Spectra of Bernoulli convolutions and random convolutions

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Spectra of Bernoulli convolutions and random convolutions ✩ Yan-Song Fu a , Xing-Gang He b,∗ , Zhi-Xiong Wen c a

School of Science, China University of Mining and Technology, Beijing, 100083, PR China School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, PR China c School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China b

a r t i c l e

i n f o

Article history: Received 16 October 2016 Available online xxxx MSC: 28A80 42C05 42A85 42B05 Keywords: Bernoulli convolutions Random convolutions Spectral measures Spectra

a b s t r a c t In this work we study the harmonic analysis of infinite convolutions generated by compatible pairs. We first give some sufficient conditions so that a random infinite convolution μ becomes a spectral measure, i.e., there exists a countable set Λ ⊆ Rn such that E(Λ) = {e2πiλ,x : λ ∈ Λ} forms an orthonormal basis for L2 (μ). As applications, we settle down the spectral eigenvalue problem for spectral Bernoulli convolutions. © 2018 Published by Elsevier Masson SAS.

r é s u m é Dans ce travail nous étudions l’analyse harmonique de convolutions infinies engendrées par des paires compatibles. Nous donnons d’abord des conditions suffisantes telles qu’une convolution infinie aléatoire μ devient une mesure spectrale, i.e., il existe un ensemble dénombrable Λ ⊆ Rn tel que E(Λ) = {e2πiλ,x : λ ∈ Λ} offre une base orthonormée pour L2 (μ). Comme application, nous tranchons le problème de valeurs propres spectrales pour les convolutions spectrales de Bernoulli. © 2018 Published by Elsevier Masson SAS.

1. Introduction For a compactly supported Borel probability measure μ on Rn , we call μ a spectral measure if there exists a countable set Λ ⊆ Rn such that the family of complex exponentials E(Λ) := {eλ (x) = e2πiλ,x : λ ∈ Λ} forms an orthonormal basis (Fourier basis) for L2 (μ). In this case, the set Λ is called a spectrum for μ, and we also say that (μ, Λ) forms a spectral pair. When μ is the normalized Lebesgue measure supported on a ✩ This work is supported by School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences of Central China Normal University and the National Natural Science Foundation of China Nos. 11371055 and 11431007. * Corresponding author. E-mail addresses: [email protected], [email protected] (Y.-S. Fu), [email protected] (X.-G. He), [email protected] (Z.-X. Wen).

https://doi.org/10.1016/j.matpur.2018.06.002 0021-7824/© 2018 Published by Elsevier Masson SAS.

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Borel set Ω, the existence of a spectrum is closely related to the famous Fuglede conjecture, which asserts that there exists a spectrum for μ if and only if Ω is a translational tile [21]. This conjecture has been proved to be false by Tao and others in both directions in dimension three or higher, see, e.g., [30,40], but it is still open in dimension 1 and 2. The first non-atomic, singular continuous spectral measure was found by Jorgensen and Pedersen [28] in 1998, which opened up a new field in researching the orthogonal harmonic analysis of fractal measures including self-similar/self-affine measures and generally Moran measures, see [35,36,29,38]. Nowadays, there has been a wide range of interests in the study of harmonic analysis on fractals. Many interesting spectral measures have been found, at the same time some singular phenomena different from the spectral theory of Lebesgue measures have been discovered, please see [1–3,5,7,9,10,13,8,19,14–16,39,26,25,6,22] and the references therein. In all these research, the following two types of problems are basic for the spectral measure theory: I. Spectral Problem: For what measure μ does L2 (μ) admit a Fourier basis? II. Spectral Eigenvalue Problem: This is a particular problem for singular spectral measures. There are two basic forms: (1) Let Λ be a spectrum for a spectral measure μ. Find all real numbers p such that pΛ is also a spectrum for μ; (2) Let μ be a spectral measure. Find all real numbers p for which there exists a set Λ such that both Λ and pΛ are spectra for μ. In the two cases, p is called a spectral eigenvalue of μ and Λ is called an eigen-spectrum of μ corresponding to p. In this paper, we will deal with the above two problems for some infinite convolutions generated by finite discrete measures, in particular, for Bernoulli convolutions μ2k , k ∈ N. Recall that the Bernoulli convolution ∞ μR for each R > 1 is the distribution of the random variable n=1 ±R−n , where the signs “+” and “−” are chosen independently with probability 1/2. Bernoulli convolutions have been studied extensively in many areas of mathematics including Fourier analysis, dynamical system, integer tile, wavelet theory, algebraic number theory and fractal geometry since 1930s (e.g., see [41] and the references therein) and have the following expression of convolutions: μR := μR,D = δR−1 D ∗ δR−2 D ∗ · · · ∗ δR−n D ∗ · · · ,

where D = {−1, 1}.

(1.1)

Here, the symbol δE for a finite set E denotes the atomic measure δE =

1  δe , #E e∈E

where δe is the Dirac point mass measure at the point e, rE = {re : e ∈ E} and #E is the cardinality of E. Moreover, the measure μR is the unique probability one with compact support satisfying that μR (·) =

1 1 −1 −1 μR ◦ τ− (·) + μR ◦ τ+ (·), 2 2

where {τ+ (x) = R−1 (x + 1), τ− (x) = R−1 (x − 1)} is an iterated function system (IFS) on R [24]. The study on the spectral property of Bernoulli convolutions dates back to the work of Jorgensen and Pedersen [28], and many spectral properties of Bernoulli convolutions have been found in [26,14,9,3,8,5] and references cited therein. An outstanding result of Jorgensen and Pedersen [28], Dai [5] shows that Theorem A. A Bernoulli convolution μR is a spectral measure if and only if R is a positive even integer. Moreover, let C = {0, k2 } for integer k ≥ 1, then the set ⎧ ⎫ m ⎨ ⎬ Λ(2k, C) := (2k)j cj : cj ∈ C, m ≥ 0 (1.2) ⎩ ⎭ j=0

is a spectrum for μ2k .

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The primary aim of this paper is to answer the spectral eigenvalue problem for Bernoulli convolution μ2k . It is known that the spectral eigenvalue problem (1) for μ2k and Λ(2k, C) as in (1.2) has been studied extensively, see e.g. [6,27,33,17,12], which is very challenging and has not been solved completely. Our principle result in this paper is the following (see also Theorem 4.2): Theorem 1.1. Let μ2k be the Bernoulli convolution for integer k ≥ 1. Then a real number p is a spectral eigenvalue of μ2k if and only if p is the quotient of two odd integers. It is noted that when k = 1, the measure μ2 is the normalized Lebesgue measure restricted on the interval [−1, 1]. In this case, the set Λ = 12 Z is the unique spectrum satisfying 0 ∈ Λ for the measure μ2 , and hence p = 1 is the unique eigenvalue of μ2 . Theorem 1.1 gives a complete answer to the spectral eigenvalue problem (2) for the Bernoulli convolution μ2k . This is the first example for which the spectral eigenvalue problem (2) has been settled completely. A key point in the proof of Theorem 1.1 is to introduce the following notation:

Λw (2k, C) =

⎧ m ⎨ ⎩

j=0

wj (2k)j cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0

⎫ ⎬ ⎭

where w = w0 w1 w2 · · · with all wi ∈ {−1, 1} and C = {0, k2 }. It is worthy noting that these sets and their scaling sets pΛw (2k, C) = Λw (2k, pC) for odd integers p will provide us more candidates of spectra for μ2k , see Proposition 4.4. So a natural question left is to check under what conditions both Λw (2k, C) and Λw (2k, pC) are spectra for μ2k . In order to solve the above question mentioned and construct more spectral measures (Spectral Problem), we will study the spectral property of infinite convolutions of discrete measures generated by an n × n expanding matrix R (all eigenvalues of R are larger than 1 in module) and a family {D(1), D(2), . . . , D(N )} of finite sets in Rn , which is defined by μR,X := δR−1 D(X(1)) ∗ δR−2 D(X(2)) ∗ · · · ∗ δR−n D(X(n)) ∗ · · · , where X is an index mapping from N to {1, 2, . . . , N }. Some results concerning the spectral property of measures with the above forms were obtained in [37,38,1–3,18,20]. In Section 3 we investigate the spectrality of random convolutions (Definition 3.1) including Bernoulli convolutions, and give some sufficient conditions for a random convolution to be a spectral measure. To some extent, the results given here extend the criterions due to Strichartz [37,38] and complement the corresponding results recently obtained in [2,18]. The proof of Theorem 1.1 is one of the applications of the theory on random convolutions given in Section 3. The paper is organized as follows. In Section 2, we introduce preliminary results. In Section 3, we investigate the spectrality of random convolutions and provide some examples to illustrate the theory. In Section 4, we prove Theorem 1.1 and construct new spectra for μ2k . In Section 5, we will give some remarks and open questions relating to spectral Bernoulli convolutions μ2k . Notations. In the present paper we use N to denote the set of positive integers and N0 = N ∪ {0}. 2. Definitions and preliminaries The aim in this section is to collect some necessary definitions and basic facts for self-affine measures and compatible pair conditions. The data of compatible pairs (Definition 2.3) allows us to create a great deal of orthogonal sets and then to check the conditions in Proposition 2.1 for a measure to be a spectral one in the following sections.

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The Fourier transform of a Borel probability measure μ on Rn is defined by  μ

(ξ) = e−2πiξ,x dμ(x) (ξ ∈ Rn ). Rn

It is easy to check that the family E(Λ) := {e2πiλ,x : λ ∈ Λ} forms an orthogonal set for L2 (μ) is equivalent to the condition that  |

μ(ξ + λ)|2 ≤ 1 (ξ ∈ Rn ), (2.1) λ∈Λ

as well as to that (Λ − Λ) \ {0} ⊆ Z(

μ),

(2.2)

where Z(f ) denotes the zero set of the function f , i.e., Z(f ) = {x ∈ Rn : f (x) = 0}. We also call Λ an orthogonal set (respectively, spectrum) for μ if E(Λ) forms an orthogonal system (respectively, Fourier basis) for L2 (μ). Since the properties of spectra are invariant under translation, it will be convenient to assume that 0 ∈ Λ, and hence Λ ⊆ Λ − Λ. Let  QΛ (ξ) = |

μ(ξ + λ)|2 (ξ ∈ Rn ). λ∈Λ

The well known result of Jorgensen and Pedersen [28, Lemma 4.2] shows that QΛ is an entire function if Λ is an orthogonal set for μ. The following provides a universal test which allows us to decide whether an orthogonal set Λ is a spectrum for the measure μ. Proposition 2.1. (See [28].) Let μ be a compactly supported Borel probability measure on Rn and let Λ be an orthogonal set for μ. Then the following statements are equivalent: (i) The orthogonal set Λ is a spectrum for μ;  (ii) QΛ (ξ) = λ∈Λ |

μ(ξ + λ)|2 = 1, ∀ ξ ∈ Rn ;  μ(ξ + λ)|2 = 1, ∀ ξ ∈ B(0, r), where B(0, r) denotes the open ball with center 0 and (iii) QΛ (ξ) = λ∈Λ |

radius r > 0. We remark that the equivalence of the statements (i) and (ii) is a direct consequence of Stone–Weierstrass theorem and Parseval’s identity, while the equivalence of (ii) and (iii) is due to the analyticity property of the function QΛ . The following lemma was proved in [8, Lemma 2.2] and will be used to prove Theorem 4.2 and Theorem 4.3. Since the proof is simple, we give it here. Lemma 2.2. Let μ = μ0 ∗ μ1 be the convolution of two probability measures μi , i = 0, 1, and they are not Dirac measures. Suppose that Λ is an orthogonal set for μ0 with 0 ∈ Λ, then Λ is also an orthogonal set for μ, but cannot be a spectrum for μ. Proof. Note that μi is not a Dirac measure is equivalent to |

μi (ξ)| ≡ 1. Since μ

0 (0) = 1, there exists ξ0 such that |

μ0 (ξ0 )| = 0 and |

μ1 (ξ0 )| < 1. Hence,    Q(ξ0 ) = |

μ(ξ0 + λ)|2 = |

μ0 (ξ0 + λ)|2 |

μ1 (ξ0 + λ)|2 < |

μ0 (ξ0 + λ)|2 ≤ 1. λ∈Λ

λ∈Λ

The desired result follows from Proposition 2.1(i) and (ii). 2

λ∈Λ

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Definition 2.3. Let R ∈ Mn (Z) be an n × n expanding integer matrix, and let D, C ⊆ Rn be two finite subsets of Rn with the same cardinality. We say that the system (R−1 D, C) forms a compatible pair (or (R, D, C) is a Hadamard triple) if the matrix HR−1 D,C := √

1 2πiR−1 d,c e d∈D,c∈C #D

(2.3)

∗ is unitary, i.e., HR −1 D,C HR−1 D,C = I.

The following properties of integrally compatible pairs can be found, explicitly or implicitly, in [28,31, 37,38]. We will denote the transpose of a matrix R by RT , and the k-times products of the matrix R by Rk := R · · R.  · k times

Proposition 2.4. Let R ∈ Mn (Z) be an n × n expanding matrix, and let D, C ⊆ Zn be two finite subsets of Zn with the same cardinality such that (R−1 D, C) forms a compatible pair. Then the following statements hold. (i) δR−1 D is a spectral measure with a spectrum C. / RZn for distinct elements di , dj ∈ D), (ii) No two elements in D are congruent modulo R (i.e., di − dj ∈ T and no two elements in C are congruent modulo R . (iii) (R−1 D, −C) forms a compatible pair. (iv) Assume that (R−1 Dk , Ck ) are compatible pairs and define Dk := R−1 D1 + · · · + R−k Dk , Ck := C1 + RT C2 + · · · + (RT )k−1 Ck . Then (Dk , Ck ) is a compatible pair for each k ≥ 1. Definition 2.5. Let R ∈ Mn (R) be an expanding matrix, and let D be a finite subset of Rn . Define the self-affine iterated function system (IFS) τd (x) = R−1 (x + d)

(x ∈ Rn , d ∈ D).

(2.4)

Since R is expanding, there exists a norm on Rn such that the maps τd are contractions, and therefore Hutchinson’s theorem in [24] can be applied: Theorem 2.6. (See [24].) For the above IFS {τd : d ∈ D}, there is a unique Borel probability measure μR,D , which is called a self-affine measure, satisfying the following invariant equation μR,D =

1  μR,D ◦ τd−1 , #D

(2.5)

d∈D

and having the compact support T (R, D) ⊆ Rn , called a self-affine set, with T (R, D) =



τd (T (R, D)).

(2.6)

d∈D

Moreover, T (R, D) can be expressed by the following radix expansion

T (R, D) =

⎧ ∞ ⎨ ⎩

j=1

R−j dj : dj ∈ D for all j ∈ N

⎫ ⎬ ⎭

.

(2.7)

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From (2.5), the Fourier transform μ R,D of the measure μR,D is easily computed by μ R,D (ξ) =

∞ 

δ R−j D (ξ) =

j=1

∞ 

δ R−1 D ((RT )−j ξ)

(ξ ∈ Rn ),

(2.8)

j=0

where 1  −2πid,ξ e δ D (ξ) = #D

(ξ ∈ Rn ),

(2.9)

d∈D

which is the mask function of D. Hence, we obtain the following relationship Z(μ R,D ) =

∞ 

(RT )j Z(δ R−1 D ).

(2.10)

j=0

Standing Assumptions. Throughout the paper, we will use the symbol μR,A (resp. T (R, A)) to denote the invariant measure (resp. attractor) generated by the IFS in the form (2.4) with an expanding matrix R and a finite digit set A ⊆ Rn . The following is an elementary but useful fact in our investigation. Proposition 2.7. Let R ∈ Mn (Z) be an n × n expanding matrix, and let D and A be two finite subsets of Zn such that 0 ∈ A, then the following statements are equivalent: (i) Z(δ R−1 D ) ∩ T (RT , A) = ∅; T (ii) Z(μ R,D ) ∩ T (R , A) = ∅. Proof. (ii) ⇒ (i) It follows since (2.10). (i) ⇒ (ii) Suppose Z(δ R−1 D ) ∩ T (RT , A) = ∅. It follows from (2.7) and 0 ∈ A that (RT )−j T (RT , A) ⊆ T (RT , A) for j ∈ N. Then Z(δ R−1 D ) ∩ (RT )−j T (RT , A) = ∅ for all j ∈ N and (ii) holds by (2.10). 2 Remark 2.8. The cardinality of the sets D and A in Proposition 2.7 may be not equal. 3. Spectra of random convolutions In the present section, we study the spectral property of random convolutions generated by compatible pairs. Definition 3.1. Let R ∈ Mn (Z) be an expanding integer matrix, and let D(1), · · · , D(N ) be a finite collection of subsets of Zn with the same cardinality. A random convolution of discrete measures scaled by R is defined by μR,X := δR−1 D(X(1)) ∗ δR−2 D(X(2)) ∗ · · · ∗ δR−n D(X(n)) ∗ · · · ,

(3.1)

where X : N → {1, 2, · · · , N } is an index mapping. We notice that if all D(i) = D for some D ⊆ Zn , then the measure μR,X is reduced to the self-affine measure μR,D generated by the IFS {τd (x) = R−1 (x + d) : d ∈ D}.

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Clearly the Fourier transform of μR,X is μ R,X (ξ) =

∞ 

δ R−j D(X(j)) (ξ).

(3.2)

j=1

The spectral property of these measures, to the best of our knowledge, was first studied by Strichartz [38,39], and extensively studied by An et al. [1–3], and Dutkay and Lai [18]. In particular, the main result of An, He and Lau [2] is the following. Theorem B. Let R ≥ 2 be an integer, and 0 ∈ C ⊆ Z+ such that C + C ⊆ {0, 1, . . . , R −1} and (R−1 D(i), C) is a compatible pair for each i = 1, 2, · · · , N . Then μR,X is a spectral measure for any index mapping X. The rest of this section is devoted to find additional conditions to guarantee random convolutions to be spectral measures in higher dimension, which will be used in the following sections. For this purpose, we set ⎫ ⎧ m ⎬ ⎨ (3.3) Λw (R, C) = wj (RT )j cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0 ⎭ ⎩ j=0

for any infinite word w = w0 w1 w2 · · · in {−1, 1}N0 and C ⊆ Zn with 0 ∈ C. In the special case that w = 111 · · · ∈ {−1, 1}N0 or C = −C, the set Λw (R, C) becomes ⎫ ⎧ m ⎬ ⎨ Λ(R, C) = (RT )j cj : cj ∈ C for 0 ≤ j ≤ m, m ∈ N0 = C + RT Λ(R, C), ⎭ ⎩

(3.4)

j=0

which has the self-affine property. For the above set Λ(R, C), Jorgensen and Pedersen [28], Strichartz [37,38], Łaba and Wang [31], Dutkay and Jorgensen [14,15], Li [32,34] obtained certain conditions to ensure the set E(Λ(R, C)) := {e2πiλ,x : λ ∈ Λ(R, C)} to be an orthonormal basis for L2 (μR,D ). In this section, we are interested in conditions under which the set Λw (R, C) forms a spectrum for μR,X . The first question we shall answer is the following. Lemma 3.2. Let R ∈ Mn (Z) be an n × n expanding matrix, and let D(1), D(2), · · · , D(N ) be a finite subset of Zn with the same cardinality. Let 0 ∈ C and C ⊆ Zn be a finite set such that (R−1 D(i), C) forms a compatible pair. Then, for any infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 , the set Λw (R, C) as in (3.3) forms an orthogonal set for μR,X . Proof. Fix X : N → {1, 2, · · · , N } and an infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 . For each k ∈ N, we set μk = δR−1 D(X(1)) ∗ δR−2 D(X(2)) ∗ · · · ∗ δR−k D(X(k)) , T T k−1 C. Λw k = w0 C + w1 R C + · · · + wk−1 (R )

From the properties of compatible pairs (see Proposition 2.4 (i), (iii) and (iv)), we know that μk is a spectral measure with a spectrum Λw k , which yields that w (Λw μk ). k − Λk ) \ {0} ⊆ Z(

By (3.2), μ k (ξ) R,X (ξ) = μ

∞  j=k+1

δ R−j D(X(j)) (ξ).

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w Combining with the relation Λw (R, C) = ∪∞ k=1 Λk , which holds since 0 ∈ C, we see that for any distinct  λ, λ ∈ Λw (R, C), there exists k ∈ N such that λ, λ ∈ Λw k and w λ − λ ∈ (Λw μk ) ⊆ Z(μ R,X ). k − Λk ) \ {0} ⊆ Z(

Using (2.2), the set Λw (R, C) forms an orthogonal set for μR,X . This ends the proof. 2 The criterions of Strichartz in [37] and [38, Theorem 2.8] can be generalized as follows. Theorem 3.3. Let R ∈ Mn (Z) be an n × n expanding matrix, and let D(1), D(2), · · · , D(N ) be a finite subset of Zn with the same cardinality. Let 0 ∈ C and C ⊆ Zn be a finite set such that (R−1 D(i), C) forms a compatible pair and Z(δ R−1 D(i) ) ∩ T (RT , C ∪ (−C)) = ∅ for each i = 1, 2, · · · , N . Then (μR,X , Λw (R, C)) is a spectral pair for all X : N → {1, 2, · · · , N } and w ∈ {−1, 1}N0 . Proof. The proof is essentially identical to that of Theorem 2.8 in [38]. We first know, from Proposition 2.7, that the condition Z(δ R−1 D(i) ) ∩ T (RT , C ∪ (−C)) = ∅ implies that T Z(μ R,D(i) ) ∩ T (R , C ∪ (−C)) = ∅,

∀ i = 1, 2, · · · , N.

T Since Z(μ R,D(i) ) is a closed set and T (R , C ∪ (−C)) is a compact set, there is a δ > 0 such that T d(Z(μ R,D(i) ), T (R , C ∪ (−C))) > δ for 1 ≤ i ≤ N , where d denotes the usual metric in Euclidean space Rn . Hence there is an ε > 0 such that 2 |μ R,D(i) (ξ)| > ε

(3.5)

if ξ ∈ {x ∈ Rn : d(x, T (RT , C ∪ (−C))) ≤ δ/2}. Fix X : N → {1, 2, · · · , N } and an infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 . Define μk , Λw k as in the proof of Lemma 3.2. Then, by Proposition 2.1 we obtain that 

|

μk (ξ + λ)|2 = 1

λ∈Λw k



and

2 |μ R,X (ξ + λ)| ≤ 1

(ξ ∈ Rn ).

(3.6)

λ∈Λw (R,C)

Fix ξ ∈ B(0, 1) and define  fk (λ) =

|

μk (ξ + λ)|2 , λ ∈ Λw k; f (λ) = 0, otherwise,



2 |μ R,X (ξ + λ)| , 0,

λ ∈ Λw (R, C); otherwise.

Then, lim fk (λ) = f (λ) for each λ ∈ Λw (R, C). It follows from (3.2) and the simple observation |δ D | ≤ 1 k→∞

that 2 |μ μk (ξ + λ)|2 R,X (ξ + λ)| = |

∞  2  

 δR−(k+j) D(X(k+j)) (ξ + λ) j=1

= |

μk (ξ + λ)|2

∞  2  

 δR−j D(X(k+j)) ((RT )−k (ξ + λ)) j=1

N  ∞  2  

 ≥ |

μk (ξ + λ)| δR−j D(i) ((RT )−k (ξ + λ)) 2

i=1 j=1

= |

μk (ξ + λ)|2

N  2  T −k μ (ξ + λ)) . R,D(i) ((R ) i=1

(3.7)

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If λ ∈ Λw k , we have (RT )−k λ ∈ wk−1 (RT )−1 C + · · · + w1 (RT )−(k−1) C + w0 (RT )−k C ⊆ T (RT , C ∪ (−C)). Since RT is an expanding integer matrix, we can choose k large enough such that |(RT )−k (ξ)| < δ/2 holds for all ξ ∈ B(0, 1). Whence, we obtain, from (3.5) and (3.7), that f (λ) ≥ fk (λ)εN for each λ ∈ Λw k . Now, (3.6) yields that 

ε−N f (λ) ≤ ε−N < ∞.

λ∈Λw (R,C)

Applying Lebesgue’s dominated convergence theorem to {fk }∞ k=1 in (3.6), we have that 

2 |μ R,X (ξ + λ)| = 1 (ξ ∈ B(0, 1)).

λ∈Λw (R,C)

So Proposition 2.1 implies that (μR,X , Λw (R, C)) is a spectral pair. This ends the proof. 2 As a consequence of Theorem 3.3 we have the following result. Theorem 3.4. Let R ∈ Mn (Z) be an n × n expanding integer matrix, and let D, C be two finite subsets of Zn such that 0 ∈ C and (R−1 D, C) forms a compatible pair. Suppose that Z(δ R−1 D ) ∩ T (RT , C ∪ (−C)) = ∅. Then μR,D is a spectral measure with the spectra Λw (R, C) as in (3.3) for all w ∈ {−1, 1}N0 . For K ∈ N and any finite words w = w0 w1 · · · wK−1 ∈ {−1, 1}K , we define w CK = w0 C + w1 RT C + · · · + wK−1 (RT )K−1 C.

(3.8)

Then by (3.4) we have

w Λ(RK , CK )=

⎧ m ⎨ ⎩

w (RT )jK cj : cj ∈ CK for 0 ≤ j ≤ m, m ∈ N0

j=0

⎫ ⎬ ⎭

.

(3.9)

w It follows from Lemma 3.2 that the set Λ(RK , CK ) forms an orthogonal set for the measure μR,X as in (3.1), which is equivalent to say that



2 n |μ R,X (ξ + γ)| ≤ 1(ξ ∈ R ).

w) γ∈Λ(RK ,CK

The following results will be used to show the Theorem 4.2 in this paper. Theorem 3.5. Let R ∈ Mn (Z) be an n × n expanding matrix, and let D(1), D(2), · · · , D(N ) be a finite subset of Zn with the same cardinality. Let 0 ∈ C and C ⊆ Zn be a finite set such that (R−1 D(i), C) forms T K w a compatible pair and Z(μ R,D(i) ) ∩ T ((R ) , CK ) = ∅ for each i = 1, 2, · · · , N . Then for all X : N → w {1, 2, · · · , N } the measure μR,X is a spectral measure with a spectrum Λ(RK , CK ) given in (3.9).

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Proof. The proof is straightforward in the spirit of that in Theorem 3.3, but we write it down for the k = K−1 Rj D(X(kK − j)) for all k ∈ N. Then, from Proposition 2.4 (iv), sake of completeness. Let D j=0 k , C w ) forms a compatible pair for each k ≥ 1. Furthermore, the measure μR,X can be expressed as (R−K D K

μR,X = δR−K D 1 ∗ δR−2K D 2 ∗ · · · ∗ δR−kK D k ∗

∞ 

δR−(kK+j) D(X(kK+j)) .

(3.10)

j=1

For any k ∈ N, denote w w w νk := δR−K D 1 ∗ δR−2K D 2 ∗ · · · ∗ δR−kK D k and Γk := CK + (RT )K CK + · · · + (RT )(k−1)K CK .

Using Proposition 2.4 again, we get that Γk is a spectrum for the measure νk . That is, 

|ν k (ξ + γ)|2 = 1 (ξ ∈ Rn ).

γ∈Γk

On the other hand, it follows from (3.10) that 2 |μ νk (ξ + γ)|2 R,X (ξ + γ)| = |

∞  2  

 δR−(kK+j) D(X(kK+j)) (ξ + γ) j=1

= |

νk (ξ + γ)|2

∞  2  

 δR−j D(X(kK+j)) ((RT )−kK (ξ + γ)) j=1

≥ |

νk (ξ + γ)|2

N  2  T −kK μ (ξ + γ)) . R,D(i) ((R ) i=1

In order to use Lebesgue’s dominated convergence theorem, it is sufficient to show that there is a positive lower bound η > 0 such that N   2 T −kK μ (ξ + γ)) > η for γ ∈ Γk and ξ ∈ B(0, 1)) R,D(i) ((R )

(3.11)

i=1

holds for k large enough. T K w Indeed, since Z(μ R,D(i) ) ∩ T ((R ) , CK ) = ∅, there exist positive numbers ε, δ such that d(Z(μ R,D(i) ), T K w T ((R ) , CK )) > δ and 2 |μ R,D(i) (t)| > ε

(3.12)

w holds for i = 1, 2, · · · , N and t ∈ {x ∈ Rn : d(x, T ((RT )K , CK )) ≤ δ/2}. Note that for any γ ∈ Γk , we have w w w (RT )−kK γ ∈ (RT )−kK CK + (RT )−(k−1)K CK + · · · + (RT )−K CK w ⊆ T ((RT )K , CK ).

(3.13)

Whence, one can choose K0 large enough such that k ≥ K0 implies that |(RT )−kK ξ| < δ/2 holds for all ξ ∈ B(0, 1). Hence, by taking η = εN , we obtain, from (3.12) and (3.13), that the desired result (3.11) w follows. By Proposition 2.1, Λ(RK , CK ) is a spectrum for μR,X . The proof is complete. 2 As a consequence of Theorem 3.5 we have the following result.

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Theorem 3.6. Let R ∈ Mn (Z) be an n × n expanding integer matrix, and let D, C be two finite subsets of T K w Zn such that 0 ∈ C and (R−1 D, C) forms a compatible pair. Suppose Z(μ R,D ) ∩ T ((R ) , CK ) = ∅. Then K w the measure μR,D is a spectral measure with a spectrum Λ(R , CK ) as in (3.9). As a special case of Theorem 3.6, we have the classical result of Strichartz [37]. Corollary 3.7. Let R ∈ Mn (Z) be an n × n expanding integer matrix, and let D, C be two finite subsets T of Zn such that 0 ∈ C and (R−1 D, C) forms a compatible pair. Suppose Z(μ R,D ) ∩ T (R , C) = ∅. Then (μR,D , Λ(R, C)) is a spectral pair. We end this section by giving some simple examples to illustrate Theorem 3.3 and Theorem 3.4. We would like to point out that Theorem B cannot cover Example 3.9 and 3.10. Example 3.8 provides new spectra for Cantor measures with consecutive digits (see [7,6] for other spectra classified by maximal mapping). Example 3.11 is a natural generalization of [33, Example 4.3]. Example 3.8. Let R = qr be an integer with q, r ≥ 2 and let D = {0, 1, · · · , q − 1}. Then the measure μR,D is a spectral measure with spectra

Λw (R, C) =

⎧ m ⎨ ⎩

wj Rj cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0

j=0

⎫ ⎬ ⎭

,

for any infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 , where C = rD. Proof. It is easy to check that the matrix in (2.3)     jk jk 1 1 HR−1 D,C := √ e2πi R = √ e2πi q q q j∈D,k∈C j,k∈D is a unitary matrix. Then (R−1 D, C) forms a compatible pair. On the other hand, δ R−1 D (ξ) = 0

⇔

ξ ∈ r(qZ + {1, 2, · · · , q − 1}),

and T (R, C ∪ (−C)) =

⎧ ∞ ⎨ ⎩

R

j=1

−j

⎫ ⎬

 R−r R−r , ⊆ [−1, 1]. cj : cj ∈ C ⊆ − ⎭ R−1 R−1 

Then Z(δ R−1 D ) ∩ T (R, C ∪ (−C)) = ∅. The result follows by Theorem 3.4.

2

Example 3.9. Let R = 8, D(1) = {0, 1, 2, 3} and D(2) = {0, 1, 2, −1}, and let μR,X be a measure for any X : N → {1, 2} as in (3.1). Then μR,X is a spectral measure with spectra

Λw (R, C) =

⎧ m ⎨ ⎩

wj 8j cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0

j=0

for all w = w0 w1 w2 · · · ∈ {−1, 1}N0 , where C = {0, 2, 4, 6}.

⎫ ⎬ ⎭

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Proof. Note that the matrix in (2.3) ⎡

H8−1 D(1),C = H8−1 D(2),C

⎤ 1 1 −1 −i ⎥ ⎥ ⎥ 1 −1 ⎦ −1 i

1 1 1 ⎢ ⎢1 i =√ ⎢ 4 ⎣ 1 −1 1 −i

is unitary, where i2 = −1. Then (8−1 D(j), C) forms a compatible pair for j = 1, 2. Next, we show that Z(δ 8−1 D(j) ) ∩ T (8, C ∪ (−C)) = ∅,

j = 1, 2.

In fact, from (2.7) and (2.9),

T (8, C ∪ (−C)) =

⎧ ∞ ⎨ ⎩

j=1

8

−j

⎫ ⎬

  6 6 , cj : cj ∈ C ∪ (−C) ⊆ − , ⎭ 7 7

and Z(δ 8−1 D(j) ) = 8Z + {2, 4, 6}. The assertion follows by Theorem 3.3.

2

Example 3.10. Let R = 3k with k ∈ N and k ≥ 2. Let Di = {0, ai , bi }, 1 ≤ i ≤ N , be a finite collection of subsets of Z such that gcd(Di ) = 1 and {ai , bi } ≡ {1, 2}(mod 3) for i = 1, 2, · · · , N . Then for each X : N → {1, 2, · · · , N } the measure μR,X as in (3.1) is a spectral measure with spectra

Λw (R, C) =

⎧ m ⎨ ⎩

wj Rj cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0

j=0

⎫ ⎬ ⎭

for all infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 and C = {0, k, 2k}. Proof. According to the definition of compatible pair, it is easy to check that (R−1 Di , C) is a compatible pair for each 1 ≤ i ≤ N . By Theorem 3.3, it suffices to show that Z(δ (3k)−1 D(i) ) ∩ T (3k, C ∪ (−C)) = ∅,

∀ i = 1, 2, · · · , N.

This is really true since Z(δ (3k)−1 D(i) ) = k(±1 + 3Z),

∀ i = 1, 2, · · · , N,

and

T (3k, C ∪ (−C)) =

⎧ ∞ ⎨ ⎩

j=1

−j

(3k)

⎫ ⎬

  2k 2k , ⊆ (−1, 1). cj : cj ∈ C ⊆ − ⎭ 3k − 1 3k − 1

2

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Example 3.11. Let R =  D(1) =

2k1 0

13

! 0 , where k1 , k2 ∈ Z with |k1 | > 1 and |k2 | > 1. Set 2k2

! !" ! ! 0 k1 0 k1 , , , 0 k2 0 k2

 and

D(2) =

! !" ! ! 0 k1 0 k1 , . , , 0 −k2 0 −k2

Then the measure μR,X as in (3.1) is a spectral measure for any X : N → {1, 2} with spectrum

Λw (R, C) =

⎧ m ⎨ ⎩

wj Rj cj : wj ∈ {−1, 1}, cj ∈ C, m ∈ N0

j=0

⎫ ⎬ ⎭

,

where w = w0 w1 w2 · · · ∈ {−1, 1}N0 and  C=

! ! ! !" 0 1 0 1 , , , . 0 0 1 1

Proof. It is easy to show that (R−1 D(i), C) forms a compatible pair for each i = 1, 2. According to (2.7) and (2.9), we have  T (RT , C ∪ (−C)) = T (R, C ∪ (−C)) ⊆ −

   1 1 1 1 , × − , 2k1 − 1 2k1 − 1 2k2 − 1 2k2 − 1

⊆ (−1, 1)2 ,

(3.14)

and Z(δ R−1 D(i) ) = RZ(δ D(i) ) ⊆ R2 \ (−1, 1)2 .

(3.15)

Therefore, from (3.14) and (3.15), we get that T (RT , C ∪ (−C)) ∩ Z(δ R−1 D(i) ) = ∅. Hence, the result follows by Theorem 3.3. 2 4. Spectra of Bernoulli convolutions In this section, we study the spectra of the Bernoulli convolutions μR (R > 1). It is well known that the Fourier transform plays an important role in the study of spectral theory. From (1.1) and (2.9), for each R > 1, the Fourier transform of μR is μ R (ξ) =

∞ 

# cos

j=1

2πξ Rj

$ (ξ ∈ R),

(4.1)

and the zero set Z(μ  R ) of the Fourier transform μ R is Z(μ R) =

∞  j=1

Rj

2Z + 1 . 4

(4.2)

Based on Theorem A and the remark following Theorem 1.1, we consider the spectral eigenvalues and eigen-spectra of the Bernoulli convolutions μ2k for each integer k > 1 in this section. We shall introduce the precise definition of spectral eigenvalues and eigen-spectra of a probability measure.

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Definition 4.1. 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 Λ such that both Λ and pΛ are spectra for μ. The set Λ is called an eigen-spectrum of μ corresponding to the eigenvalue p. The following is our main result. Theorem 4.2. Let p be a real number and let k > 1 be a positive integer. Then p is an eigenvalue of μ2k if and only if p = pp12 , where p1 , p2 ∈ 2Z + 1. We will later see that the eigen-spectrum corresponding to an odd integer p is not unique and that is chosen from the sets ⎫ ⎧ m ⎬ ⎨ (4.3) Λw (2k, C) = wj (2k)j cj : wj ∈ {−1, 1} , cj ∈ C, m ∈ N0 , ⎭ ⎩ j=0

where w = w0 w1 w2 · · · ∈ {−1, 1}N0 and C = {0, k2 }. Note that C = {0, k2 } is not an subset of Z when k is odd. We claim that the theory in Section 3 is suitable for this non-integral case with respect to μ2k . In fact, let D = {−1, 1}, C  = {0, k} and D = {0, 1}. Then (1) ((2k)−1 D, C) is a compatible pair if and only if ((2k)−1 D , C  ) is a compatible pair; (2) Z(δ (2k)−1 D ) ∩ T (2k, C ∪ (−C)) is an empty set if and only if Z(δ (2k)−1 D ) ∩ T (2k, C  ∪ (−C  )) is an empty set; and (3) (μ2k , Λw (2k, C)) is a spectral pair if and only if (μ2k,D , Λw (2k, C  )) is a spectral pair. These guarantee the claim. The assertions (1) and (2) are easy to check. Now we show the assertion (3). From (2.8) and the definition of μ2k , we obtain that |% μ2k (ξ)| = |μ 2k,D  (2ξ)|,

∀ξ ∈ R.

It follows that  λ∈Λw (2k,C)

|% μ2k (ξ + λ)|2 =

 λ∈Λw (2k,C)

2 |μ 2k,D  (2ξ + 2λ)| =



2 |μ 2k,D  (2ξ + λ)| ,

∀ξ ∈ R.

λ∈Λw (2k,C  )

Thus the assertion (3) follows by Proposition 2.1. The following Theorem 4.3 plays a crucial role in the proof of Theorem 4.2. Theorem 4.3. Let p be an integer number. Then p is an eigenvalue of μ2k if and only if p is an odd integer. Moreover, if p ∈ 2Z + 1, there exists an infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 such that both Λw (2k, C) and Λw (2k, pC) are spectra for μ2k . Assumptions in the following of this section: we assume that 1 < k ∈ N, D = {−1, 1}, C = {0, k2 } and ±C := C ∪ (−C). To prove Theorem 4.3, we first obtain, from the proof of Lemma 3.2, that all the sets Λw (2k, C) in (4.3) are orthogonal sets for the Bernoulli convolution μ2k . The following Proposition 4.4 provides a characterization on integers p which determine whether the scaling sets pΛw (2k, C) = Λw (2k, pC) form orthogonal sets for μ2k . Proposition 4.4. Let p be an integer and let w = w0 w1 w2 · · · ∈ {−1, 1}N0 be an infinite word. Then the following two statements are equivalent: (i) Λw (2k, pC) forms an infinite orthogonal set for μ2k ; (ii) p = (2k)j p for some j ≥ 0 and p ∈ 2Z + 1.

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Proof. According to (4.2) and the definition of Λw (2k, C) we have Z(% μ2k ) =

∞ 

k k (2k)j (2Z + 1) and Λw (2k, C) ⊆ Z. 2 2 j=0

(4.4)

(i) ⇒ (ii). By (2.2), Λw (2k, pC) is an infinite orthogonal set for μ2k is equivalent to that Λw (2k, pC) − Λw (2k, pC) ⊆ Z(% μ2k ) ∪ {0}. Then Λw (2k, pC) = pΛw (2k, C) ⊆ (% μ2k ) ∪ {0}. Since p is an integer, we obtain, from (4.4), that p∈

∞ 

(4.5) k 2

∈ Λw (2k, C) and

(2k)j (2Z + 1).

j=0

Then (ii) follows. (ii) ⇒ (i). According to Lemma 3.2 we have Λw (2k, C) − Λw (2k, C) ⊆ Z(% μ2k ) ∪ {0}. For w ∈ {−1, 1}N0 , j N0 we write 1 w be the word in {−1, 1} with the first j words being 1 and following by w. Then we have ' & Λw (2k, pC) − Λw (2k, pC) = p (2k)j Λw (2k, C) − (2k)j Λw (2k, C)) ⊆ p (Λ1j w (2k, C) − Λ1j w (2k, C)) (by Lemma 3.2) μ2k ) ∪ {0}) ⊆ p (Z(%

(by (4.4))

⊆ Z(% μ2k ) ∪ {0}. Then (i) follows by (4.5). 2 Proof of the necessary part of Theorem 4.3. Suppose that p is an eigenvalue of μ2k , then there exists a spectrum Λ for μ2k with 0 ∈ Λ such that pΛ is a spectrum for μ2k too. By the orthogonality of Λ, pΛ, we have Λ ⊆ Λ − Λ ⊆ {0} ∪ Z(% μ2k ),

(4.6)

and pΛ ⊆ {0} ∪ Z(% μ2k ). Denote Λ1 = k2 (2Z + 1). We claim that Λ ∩ Λ1 = ∅. Otherwise, it is easy to get, from (4.4) and (4.6), that Λ − Λ ⊆ {0} ∪

∞ 

k (2k)j (2Z + 1), 2 j=1

which implies that Λ is an orthogonal set for the measure ν = δ(2k)−2 D ∗ δ(2k)−3 D ∗ · · · , where μ2k = δ(2k)−1 D ∗ ν. This yields a contradiction by Lemma 2.2. Similarly, we obtain that pΛ ∩ Λ1 = ∅. Moreover, if p ∈ 2Z, then, from (4.4), there exists a j ≥ 0 and a ∈ 2Z + 1 such that (2k)j k2 a ∈ Λ and p(2k)j k2 a ∈ pΛ ∩ Λ1 , thus k k p(2k)j a ∈ (2Z + 1) ⇔ p(2k)j a ∈ 2Z + 1, 2 2 which is impossible. This forces p to be an odd integer. The proof is complete. 2

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To show the sufficient part of Theorem 4.3, we only need to show that for any odd number p, there exists a w ∈ {−1, 1}N0 such that both Λw (2k, C) and Λw (2k, pC) are spectra for μ2k . We complete it by the following proposition and two subsections. This is the most subtle part of this paper. Proposition 4.5. Let k be an integer larger than 1. Then the Bernoulli convolution μ2k is a spectral measure with spectra Λw (2k, C) for each w ∈ {−1, 1}N0 . Proof. According to Lemma 3.2 (or Proposition 4.4), for any infinite word w ∈ {−1, 1}N0 the set Λw (2k, C) is an orthogonal set for μ2k . By Theorem 3.4, it is enough to check that Z(δ (2k)−1 D ) ∩ T (2k, ±C) = ∅. It is true since Z(δ (2k)−1 D ) = k2 (2Z + 1) and

T (2k, ±C) =

⎧ ∞ ⎨ ⎩

(2k)−j cj : cj ∈

j=1

(

⎫ )   k 1 k 1 k ⎬ ⊆ − , , 0, ± 2 ⎭ 2 2k − 1 2 2k − 1

where the assumption that k > 1 is needed. 2 Now we use the following two subsections to show that Λw (2k, pC) is a spectrum for μ2k for some w ∈ {−1, 1}N0 . 4.1. The case Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅ where p ∈ 2Z + 1 * + Theorem 4.6. Let C = 0, k2 and p ∈ 2Z + 1. If Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅, then for any w = w0 w1 w2 · · · ∈ {−1, 1}N0 the set Λw (2k, pC) forms a spectrum for μ2k . Proof. According to Theorem 3.4, it suffices to show that ((2k)−1 D, pC) is a compatible pair. In fact, the matrices defined in (2.3)

H(2k)−1 D,pC =

1 −i 1 i

!

! (if p ∈ 4Z + 1) and H(2k)−1 D,pC =

1 i 1 −i

(if p ∈ 4Z + 3)

are unitary. The proof is complete. 2 As a consequence of Theorem 4.6, we get the following corollary. Corollary 4.7. Let k ∈ N with k > 1 and p ∈ 2N0 +1. If p < 2k −1, then for any w = w0 w1 w2 · · · ∈ {−1, 1}N0 the set Λw (2k, pC) forms a spectrum for μ2k . Proof. According to (2.9) and (2.7), we have ξ 1 , −2πi ξ 2k + e2πi 2k δ (2k)−1 D (ξ) = =0 e 2 and

⇔

ξ∈

k (2Z + 1), 2

(4.7)

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T (2k, ±pC) = p

⎧ ∞ ⎨ ⎩

(2k)−j cj : cj ∈

j=1

(

  k p k p , . ⊆ − 2 2k − 1 2 2k − 1

17

⎫ )⎬

⎧ ⎫ ∞ ⎨ ⎬  k k =p 0, ± (2k)−j cj : cj ∈ {0, ±1} ⎭ 2 ⎭ 2 ⎩ j=1 (4.8)

Thus, from (4.7) and (4.8), we have Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅ if p < 2k − 1. Then Theorem 4.6 implies the desired result. The proof is complete. 2 4.2. The case Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅ where p ∈ 2Z + 1 * + Theorem 4.8. Let C = 0, k2 and p ∈ 2Z + 1. If Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅, then there exists an infinite word w = w0 w1 w2 · · · ∈ {−1, 1}N0 such that the set Λw (2k, pC) forms a spectrum for μ2k . Before giving the proof of Theorem 4.8, it is necessary for us to analyze the finer structure of the compact set T (2k, ±pC). Observe that, for any x ∈ T (2k, ±pC), by (2.7) we have x=

∞  (2k)−j aj ,

aj ∈ ±pC.

(4.9)

j=1

Definition 4.9. With the above expansion of x ∈ T (2k, ±pC), we give the following definition: (i) We say that the expansion (in base 2k) of x is unique if there exists an unique infinite word a1 a2 · · · ∈ N (±pC) such that (4.9) holds; (ii) We say that the expansion of x is finite if the infinite word a1 a2 · · · ends with 0∞ ; (iii) We say that the expansion of x is ultimately periodic (resp., periodic) if the infinite word a1 a2 · · · is ultimately periodic (resp., periodic), that is, there exist constants m,  ∈ N such that aj+ = aj for j ≥ m (resp., aj+ = aj for all j ∈ N). The following three lemmas will be needed in the proof of Theorem 4.8. * + Lemma 4.10. Let C = 0, k2 with k > 1 and p ∈ 2Z +1. Then each element x in the compact set T (2k, ±pC) has a unique expansion in base 2k. Proof. Assume that there are two distinct infinite words c1 c2 · · · and c1 c2 · · · in {−1, 0, 1}N such that x=p

∞ ∞ k  cj k  cj = p , 2 j=1 (2k)j 2 j=1 (2k)j

cj , cj ∈ {−1, 0, 1}.

Let t ≥ 1 be the smallest integer such that ct = ct . Then ct − ct =

 cj − cj . (2k)j−t j=t+1

(4.10)

Since the value of the term on the left hand side in (4.10) must be chosen from {±1, ±2}, while the absolute 2 value of the expression on the right hand side in (4.10) is controlled by 2k−1 (< 1), which is impossible. This ends the proof. 2 * + Lemma 4.11. Let C = 0, k2 with k > 1 and p ∈ 2Z + 1. Then the expansion of each rational number x ∈ T (2k, ±pC) is ultimately periodic.

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Proof. Since each number x ∈ T (2k, ±pC) can be represented as x=p

∞ k cj (2k)−j , 2 j=1

cj ∈ {−1, 0, 1}.

Then for any m ∈ N, we have ∞ k k k (2k)m x − p c1 (2k)m−1 − · · · − p cm = p cm+j (2k)−j . 2 2 2 j=1

(4.11)

Notice that the series on the right hand of the equality (4.11) is contained in the compact set T (2k, ±pC). Therefore, if x is a rational number, there exist m,  ∈ N such that ∞ 

cm+j (2k)−j =

j=1

∞ 

cm++j (2k)−j .

j=1

Lemma 4.10 implies that cm+j = cm++j for all j ∈ N, so the expansion of rational number x is ultimately periodic. This ends the proof. 2 + * Lemma 4.12. Let C = 0, k2 with k > 1 and p ∈ 2Z + 1. If x ∈ T (2k, ±pC) ∩ Z(% μ2k ), then the expansion of x cannot be finite. Proof. Suppose x ∈ T (2k, ±pC) ∩ Z(% μ2k ). Then, from (4.2), there is a j ∈ N0 and p ∈ 2Z + 1 such that jk  x = (2k) 2 p . On the contrary there exist c1 , c2 , · · · , cn−1 ∈ {−1, 0, 1} and cn ∈ {−1, 1} for some n ∈ N such that n k k  x = (2k) p = p (2k)−j cj . 2 2 j=1 j

Then n j 

(2k) p = p

n−j cj j=1 (2k) n (2k)

.

Since cn ∈ {−1, 1}, then the numerator on the right hand is odd and thus p is even, which contradicts to the assumption. This ends the proof. 2 Now, let us return to the proof of Theorem 4.8. Proof of Theorem 4.8. First, from Proposition 2.7, we know that the following two statements are equivalent: (i) Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅ and μ2k ) ∩ T (2k, ±pC) = ∅. (ii) Z(% Fixed 1 < k ∈ N and an odd integer p. We have obtained that ((2k)−1 D, pC) forms a compatible pair in the proof of Theorem 4.6. The idea of the proof left is to find a positive integer K and a finite word w = w0 w1 · · · wK−1 ∈ {−1, 1}K such that the following set w CK = w0 pC + w1 2kpC + · · · + wK−1 (2k)K−1 pC

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w w satisfies the condition Z(% μ2k ) ∩ T ((2k)K , CK ) = ∅, where T ((2k)K , CK ) is the attractor of the IFS {τc (x) = −K w and is a subset of T (2k, ±pC). Therefore, by Proposition 4.4, the set (2k) (x + c)}c∈CK

w Λ((2k)K , CK )=

⎧ m ⎨ ⎩

w (2k)jK cj : cj ∈ CK for 0 ≤ j ≤ m, m ∈ N0

j=0

⎫ ⎬ ⎭

is an orthogonal set for μ2k , and hence forms a spectrum for μ2k by Theorem 3.6. As the zero set Z(% μ2k ) is uniformly discrete, then its intersection with the compact set T (2k, ±pC) contains at most finitely many points, say that x1 , x2 , · · · , xm . So by Lemma 4.10, each number xi has a unique expansion in base 2k, xi = p

∞ k ci,j (2k)−j , 2 j=1

ci,j ∈ {−1, 0, 1},

i = 1, 2, · · · , m.

(4.12)

Clearly, one can neglect the action of p k2 in the above expansion in base 2k of xi . Since all xi are rational, by Lemma 4.11 and 4.12 each expansion of xi are ultimately periodic and not finite. Without loss of generality, we divide the xi s, i = 1, 2, · · · , m, into the following three classes and assume that each class is non-empty: (a) for 1 ≤ i ≤ s, there exist ij , ij  ∈ N such that ci, (b) for s + 1 ≤ i ≤ t, all terms ci,

j

ij

= 1 and ci,

ij 

= −1 in the expansion of xi .

≥ 0 in the expansion of xi .

(c) for t + 1 ≤ i ≤ m, all terms ci, j ≤ 0 in the expansion of xi . Choose ki = max{ij , ij } for 1 ≤ i ≤ s in the case (a) and choose ki such that ci, ki = 1 for s + 1 ≤ i ≤ t in the case (b). Set M := max{k1 , k2 , · · · , kt }. According to (c), for each i (t + 1 ≤ i ≤ m), there exist infinitely many terms ci, j = −1 for j > M . Hence, we can choose ki > M such that ci, ki = −1 for t + 1 ≤ i ≤ m. Denote K = max{kt+1 , kt+2 , · · · , km } > M . Now we set w = 11  · · · 1 (−1)(−1) · · · (−1) and define    K−M times

M times

. / . / w CK := pC + · · · + (2k)K−(M +1) pC − (2k)K−M pC + · · · + (2k)K−1 pC .

(4.13)

w We claim that xi ∈ / T ((2k)K , CK ) for i = 1, 2, · · · , m. w ) ⊆ T (2k, ±pC), then, by Theorem 2.6, the xi has the Indeed, if there exists a point xi ∈ T ((2k)K , CK following expansion in base 2k

xi =

∞  (2k)−jK cij ,

w where cij ∈ CK

j=1

=p

k, −(2k)−1 ci,K−1 − · · · − (2k)−M ci,K−M + (2k)−(M +1) ci,K−(M +1) + · · · + (2k)−K ci,0 2 ∞  (2k)−jK cij , (4.14) + j=2

where ci1 = p

k, ci,0 + · · · + (2k)K−(M +1) ci,K−(M +1) − (2k)K−M ci,K−M − · · · − (2k)K−1 ci,K−1 2

with ci, ∈ {0, 1} for 0 ≤  ≤ K − 1.

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By comparing the leading K terms in the expansion of xi as in (4.12), which satisfies that ci, ki = 1 if 1 ≤ i ≤ t and ci, ki = −1 if t + 1 ≤ i ≤ m, with that of xi as in (4.14), we get that the above xi has two different expansions. It is a contradiction to Lemma 4.10. Thus, the claim follows. w Now we obtain the desired integer K and word w such that Z(

μ2k ) ∩ T ((2k)K , CK ) = ∅. This finishes the proof of Theorem 4.8. 2 Whence, we can complete the proof of Theorem 4.3. Proof of the sufficient part of Theorem 4.3. By Proposition 4.5, we know, for any w = w0 w1 w2 · · · ∈ {−1, 1}N0 , that the set Λw (2k, C) forms a spectrum for μ2k . Together with Theorem 4.6 and 4.8, Λw (2k, pC) is also a spectrum for μ2k for some w ∈ {−1, 1}N0 depended on integer p. 2 Let us say some words about the details and method used in the proof of Theorem 4.8. μ2k ) and the • The classes (a), (b) and (c) may not appear at the same time. Note that the zero set Z(% compact set T (2k, ±pC) are both symmetric to the origin point. Then the classes (b) and (c) must appear or disappear at the same time, and thus s ∈ 2N, m ∈ 2N. In the case when (a) disappear and (b) appear, then the proof in Theorem 4.8 can be repeated step by step. In the case when (a) appear and (b) disappear, then s = m and the intersection relation Z(% μ2k ) ∩ T (2k, pC) = ∅ holds. Hence, by Corollary 3.7, the set Λ(2k, pC) forms a spectrum for μ2k . Moreover, if w we replace CK in (4.13) by the following general set . / . / K−(M +1) w C − (2k)K−M C + · · · + (2k)K−1 C , C0 K := w0 C + · · · + wK−(M +1) (2k) where K > M and w0 w1 · · · wK−(M +1) ∈ {−1, 1}K−M is any finite word, then one can conclude, by w taking the similar arguments in the proof of Theorem 4.8, that Z(

μ2k ) ∩ T ((2k)K , C0 K ) = ∅. And hence, by Proposition 4.4 and Theorem 3.6, ⎫ ⎧ m ⎬ ⎨ w w Λ((2k)K , C0 (2k)Kj cj : cj ∈ C0 K) = K for 0 ≤ j ≤ m, m ∈ N0 ⎭ ⎩ j=0

forms a spectrum for μ2k . • The eigen-spectrum corresponding to an odd integer p is not unique. Moreover, the ‘period’ in the w eigen-spectrum Λ((2k)K , CK ) constructed in Theorem 4.8 can be enlarged as long as possible. More w precisely, for the set CK defined as in (4.13), and for any L ∈ N and u = u0 u1 · · · uL−1 ∈ {−1, 1}L , we define new sets uw w CL+K = u0 pC + u1 2kpC + · · · + uL−1 (2k)L−1 pC + (2k)L CK , uw uw . From the proof and T ((2k)K+L , CL+K ) is the attractor generated by {τc (x) = R−(K+L) (x + c)}c∈CL+K of Theorem 4.8, one can easily get that

uw T ((2k)K+L , CL+K ) ∩ Z(% μ2k ) = ∅.

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Hence, by Proposition 4.4 and Theorem 3.6, the following set

uw Λ((2k)L+K , CL+K )=

⎧ m ⎨ ⎩

uw (2k)(L+K)j cj : cj ∈ CL+K for 0 ≤ j ≤ m, m ∈ N0

j=0

⎫ ⎬ ⎭

also forms a spectrum for μ2k . The procedure done in the proof of Theorem 4.8 can be proceeded simultaneously for finitely many distinct odd integers. We illustrate it for two distinct integers p1 and p2 . Consider the following intersection A := Z(% μ2k ) ∩ (T (2k, ±p1 C) ∪ T (2k, ±p2 C)). Thus, by Lemma 4.10, each element x ∈ A has a unique expansion in base 2k in the form (4.12) for which p = p1 or p2 . Noting that the choice of p does not make a substantial contribution to the whole proof of Theorem 4.8. So by adjusting the proof slightly, we will get positive integers M, K, a finite word w and a w w set CK, μ2k ) ∩ T ((2k)K , CK, i like in (4.13) such that Z(

i ) = ∅ for i = 1, 2 where . / . / w K−(M +1) CK, pi C − (2k)K−M pi C + · · · + (2k)K−1 pi C . i = pi C + · · · + (2k) By Proposition 4.4 and Theorem 3.6 again, we obtain that

w Λ((2k)K , CK, i) =

⎧ m ⎨ ⎩

w (2k)Kj cj : cj ∈ CK, i for 0 ≤ j ≤ m, m ∈ N0

j=0

⎫ ⎬ ⎭

are both spectra for μ2k . Combining with Proposition 4.5, we have Theorem 4.13. Let k be an integer larger than 1. Then for two distinct odd integers p1, p2 , there exists a discrete set Λ such that all Λ, p1 Λ, p2 Λ are spectra for the Bernoulli convolution μ2k . Moreover, both p1 /p2 and p2 /p1 are spectral eigenvalues of μ2k . Proof. We only need to show the last assertion for p1 /p2 . Write p = p1 /p2 . Then p2 Λ is the eigen-spectrum corresponding to p because both p1 Λ and p2 Λ are spectra for μ2k . 2 Corollary 4.14. Let k be an integer larger than 1. Then for finite distinct odd integers p1 , p2 , · · · , pn , there exists a discrete set Λ such that all Λ, p1 Λ, · · · , pn Λ are spectra for the Bernoulli convolution μ2k . Moreover, all pi /pj are spectral eigenvalues of μ2k . Proof of Theorem 4.2. The sufficiency follows from Theorem 4.13. Now we show the necessity. Suppose that 0 ∈ Λ, pΛ are both spectra for the measure μ2k . Notice that Λ, pΛ ⊆ {0} ∪ Z(% μ2k ) = {0} ∪

∞ 

k (2k)j (2Z + 1). 2 j=0

(4.15)

So p is a rational number and is of the form pp12 , where p1 , p2 ∈ Z \ {0} and gcd(p1 , p2 ) = 1. Recall that Λ1 = k2 (2Z + 1). According to the proof of the necessary part of Theorem 4.3, we easily know that Λ1 ∩ Λ = ∅

and Λ1 ∩ pΛ = ∅.

(4.16)

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When p1 is even and p2 is odd, one can yield that Λ1 ∩ pΛ = ∅, which contradicts to (4.16). In fact, if Λ1 ∩ pΛ = ∅, by (4.15), there is a j ≥ 0 and a ∈ 2Z + 1 such that k k p(2k)j a ∈ (2Z + 1) 2 2

⇔

p1 (2k)j a ∈ 2Z + 1. p2

It is a contradiction to the assumptions of p1 , p2 and a. / Z(

μ2k ) if λ ∈ Λ1 ∩ Λ, which means that pΛ When p1 is odd and p2 is even, one can conclude that pλ ∈ is not a spectrum for the measure μ2k . In fact, if pλ ∈ Z(

μ2k ) for some λ ∈ Λ1 ∩ Λ, then there is a ∈ 2Z + 1 and j ≥ 0 such that k k p a ∈ (2k)j (2Z + 1) 2 2

⇔

p1 a ∈ (2k)j (2Z + 1). p2

It is also a contradiction to the assumptions of p1 , p2 and a. The desired result follows. 2 Remark 4.15. It follows from the properties of Fourier transform, we have |μ μ2k (−ξ)| = |% μ2k (ξ)| (ξ ∈ R). −2k (ξ)| = |% This means that the spectral property of the measure μ−2k is the same with μ2k . Hence, using Proposition 2.1, one can show that if Λ is a spectrum for μ2k , then Λ is a spectrum for μ−2k . Whence the above Theorem 4.2 is also suitable to the case that k < 0. The following example is devoted to display the proof of Theorem 4.2. Example 4.16. Let C = {0, 1}. Then the sets Λ(43 , Cp ) =

⎧ m ⎨ ⎩

43j cj : cj ∈ Cp for m ∈ N0

j=0

⎫ ⎬ ⎭

,

where Cp = −pC + 4pC + 42 pC,

(4.17)

is a spectrum for the measure μ4 for each p = 3, 5. Consequently, all 3, 5, 3/5, 5/3 are spectral eigenvalues of the measure μ4 . Proof. For p = 3 it is easy to show that Z( μ4 ) ∩ T (4, 3{0, ±1}) = {±1}, and ±1 = ±3

∞  1 . j 4 j=1

Similarly, for p = 5 we have that Z( μ4 ) ∩ T (4, {0, ±5})) = {±1},

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and ±1 = ±5

∞ 

1 1 4−2j ( − 2 ). 4 4 j=0

Like (4.13), we define Cp satisfying (4.17). Let T (43 , Cp ) be the attractor generated by the IFS {4−3 (x + c) : c ∈ Cp } for p = 3, 5. Then ±1 ∈ / T (43 , Cp ) by Proposition 4.10, in other words, we have that Z( μ4 ) ∩ 3 −1 T (4 , Cp ) = ∅. Since (4 {−1, 1}, pC) form compatible pairs for all p = 3, 5, then the first statement follows from Theorem 3.6 and the second follows from Theorem 4.13. 2 5. Concluding remarks In this section, we will give some further remarks on the results obtained in the previous sections and list some open questions about Bernoulli convolutions μ2k . 5.1. The well known result of Jorgensen and Pedersen [28] proved that for each k ∈ N with k > 1, the following set ⎫ ⎧ ( ) m ⎬ ⎨ k , (5.1) Λ(2k, C) = (2k)j cj : cj ∈ C for 0 ≤ j ≤ m, m ∈ N0 , where C = 0, ⎭ ⎩ 2 j=0

forms a spectrum for the Bernoulli convolution μ2k . In 2011, Jorgensen, Kornelsen and Shuman [27], Li [33, Section 3] found a condition on p and k which determines whether Λ(2k, pC) is a spectrum for μ2k . Their main results are the following. Theorem C. For k ∈ N with k > 1 and p ∈ 2N0 + 1, if p < 2k − 1, then Λ(2k, pC) forms a spectrum for μ2k . Theorem D. Let k ∈ N with k > 1 and p = 2k − 1. Then Λ(2k, pC) is not a spectrum for μ2k . We remark here that Corollary 4.7 is an effective complement to Theorem C, and the following theorem is an improvement to Theorem D. Theorem 5.1. Let k ∈ N with k > 1 and p = 2k − 1. Then the set ⎫ ⎧ m ⎬ ⎨ Λ((2k)2 , pC − 2kpC) := (2k)2j (cj,1 − 2kcj,2 ) : cj,1 , cj,2 ∈ pC, m ∈ N0 ⎭ ⎩ j=0

forms a spectrum for μ2k .  be the following set Proof. Let C  = pC − 2kpC = C

(

k k k 0, p , −2kp , −(2k − 1)p 2 2 2

) .

We have, from (2.7), that ⎧ ⎫  ∞ ⎨ ⎬  k k2 2  −2j  , T ((2k) , C) = (2k) cj : cj ∈ C ⊆ − ⎩ ⎭ 2k + 1 2(2k + 1) j=1

# ⊆

k k − , 2 2

$ .

(5.2)

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 = ∅. Hence, the desired result follows from TheoIt follows from (4.2) and (5.2) that Z(

μ2k ) ∩ T ((2k)2 , C) rem 3.6. 2 Furthermore, in Corollary 4.7, there are many odd integers p > 2k − 1 such that Λw (2k, pC) form spectra for the measure μ2k . We now talk about it in terms of μ4 . Example 5.2. Let C = {0, 1} and let w = w0 w1 w2 · · · ∈ {−1, 1}N0 be an infinite word. If p is chosen from the following positive integers {1, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97}, then all the sets

Λw (4, pC) = p

⎧ m ⎨ ⎩

wj 4j cj : wj ∈ {−1, 1} , cj ∈ C, m ∈ N0

j=0

⎫ ⎬ ⎭

are spectra for the measure μ4 . Proof. Recall that D = {−1, 1}. As before, the crucial point is to check the condition that Z(δ 4−1 D ) ∩ T (4, {−p, 0, p}) = ∅ in Theorem 4.6 for the above integers p ≤ 99. Here, we give a special algorithm from fractal geometry to check the condition. More precisely, let c = c1 c2 · · · cn ∈ {−p, 0, p}n be a finite word and let σc = σc1 ◦ σc2 ◦ · · · ◦ σcn be the compositions of σc (x) = 4−1 (x + c), c ∈ {−p, 0, p}. Then 

T (4, {−p, 0, p}) ⊆

c1 c2 ···cn ∈{−p,0,p}

⊆

 c1 c2 ···cn ∈{−1,0,1}

p p σc1 ◦ σc2 ◦ · · · ◦ σcn − , 3 3 n

- , -⎤ ⎡ , n n p −1 + 3 j=1 cj 4n−j p 1 + 3 j=1 cj 4n−j ⎣ ⎦. , 3 · 4n 3 · 4n n

Note that Z(δ 4−1 D ) = 2Z + 1. By calculations, we show that, for each of the above integers p ≤ 99, the desired condition that Z(δ 4−1 D ) ∩ T (4, {−p, 0, p}) = ∅ holds for each p. 2 Combining these with the results in [12] and relating to the spectral eigenvalue problem (1), it is of interest to consider the following question. Question 1. For what odd integers p ∈ 2Z + 1, the sets

Λw (4, pC) = p

⎧ m ⎨ ⎩

wj 4j cj : wj ∈ {−1, 1} , cj ∈ C, m ∈ N0

j=0

for all w = w0 w1 w2 · · · ∈ {−1, 1}N0 are spectra for the measure μ4 ?

⎫ ⎬ ⎭

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5.2. Fix 1 < k ∈ N, C = {0, k2 } and D = {−1, 1}. Assume Akp is the eigen-subspace of the Bernoulli convolution μ2k corresponding to an odd integer p. That is, Akp = {Λ ⊆ R : Λ and pΛ are spectra for μ2k }. When p satisfies Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅, Corollary 4.7 implies that Λw (2k, C) ⊆ Akp . Hence the cardinal number of the set Akp is equal to the cardinality of the continuum, which is denoted by ℵ. On the other hand, when Z(δ (2k)−1 D ) ∩ T (2k, ±pC) = ∅, the arguments in Theorem 4.8 show that the set Akp has at least infinite countable cardinal, which is denoted by ℵ0 (< ℵ). A natural question is the following: Question 2. Is the cardinal number of Akp equal to ℵ? 5.3. Theorem 4.3 shows that the spectra corresponds to Bernoulli convolution μ2k can have arbitrarily small upper Beurling density. Recall that for a discrete set Λ in R and r > 0, the upper r-Beurling density of Λ is defined by Dr+ (Λ) := lim sup h→∞ x∈R

#(Λ ∩ (x + [−h, h])) , (2h)r

and the upper Beurling dimension of Λ is defined by dim+ (Λ) = sup{r > 0 : Dr+ (Λ) > 0} = inf{r > 0 : Dr+ (Λ) < ∞}. Fix 1 < k ∈ N and C = {0, k2 }, we have proved in Proposition 4.5 that Λw (2k, C) is a spectrum for μ2k for + each w ∈ {−1, 1}N0 . By [11, Theorem 3.5 (a)], we have that Dlog (Λw (2k, C)) < ∞. 2k 2 k k On the other hand, by letting h = 2 (1 + 2k +· · ·+ (2k)n−1 )) =: a((2k)n − 1) < a(2k)n , where a = 2(2k−1) , n we can easily get that the interval [−h, h] contains at least 2 points in each Λw (2k, C). Thus + Dlog (Λw (2k, C)) ≥ lim 2k 2

h→∞

#(Λw (2k, C) ∩ [−h, h]) (2h)log2k 2

2n n→∞ [2a(2k)n ]log2k 2

≥ lim

= (2a)− log2k 2 > 0. Moreover, for each p ∈ 2Z + 1, + Dlog (Λw (2k, pC)) = 2k 2

1 |p|log2k 2

+ Dlog (Λw (2k, C)). 2k 2

Hence, from Theorem 4.3, there exists a spectrum Λw (2k, pC) for μ2k such that + lim Dlog (Λw (2k, pC)) = 0. 2k 2

|p|→∞

According to [7, Theorem 2.9] and [11, Theorem 3.5 (a)], we have 0 ≤ dim+ (Λ) ≤ log2k 2 for any spectrum Λ for μ2k . By taking advantage of spectra obtained in this paper we hope to show the following conjecture.

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Conjecture 5.3. The upper Beurling dimension of spectra for the Bernoulli convolutions μ2k take values over the closed interval [0, log2k 2]. That is, for any 0 ≤ s ≤ log2k 2, there is a spectrum Λ for μ2k such that dim+ (Λ) = s. 5.4. Let 1 < k ∈ N, C = {0, k2 } and D = {−1, 1}. For the spectral pair (μ2k , Λw (2k, C)), we define the sets Λw n as follows: n−1 Λw C, n := w0 C + w1 RC + · · · + wn−1 R

n ∈ N0 ,

and define the partial Fourier series, by summing over the sets Λw n, sn (f ; x) =



f (λ)e2πiλx ,

(5.3)

λ∈Λw n

where f (λ) =



f (y)e−2πiλy dμ2k (y),

∀λ ∈ Λw (2k, C).

A fundamental result about Fourier series, called Carleson’s theorem which is proved by Carleson in 1966, see [4], is that the Fourier series for L2 functions on the circle converges almost everywhere with respect to the normalized Lebesgue measure. The extension version of Carleson’s theorem to Lq for q > 1, called Carleson-Hunt theorem, was proved by Hunt in 1968, see [23]. So what happened to the Fourier series if we replace the Lebesgue measure by μ2k ? This is the following question. Question 3. Let f ∈ Lq (μ2k ) for q > 1. Does the Fourier series Sn (f, x) in (5.3) converge to f for μ2k -almost everywhere x ∈ T (2k, D)? It is noted that a special case related to Question 3 has been studied by Strichartz in [39], which proved that the Fourier series Sn (f, x) of continuous function f uniformly converge to itself with respect to Λw (2k, C) where w = 111 · · · . In fact, using the method in [39, Section 2], one can easily get that the above result of Strichartz also holds for all infinite word w ∈ {−1, 1}N0 . Also, for the scaling spectra Λw (2k, pC) for μ2k , one can similarly define the Fourier series Sn (f ; x) as in (5.3) and consider the above Question 3. In particular, due to Dutkay, Han and Sun [10, pp. 2197–2199] and Example 5.2, one can easily show that for the measure μ4 and any infinite word w ∈ {−1, 1}N0 , the corresponding Fourier series with respect to Λw (4, p{0, 1}) diverge at zero for some continuous function, where p = 23, 29. We will not include the proof of this fact and please refer to [10] for more detail. Acknowledgements The authors would like to thank the anonymous referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8]

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