of Theorem 3.2 step by step to the finite set
T b, ± p1 C0 ∪ T b, ± p2 C0 ∩ Z μ b, q instead of (3.1), where gcd( p1 , p2 ) = 1 and gcd( pi , q) = 1 for i = 1, 2, one can get the following result. Here, we omit the details of the proof. Theorem 3.3 For any p1 , p2 ∈ Z with gcd( pi , q) = 1 for i = 1, 2 and gcd( p1 , p2 ) = 1, then A p1 , p2 := ⊆ R : , p1 and p2 are both spectra for μb, q and 0 ∈ has the cardinality of the continuum, that is, Card(A p1 , p2 ) = c. If we combine Theorem 3.3 with Theorem B, it yields the following result. Theorem 3.4 For any p ∈ R and q|b, the following three statements are equivalent: (i) p is a spectral eigenvalue of μb, q ; (ii) p = pp21 , where p1 , p2 and q are pairwise co-prime; (iii) Card({ ⊆ R : and p are spectra for μb, q and 0 ∈ }) = c. Proof The equivalence of (i) and (ii) is due to Theorem B and (iii) clearly implies (i) by the definition of spectral eigenvalue. It suffices to show that (ii) implies (iii). Indeed, the discussions in the first part of the proof of Theorem 3.2 always implies that Card({ ⊆ R : and p are spectra for μb, q and 0 ∈ }) ≤ c. Moreover, if (ii) holds, then Theorem 3.3 yields that Card({ ⊆ R : and p are spectra for μb, q and 0 ∈ }) ≥ Card({p2 ⊆ R : p1 , p2 are both spectra for μb, q and 0 ∈ }) (since p1 = pp2 ) = Card({ ⊆ R : p1 , p2 are both spectra for μb, q and 0 ∈ }) ≥ Card(Ap1 ,p2 ) = c,
Journal of Fourier Analysis and Applications
as desired. The proof is complete.
The following proposition gives new spectra for Bernoulli convolution μ2k defined in Example 2.5, and answers the question 2 proposed in [20, Sect. 5]. Proposition 3.5 For any p =
p1 p2
with p1 , p2 ∈ 2Z + 1 and gcd( p1 , p2 ) = 1, then
{ ⊆ R : and p are spectra for μ2k and 0 ∈ }. has the cardinality of the continuum. Proof Applying Theorem 3.4 to the case that b = 2k and q = 2, we get, for each p = pp21 with p1 , p2 ∈ 2Z + 1 and gcd( p1 , p2 ) = 1, that Card({ ⊆ R : and p are spectra for μ2k, 2 and 0 ∈ }) = c.
(3.9)
Next we claim that (μ2k , ) is a spectral pair if and only if (μ2k, 2 , 2) is a spectral pair, where ⊆ R is a discrete set. Indeed, an easy calculation shows that |μ2k (ξ )| = | μ2k, 2 (2ξ )|, (∀ξ ∈ R). It follows that
|μ2k (ξ + λ)|2 =
λ∈
| μ2k, 2 (2ξ + 2λ)|2 , (∀ξ ∈ R).
λ∈
By Lemma 2.4, the claim is true. By (3.9) and the claim, one get the desired result of Proposition 3.5.
We end this section by giving a simple condition on p and b which determines when the scaling set ⎧ ⎫ m ⎨ ⎬ pu j b j−1 c j : c j ∈ C0 , u j ∈ {−1, 1}, m ∈ N E( pu (b, C0 )) = ⎩ ⎭ j=1
of u (b, C0 ) [see (3.8)] is also an orthonormal basis for L 2 (μb, q ). This condition is a natural generalization on the spectra of Bernoulli convolution, see [28, Theorem 3.1], and [20, Corollary 4.7]. Theorem 3.6 Let p ∈ N with gcd( p, q) = 1. If p < the set pu (b, C0 ) is a spectrum for μb, q .
b−1 p−1 , then for any u
∈ {−1, 1}N0
Proof It is easy to check (see [17, Lemma 3.3]) that gcd( p, q) = 1 implies that (b−1 D, qC0 ) forms a compatible pair. Consequently, the set E( pu (b, C0 )) is an orthogonal set for the measure μb, q . If one can show that
Z δb−1 D ∩ T (b, pC0 ∪ (− pC0 )) = ∅,
(3.10)
Journal of Fourier Analysis and Applications
then [20, Theorem 3.4], guarantees the result of Theorem 3.6 is true. It remains to show (3.10). Since
Z δb−1 D = r (Z \ qZ), where b = qr , and T (b, pC0 ∪ (− pC0 )) =
⎧ ∞ ⎨ ⎩
b− j c j : c j ∈ pC0 ∪ (− pC0 )
j=1
$
⊆ − then the condition p <
b−1 p−1
⎫ ⎬ ⎭
%
pr (q − 1) pr (q − 1) , , b−1 b−1
implies that (3.10), which finishes the proof.
Acknowledgements The research is supported by the National Natural Science Foundation of China Nos. 11371055 and 11431007. The author would like to thank the anonymous referee for his/her many valuable suggestions which led to an improvement of this paper.
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