Asymptotic Properties of Fourier Transforms of b-Decomposable Distributions

Abstract

Erdös-Kahane numbers (EK numbers) are introduced in relation to the decay of the Fourier transforms of non-symmetric Bernoulli convolutions. The PV, PS, and EK numbers are characterized by using a certain trigonometric series H _b (u). The relations between those numbers and the asymptotic properties of the Fourier transforms of full b-decomposable distributions are shown. A sufficient condition for the absolute continuity of one-dimensional b-decomposable distributions is given. As an application, an open problem on the uniform decay of the Fourier transforms of refinable distributions, raised by Dai et al. (J. Funct. Anal. 250(1):1–20, 2007), is solved. Finally, temporal evolution on continuity properties of distributions of some Lévy processes is discussed.

1 Introduction

In what follows, denote the Euclidean inner product of z and x and the norm of x in ℝ^d by 〈z,x〉 and |x|, respectively. Let ℤ:={0,±1,±2,…}, ℤ₊:={0,1,2,…}, ℕ:={1,2,3,…}, and ℝ₊:=[0,∞). The symbol δ _a (dx) stands for the delta measure at a in ℝ^d. Denote the convolution of probability distributions ρ and μ on ℝ^d by ρμ and the characteristic function (Fourier transform) of a probability distribution μ on ℝ^d by $\widehat{\mu}(z)$, namely,

Denote by η ^n∗ the nth convolution power of a finite measure η on ℝ^d with the understanding that η ^0∗(dx)=δ ₀(dx), and by $\bar{\eta}$ the reflection of η, that is, $\bar{\eta}(dx) = \eta(-dx)$. A probability distribution on ℝ^d is said to be full if its support is not contained in any hyperplane in ℝ^d. Let 0<b<1. A probability distribution μ on ℝ^d is said to be b-decomposable if there exists a probability distribution ρ on ℝ^d such that

$$ \widehat{\mu}(z) = \widehat{\mu}(bz)\widehat{\rho}(z).$$

(1.1)

The decomposition (1.1) is equivalent to

$$\widehat{\mu}(z) = \prod_{n=0}^{\infty}{\widehat{\rho}}\bigl(b^nz\bigr).$$

The infinite product above converges if and only if ρ has a finite log-moment, namely,

$$ \int_{\mathbb{R}^d} \log\bigl(2+|x|\bigr)\rho(dx) < \infty.$$

(1.2)

See Lemma 1 of Bunge [2]. We see from Lemma 5 of Watanabe [28] that a b-decomposable distribution μ satisfying (1.1) with ρ is full if and only if ρ is full. A distribution μ on ℝ^d is called self-decomposable if it is b-decomposable for every b∈(0,1). Self-decomposable distributions are infinitely divisible. Many statistically important distributions are known to be self-decomposable and all full self-decomposable distributions on ℝ^d are absolutely continuous. See Sato [25]. A distribution μ on ℝ^d is called a homogeneous self-similar measure with contraction ratio b if it is b-decomposable with some b∈(0,1) and the support of ρ is a finite set. The idea of b-decomposability was introduced by Loève [17] and then extended by Bunge [2] and Maejima et al. [18] in the theory of limit distributions for some sequence of normalized sums of independent random variables. Denote by L(b) the totality of b-decomposable distributions on ℝ^d. The class ⋃ _b∈(0,1) L(b) is a rich class which contains the semi-stable (including Gaussian) distributions and the semi-self-decomposable distributions. Wolfe [36] proved that every full distribution in L(b) is either singular or absolutely continuous. The author [28, 29] studied continuity properties of some distributions in L(b). Applications of the class L(b) to limit theorems for shift self-similar additive random sequences, laws of iterated logarithm for Brownian motions on nested fractals, and exact Hausdorff and packing measures of random fractals on Galton-Watson trees are found in a series of papers [31–34]. Another application to generalized Ornstein-Uhlenbeck processes is found in Lindner and Sato [15, 16].

Let b and p be real numbers in (0,1). We define a Bernoulli convolution ν _b,p on ℝ by

$$ \widehat{\nu_{b,p}}(z) = \prod_{n=0}^{\infty}\bigl(p+(1-p)\exp\bigl(ib^nz\bigr)\bigr).$$

(1.3)

Note that ν _b,p is b-decomposable with ρ(dx)=pδ ₀(dx)+(1−p)δ ₁(dx) in (1.1). Let ν _b :=ν _b,p with p=2⁻¹. The probability distribution ν _b is called a symmetric Bernoulli convolution. An algebraic integer θ>1 is said to be a Pisot-Vijayaraghavan number (PV number, for short) if all its Galois conjugates θ _j satisfy |θ _j |<1. Kershner and Wintner [13] proved that, for 0<b<2⁻¹, ν _b is a distribution on a Cantor set and hence it is singular. Wintner [35] noted that, for b=2⁻¹, ν _b is the uniform distribution on [0,2]. Jessen and Wintner [11] observed that, for 2⁻¹<b<1, ν _b is either singular or absolutely continuous. Erdös [6] showed that it is singular in case b ⁻¹ in (1,2) is a PV number. Garsia [9] gave concrete examples of b in (2⁻¹,1) such that ν _b is absolutely continuous. In 1995, Solomyak [26] proved that ν _b has an L ²-density for a.e. b∈(2⁻¹,1). Peres and Solomyak [21] showed that, for 1/3≤p≤2/3, ν _b,p has an L ²-density for a.e. b∈(p ²+(1−p)²,1). We say that b in (0,1) is a Peres-Solomyak number (PS number, for short) if there exist p in (0,2⁻¹) and a positive number q such that $\widehat{\nu_{b,p}}(z)$ belongs to L ^q(ℝ), that is, $\int_{-\infty}^{\infty}|\widehat{\nu_{b,p}}(z)|^{q}dz <\infty$. We say that a nonnegative function f(z) on ℝ^d has uniform decay at infinity if there exists γ>0 such that f(z)=O(|z|^−γ) as |z|→∞. Erdös [7] proved that $|\widehat{\nu_{b}}(z)|$ has uniform decay at infinity and ν _b is absolutely continuous for a.e. b∈(a,1) with a sufficiently close to 1. Kahane [12] pointed out that the Hausdorff dimension of the set of b in (a,1) such that ν _b is singular tends to 0 as a↑1. We say that b in (0,1) is an Erdös-Kahane number (EK number, for short) if there exists p in (0,2⁻¹) such that $|\widehat{\nu_{b,p}}(z)|$ has uniform decay at infinity. EK numbers are always PS numbers, but its converse remains open. Applications of PS numbers to Lévy processes are found in Watanabe [29, 30] and Lindner and Sato [15]. They showed that the distribution of a certain Lévy process is singular for small time and absolutely continuous for large time. Using EK numbers, we see in Theorem 5.1 in Sect. 5 that more drastic temporal evolution can occur in continuities of the distribution of such a Lévy process.

In the following, we characterize the PV, PS, and EK numbers by using the function H _b (u) defined by (1.4) below and discuss the relations between those numbers and the asymptotic properties of the characteristic functions of distributions in the class L(b). Let 0<b<1. Define functions H _b (u), I _b (t), and J _b (t) on ℝ₊ as

(1.4)

Define the subsets B _j in (0,1) for j=1,2,3 as

Our main results are as follows.

Theorem 1.1

The following are equivalent:

• (1) b∈B ₁.

• (2) b ⁻¹ is not a PV number.

• (3) For some p∈(0,2⁻¹), $\lim_{|z| \to\infty}\widehat{{\nu}_{b,p}}(z) = 0$.

• (4) For every full μ∈L(b) on ℝ^d, $\lim_{|z|\to \infty } \widehat{\mu}(z)= 0$.

Remark 1.1

• (i) In Theorem II on P. 40 of [24], Salem proved that, for b≠1/2, the condition (2) in Theorem 1.1 is equivalent to $\lim_{|z| \to\infty} \widehat{\nu}_{b}(z) = 0$. An operator version of a part of Theorem 1.1 is found in Theorem 2 of Watanabe [28].

• (ii) Theorem 2.3 of Hu [10] gave a necessary and sufficient condition in order that the characteristic function $\widehat{\mu}(z)$ of a one-dimensional homogeneous self-similar measure μ tends to 0 as |z|→∞. Our Theorem 1.1 says that if the contraction ratio b is not the reciplocal of a PV number, then the characteristic function $\widehat{\mu}(z)$ of every full homogeneous self-similar measure μ on ℝ^d tends to 0 as |z|→∞.

Theorem 1.2

The following are equivalent:

• (1) b∈B ₂.

• (2) b is a PS number.

• (3) For every full μ∈L(b) on ℝ^d, $\widehat {\mu}(z)\in L^{q}(\mathbb{R}^{d})$ for some q>0.

Remark 1.2

• (i) Let n∈ℤ₊. For a.e. $b \in ((5/9)^{2^{-n}},(5/9)^{2^{-n-1}})$, we have

  $$I_b\bigl(2^{-n}\log3\bigr) <\infty.$$

• (ii) Let c>0. For all b∈(0,e ^−2c), we have I _b (c)=∞.

Theorem 1.3

The following are equivalent:

• (1) b∈B ₃.

• (2) b is an EK number.

• (3) exp(−H _b (|z|)) has uniform decay at infinity.

• (4) For every full μ∈L(b) on ℝ^d, $|\widehat {\mu }(z)|$ has uniform decay at infinity.

Remark 1.3

• (i) Let n∈ℤ₊. For a.e. $b \in(2^{-2^{-n}},2^{-2^{-n-1}})$, we have

  

• (ii) Let c>0. For all b∈(0,e ^−c), we have J _b (c)=∞ and

  

Theorem 1.4

Let μ be a full b-decomposable distribution on ℝ satisfying (1.1). Let $q:=\rho*\bar{\rho}(\{0\})$. Then we have the following.

• (i) Let N be the smallest positive integer satisfying 2^−Nlog3+q<1. Then, for a.e. $b \in((5/9)^{2^{-N}}, 1)$, μ has an L ²-density. In particular, if ρ has no point mass, then μ has an L ²-density for a.e. $b \in({\sqrt{5}}/3, 1)$.

• (ii) Let n∈ℤ₊. We have, for a.e. $b \in(2^{-2^{-n}},2^{-2^{-n-1}})$,

  $$\bigl|\widehat{\mu}(z)\bigr|= O\bigl(|z|^{(-0.0013)(1-q)2^n}\bigr).$$

Remark 1.4

Theorem 1.4 holds for μ=ν _b,p with 1−q=2p(1−p). We apply Theorem 1.4 to a one-dimensional homogeneous self-similar measure in Proposition 4.1 in Sect. 4. Peres and Solomyak [21] discussed the absolute continuity of a one-dimensional homogeneous self-similar measure under the assumption of the transversality condition. Ngai and Wang [19] extended the results of [21] to the inhomogeneous case. But, we do not know whether the transversality condition holds on an interval (a,1) with a close to 1.

We do not yet know an explicit example of EK numbers or PS numbers. However, modifying the Erdös-Kahane argument for symmetric Bernoulli convolutions, we have the following.

Theorem 1.5

All real numbers in (0,1) outside a set of Hausdorff dimension 0 are EK numbers and hence PS numbers.

Corollary 1.1

If a full distribution μ on ℝ^d is b-decomposable for all b in a set of positive Hausdorff dimension, then $|\widehat {\mu}(z)|$ has uniform decay at infinity. In particular, for every full self-decomposable distribution μ on ℝ^d, $|\widehat {\mu}(z)|$ has uniform decay at infinity.

In Sect. 2, we prove our main results. In Sect. 3, we discuss the structures of the PS numbers and the EK numbers. In Sects. 4 and 5, we give applications of our main results to refinable distributions and to some Lévy processes, respectively.

2 Proofs of the Main Results

First we give several preliminary lemmas for the proofs of the main results mentioned in Sect. 1.

Lemma 2.1

Let 0<b<1. Then we have the following.

• (i) For p∈(0,1),

  $$\bigl|\widehat{{\nu}_{b,p}}(z)\bigr|^2 \leq\exp\bigl(-2p(1-p)H_b\bigl(|z|\bigr)\bigr).$$

• (ii) For p∈(0,2⁻¹)∪(2⁻¹,1),

  $$\bigl|\widehat{{\nu}_{b,p}}(z)\bigr|^2 \geq\exp\bigl(-\bigl|\log|2p-1|\bigr|H_b\bigl(|z|\bigr)\bigr).$$

Proof

Let p∈(0,1). Note that 1+2p(1−p)(cos(b ^k z)−1)≥0. Thus, using the inequality x≤e ^x−1 for x≥0, we have by (1.3)

Next let p∈(0,2⁻¹)∪(2⁻¹,1). Using the inequality 1−x≥exp(a ⁻¹ xlog(1−a)) for 0≤x≤a<1, we see that

Thus we have proved the lemma. □

Lemma 2.2

Let μ be a full b-decomposable distribution μ on ℝ^d satisfying (1.1).

• (i) There are positive constants K ₁ and K ₂ depending only on ρ such that

  $$\bigl|\widehat{\mu}(z)\bigr|^2 \leq\sup_{u \ge K_2 |z|}\exp \bigl(-K_1H_b(u)\bigr). $$

  (2.1)

• (ii) Let d=1. Let $C_{\delta}:= \rho*\bar{\rho}(|x| >\delta)$ for δ>0. Then we have

  $$\bigl|\widehat{\mu}(z)\bigr|^2 \leq C_{\delta}^{-1}\int _{|x| >\delta}\exp \bigl(-C_{\delta} H_b\bigl(|zx|\bigr)\bigr)\rho*\bar{\rho}(dx).$$

  (2.2)

Proof

Note that $\int_{\mathbb{R}^{d}} \cos\langle z,x\rangle\rho*\bar{\rho}(dx)=|\widehat {\rho}(z)|^{2} \ge0$. Using the inequality x≤e ^x−1 for x≥0, we have

(2.3)

We find from Lemma 5 of Watanabe [28] that ρ and also $\rho *\bar{\rho}$ are full on ℝ^d. Thus there exist disjoint closed balls D _j for 1≤j≤d in ℝ^d such that $C_{j} :=\rho *\bar {\rho }(D_{j}) > 0$ and, for any choice of x _j ∈D _j , $\{x_{j}\}_{j=1}^{d}$ is a basis of ℝ^d. Let K ₁:=min_1≤j≤d C _j . There exists K ₂>0 independent of any choice of {x _j } such that max_1≤j≤d |〈z,x _j 〉|≥K ₂|z| for z∈ℝ^d. Applying Jensen’s inequality, we conclude from (2.3) that

(2.4)

Thus we have proved (i). Let d=1. Applying Jensen’s inequality again, we have by (2.3)

Thus we have proved (ii). □

Lemma 2.3

Let 0<b<1. Then we have the following.

• (i) If u,v≥0 and |u−v|≤1, then |H _b (u)−H _b (v)|≤(1−b)⁻¹.

• (ii) There are positive constants C ₁ and C ₂ such that, for u≥0,

  $$H_b(u) \le\frac{2 }{|\log b|}\log(2 +u)+ C_1\le C_2\log(2 +u).$$

Proof

Let u,v≥0 and |u−v|≤1. We have

Thus (i) is true. Let N:=logu/|logb| for u≥1. Then we have, for u≥1,

$$H_b(u) \le\sum_{0 \le k \le N} 2 + \sum _{k >N } 2^{-1}u^2 b^{2k} \le 2\biggl(\frac{\log u }{|\log b|}+1\biggr) + \frac{1}{2(1-b^2 )}.$$

Thus we have proved (ii). □

The following lemma is due to Pisot [22].

Lemma 2.4

A real number θ>1 is a PV number if and only if there exists t≠0 such that $\sum_{n=1}^{\infty}\sin^{2}( t\theta^{n})$ converges.

Let θ>1. For 1≤t<θ and n∈ℕ, we can uniquely represent θ ⁿ t as

where C _n ∈ℕ and ϵ _n ∈[−2⁻¹,2⁻¹). The symbol [x] stands for the largest integer not exceeding x∈ℝ and ♯(A) does for the cardinality of a finite set A. We denote by dimE the Hausdorff dimension of a Borel set E in ℝ^d. The following lemma is due to Lemma 6.3 of Peres et al. [20]. Its idea goes back to Erdös [7].

Lemma 2.5

Fix a>1 and δ>0. For any ϵ>0, the following holds for all sufficiently large integers n≥n ₀(a,δ,ϵ).

• (i) Given C _n and C _n+1, there are at most [1+(a+δ+1)²+ϵ] possibilities for C _n+2, independent of θ∈[a,a+δ] and t∈[1,θ).

• (ii) If

  

  then C _n+2 is uniquely determined by C _n and C _n+1, independent of θ∈[a,a+δ] and t∈[1,θ).

Lemma 2.6

Fix a>1 and δ>0. Let K≥3. Suppose that

$$0 < B < \frac{1-\cos(\pi/(a + \delta+1)^2)}{\log(a + \delta)}. $$

(2.5)

Then we have

(2.6)

Proof

Let N∈ℕ and ϵ>0. Define τ>0 and a set E _N as τ:=(2(a+δ+1)²+ϵ)⁻¹ and

Let E:=lim sup _N→∞ E _N . Let θ∈[a,a+δ]. Choose ϵ>0 such that

$$B < \frac{1-\cos(2\pi\tau)}{\log(a + \delta)}.$$

Note that there exists M>0 independent of N such that, for N≥2,

We see from Lemma 2.5 that, for θ∈E _N , the possibilities of $\{C_{j}\}_{j=1}^{N}$ are at most $\mathrm{Const}\cdot{N\choose[N/K]} [1+(a +\delta+1)^{2}+\epsilon]^{3N/K}$. Thus E _N can be covered by at most $\mathrm{Const}\cdot{N\choose[N/K]} [1+(a +\delta+1)^{2}+\epsilon]^{3N/K}$ intervals with length 2Ma ^−N. Hence, using Stirling’s formula, we have

(2.7)

Let b=θ ⁻¹ and represent u∈[2πθ,∞) as u=2πtθ ^N with some t∈[1,θ) and N∈ℕ. Then we see that

Thus if $b \in[\frac{1}{a + \delta},\frac{1}{a}]$ and

then we have b ⁻¹=θ∈E. Hence, letting ϵ→0, we obtain (2.6) from (2.7). □

Proof of Theorem 1.1

The equivalence (1)⟺(3) is clear from Lemma 2.1. It is obvious that (4)⟹(3). We see from (i) of Lemma 2.2 that (1)⟹(4). Suppose that (2) is not true, that is, b ⁻¹ is a PV number. Then we see from Lemma 2.4 that there exists t>0 such that $\sum_{n=1}^{\infty}\sin^{2}( tb^{-n})$ converges. Thus we have

That is, (1) is not true, and hence (1)⟹(2). Finally, suppose that (1) is not true. Then there exist a sequence $\{2 t_{n}b^{-m_{n}}\}_{n=1}^{\infty}$ such that 1≤t _n <b ⁻¹, m _n ∈ℕ, m _n ↑∞ as n→∞, and lim _n→∞ t _n =t∈[1,b ⁻¹] and that

We find that, for n≥N with N∈ℕ,

Letting n→∞ and then N→∞, we have

Thus, by Lemma 2.4, b ⁻¹ is a PV number, and hence (2)⟹(1). □

Proof of Theorem 1.2

The equivalence (1)⟺(2) is clear from Lemma 2.1. It is obvious that (3)⟹(2). Suppose that (1) is true. Then I _b (t ₀)<∞ for some t ₀>0. Let {D _j }, {C _j }, and {x _j } be the same as in the proof of (i) of Lemma 2.2. Let x _j =^t(x _1j ,x _2j ,…,x _dj ) and define a real d×d matrix X as X=(x _ij ). Clearly, if x _j ∈D _j for 1≤j≤d, then detX≠0. Take any positive number q satisfying 2⁻¹ qC _j ≥t ₀ for 1≤j≤d. Changing variables as 〈z,x _j 〉=u _j for 1≤j≤d, we obtain as in (2.4) that

Thus we see that (1)⟹(3). □

Proof of Remark 1.2

Let n∈ℤ₊. Corollary 1.4 of Peres and Solomyak [21] says that, for a.e. $b \in(5/9,\sqrt{5/9})$ with p=3⁻¹, $\widehat{{\nu}_{b,p}}(z) \in L^{2}(\mathbb{R})$. Thus, by (ii) of Lemma 2.1, the assertion is true for n=0. Note that, for all m∈ℕ,

$$H_{b^{1/m}}(u) = \sum_{k=0}^{m-1}H_b\bigl(b^{k/m}u\bigr). $$

(2.8)

For general n, using (2.8) with m=2ⁿ and applying the generalized Hölder’s inequality, we have, for a.e. $b \in(5/9,\sqrt{5/9})$,

Thus we proved (i). Assertion (ii) follows from (ii) of Lemma 2.3. □

Proof of Theorem 1.3

The equivalence (2)⟺(3) is clear from Lemma 2.1. Obviously, (3)⟹(1) and (4)⟹(2). We see from (i) of Lemma 2.2 that (3)⟹(4). Let t>0 be arbitrary. We obtain from (i) of Lemma 2.3 that, for v≥0,

Thus we have, for v≥1,

$$\exp\bigl(-H_b(v)\bigr) \le\exp\biggl(\frac{1}{1-b}\biggr)\bigl(J_b(t)\bigr)^{1/t} v^{-1/t}. $$

(2.9)

Hence we have proved that (1)⟹(3). □

Proof of Remark 1.3

Let n∈ℤ₊. For n=0, take K=32, $a=\sqrt{2}$, and a+δ=2− in (2.6). Then we can take B/K=0.0027 and make the dimension in (2.6) less than 1. Thus, by using (2.8) with m=2ⁿ, we find that (i) is true. By virtue of Theorem 4.5.2 of Bertin et al. [1], for a.e. t>0, the sequence $\{\theta^{n} t\}_{n=1}^{\infty}$ is uniformly distributed mod 1. Fix such a t. We have

Thus we see that, for all b∈(0,1) and any ϵ>0,

Thus the second assertion of (ii) is true. The first assertion follows from (2.9). □

Proof of Theorem 1.4

Let $C_{\delta}:=\rho*\bar{\rho}(|x| >\delta )$ for δ>0. We can choose δ>0 such that 2^−Nlog3<C _δ . We obtain from (ii) of Lemma 2.2 and (i) of Remark 1.2 that, for a.e. $b \in((5/9)^{2^{-N}}, 1)$,

Thus μ has an L ²-density. If ρ has no point mass, we have q=0 and hence N=1. Thus (i) is true. Assertion (ii) follows from (ii) of Lemma 2.2 and (i) of Remark 1.3. □

Proof of Theorem 1.5

Since B ₃⊂B ₂, it is enough to prove that

$$\dim\bigl((0,1) \setminus\mathbf{ B}_3\bigr) =0. $$

(2.10)

Let K≥3 and take B satisfying (2.5). We see from Theorem 1.3 that

Thus we obtain from Lemma 2.6 that

Letting K↑∞, and then a↓1 and δ↑∞, we have (2.10). □

Proof of Corollary 1.1

The corollary follows from Theorems 1.3 and 1.5. □

Finally, we add a theorem of the Erdös-Kahane type for b-decomposable distributions. For an integer k≥0, let C ^k(ℝ^d) be the class of real-valued functions on ℝ^d all of whose partial derivatives of order up to and including k are continuous.

Theorem 2.1

• (i) Let μ be a full b-decomposable distribution on ℝ^d. Suppose that b is an EK number. Then $|\widehat{\mu}(z)|$ has uniform decay at infinity. Thus, for every integer k≥0, the convolution power μ ^n∗ is absolutely continuous with a bounded density of class C ^k(ℝ^d) for all sufficiently large integers n.

• (ii) Fix a full probability distribution ρ on ℝ^d satisfying (1.2). Then there are a positive strictly increasing sequence $\{a_{k}\}_{k=0}^{\infty}$ and a positive strictly decreasing sequence $\{c_{k}\}_{k=0}^{\infty}$ with lim _k→∞ a _k =1 and lim _k→∞ c _k =0 such that the following statement is true for every integer k≥0: For b∈(a _k ,1) outside a set of Hausdorff dimension less than c _k , any b-decomposable distribution μ on ℝ^d satisfying (1.1) has a density of class C ^k(ℝ^d).

Proof

The first assertion of (i) follows from Theorem 1.3. The second one is a direct consequence of the first one. Next we prove (ii). We see from (2.1), (2.8), and Lemma 2.6 that we can take the two sequences $\{a_{k}\}_{k=0}^{\infty}$ and $\{c_{k}\}_{k=0}^{\infty}$. □

3 Structures of the Sets B _j

Denote by ∥x∥ the distance of x∈ℝ to the nearest integer. We define a function D(θ) for θ>1 as

$$D(\theta):= \inf_{1 \le t <\theta} \limsup_{n \to\infty} \bigl\Vert \theta^n t \bigr\Vert.$$

Let S be the totality of PV numbers. Salem [23] proved that the set S is a closed set. All positive integers bigger than 1 are PV numbers. The minimum PV number is the positive zero of x ³−x−1. An algebraic integer θ>1 is said to be a Salem number if all its Galois conjugates θ _j satisfy |θ _j |≤1 and at least one of the conjugates has modulus equal to one. Let T be the totality of Salem numbers and T ⁻¹ be the set of their reciprocals. Salem numbers are not PV numbers. We say that a real number θ>1 is a generalized Salem number if D(θ)=0. Let T ^∗ be the totality of generalized Salem numbers and (T ^∗)⁻¹ be the set of their reciprocals. Let S _⊥ be the set of b in (2⁻¹,1) such that ν _b is singular, and let $\mathbf{ S}^{*}_{\bot}(\supset\mathbf{ S}_{\bot})$ be the set of b in (2⁻¹,1) such that ν _b does not have an L ²-density. Let E be a set in (0,1). We say that E is of type A if E contains a left neighborhood of 1, that is, E contains an interval (1−ϵ,1) for some ϵ∈(0,1). We say that E is of type B if, for any nonempty open interval I in (0,1), I∖E is an uncountable set. Clearly, if E is of type B, then it is totally disconnected. We do not yet know whether S _⊥ is a countable set. However, we prove in Theorem 3.1 below that if 1 is a limit point of S _⊥, then S _⊥ is an uncountable dense set in (2⁻¹,1) and B ₂ and B ₃ are of type B.

Lemma 3.1

Suppose that a set E in (0,1) is a F _σ set and that if b∈E, then b ^1/n∈E for all n∈ℕ. Then we have the following.

• (i) E is either of type A or of type B.

• (ii) If (0,1)∖E is a countable set, then E is of type A and (0,1)∖E is nowhere dense in (0,1).

Proof

Suppose that there exists a nonempty open interval I in (0,1) such that I∖E is a countable set. Then, by virtue of the Baire-Hausdorff theorem, I∩E contains a nonempty open interval. We see that if 1 is a limit point of (0,1)∖E, then I∩E cannot include a nonempty open interval. Thus E contains a left neighborhood of 1 and the proof of (i) is complete. Suppose that (0,1)∖E is a countable set. Then E is not of type B and hence it is of type A. Let I be any nonempty open interval in (0,1). Then, as in the proof of (i), I∩E includes a nonempty open interval and thereby (0,1)∖E is nowhere dense in (0,1). Thus (ii) is true. □

Proposition 3.1

We have the following.

• (i) B ₃⊂B ₂⊂B ₁. B ₁ is an open set. B ₂ and B ₃ are F _σ sets.

• (ii) S∪T⊂T ^∗, (T ^∗)⁻¹∩B ₃=∅, and T ⁻¹⊂B ₁∖B ₃.

• (iii) Let j=1,2,3. If b∈B _j , then b ^1/n∈B _j for all n∈ℕ.

• (iv) B ₁ is of type A. Assertions (i) and (ii) of Lemma 3.1 are true for E=B ₂ and for E=B ₃.

Proof

First we prove (i). It is clear that B ₃⊂B ₂. Suppose that b∉B ₁. Then there is a sequence $\{u_{n}\}_{n=0}^{\infty}$ such that u ₀=1, u _n+1≥u _n +1 for n≥0 and that

$$\lim_{n \to\infty}\exp\bigl(-H_b(u_n)\bigr) \in(0,1].$$

Let t>0 be arbitrary. Thus we see from (i) of Lemma 2.3 that

Thus b∉B ₂ and hence B ₂⊂B ₁. Since S is a closed set, we find from Theorem 1.1 that B ₁ is an open set. Note that

$$H_b(u) \le\frac{1}{2}\sum_{n=0}^{\infty}b^{2n}u^2=\frac{u^2}{2(1-b^2)}.$$

By virtue of the dominated convergence theorem, we see that H _b (u) is continuous in b∈[1/(n+1),n/(n+1)] for n∈ℕ. We have

$$ \begin{aligned} \mathbf{B}_2 &= \bigcup _{n \in\mathbb{N}}\bigcup_{N \in\mathbb {N}}\bigcup _{t \in\mathbb{N}}\bigl\{b \in\bigl[1/(n+1),n/(n+1)\bigr]:I_b(t ) \le N \bigr\},\quad \mathrm{and}\\\mathbf{ B}_3 &= \bigcup_{n \in\mathbb{N}}\bigcup _{N \in\mathbb {N}}\bigcup_{t \in\mathbb{N}}\bigl \{b \in\bigl[1/(n+1),n/(n+1)\bigr]: J_b(t ) \le N \bigr\}.\end{aligned}$$

(3.1)

Thus B ₂ and B ₃ are F _σ sets. Next we prove (ii). It is well known that if θ∈S, then lim _n→∞∥θ ⁿ∥=0. Hence S⊂T ^∗. We see from Theorem 5.5.1 of Bertin et al. [1] that T⊂T ^∗. Thus, S∪T⊂T ^∗. Suppose that b:=θ ⁻¹∈(T ^∗)⁻¹. Then, for any δ>0, there is t∈[1,θ) such that ∥θ ⁿ t∥≤δ for all sufficiently large n∈ℕ. Let ϵ:=4π ² δ ²/logθ. Since δ>0 is arbitrary, so is ϵ>0. We obtain that

$$\limsup_{u \to\infty} \exp\bigl(- H_b(u)\bigr)u^{\epsilon} \ge \limsup_{n\to \infty}\exp\bigl(- H_b\bigl(2\pi t\theta^n\bigr)\bigr) \bigl(2\pi t \theta^n\bigr)^{\epsilon} =\infty.$$

Namely, by Theorem 1.3, b∉B ₃ and thereby (T ^∗)⁻¹∩B ₃=∅. Since Salem numbers are not PV numbers, the inclusion T ⁻¹⊂B ₁ is clear. Thus T ⁻¹⊂B ₁∖B ₃ because T⊂T ^∗. To prove (iii), note from (2.8) that, for all n∈ℕ, $H_{b^{1/n}}(u) \ge H_{b}(u)$. Thus, if b∈B _j , then b ^1/n∈B _j for all n∈ℕ and j=1,2,3. Next, we prove (iv). Since there exists the minimum PV number greater than 1, we find from Theorem 1.1 that B ₁ is of type A. Remaining assertions follow from Lemma 3.1. □

Remark 3.1

• (i) Theorem 5.6.1 of Bertin et al. [1] says that the set {θ>1:D(θ)<(2(θ+1)²)⁻¹} is a countable set and hence T ^∗ is a countable set.

• (ii) Let $M(\theta):= \sum_{k=0}^{n} |a_{k} |$ for an algebraic number θ>1 where $P(x):= \sum_{k=0}^{n} a_{k} x^{k}$ is the minimal polynomial of θ. Theorem 1 of Dubickas [4] says that if θ>1 is an algebraic number satisfying θ∉S∪T, then we have

  $$D(\theta) \ge\bigl(M(\theta)\bigr)^{-1}.$$

  Thus if θ∈T ^∗ is an algebraic number, then θ∈S∪T.

• (iii) We do not yet know whether the following equations are true: $\mathbf{ S}_{\bot}= \mathbf{ S}^{*}_{\bot}$; S _⊥∩B ₁=∅; B ₁=B ₂; B ₂=B ₃; S∪T=T ^∗; and (T ^∗)⁻¹∪B ₃=(0,1). We see from (ii) of Proposition 3.1 that B ₃⊆̷B ₂ or B ₂⊆̷B ₁ holds. Feng and Wang [8] showed that $\mathbf{ S}_{\bot} \subsetneq\mathbf{ S}^{*}_{\bot}$ or S _⊥∩B ₁≠∅ holds.

At this point, it is not easy to answer the fascinating question whether B ₂ and B ₃ are of type A. However, we can find that some strong assertions are true under the assumption that B ₂ and B ₃ are of type A.

Proposition 3.2

Suppose that B ₂ is of type A. Then we have the following.

• (i) There are N ₁∈ℕ and t ₁∈ℕ such that I _b (t ₁)≤N ₁ for all b in a left neighborhood of 1.

• (ii) Fix a full probability distribution ρ on ℝ^d satisfying (1.2). Then, for all b in a left neighborhood of 1, any b-decomposable distribution μ on ℝ^d satisfying (1.1) has a bounded continuous density.

• (iii) 1 is not a limit point of $\mathbf{ S}^{*}_{\bot}$.

Proof

Suppose that B ₂ is of type A, that is, B ₂ contains a left neighborhood I ₀ of 1. Then, by virtue of the Baire-Hausdorff theorem, we see from (3.1) that I ₀∩{b∈(0,1):I _b (t ₁)≤N ₁} includes a non-empty open interval for some N ₁∈ℕ and t ₁∈ℕ. Let B:={b∈(0,1):I _b (t ₁)≤N ₁}. Like in (iii) of Proposition 3.1, if b∈B, then b ^1/n∈B for all n∈ℕ. In case 1 is a limit point of (0,1)∖B, B cannot include a non-empty open interval. Thus we see that B contains a left neighborhood I ₁ of 1. Hence (i) is true. Next to prove (ii), let {D _j }, {C _j }, {x _j }, and the matrix X be the same as in the proof of Theorem 1.2. Choose n∈ℕ satisfying 2⁻¹ nC _j ≥t ₁ for 1≤j≤d and let b∈I ₁. Fix a full probability distribution ρ on ℝ^d satisfying ( 1.2 ). Define a b ^1/n-decomposable distribution μ _(n) on ℝ^d by

$$ \widehat{\mu_{(n)}}(z) = \prod_{k=0}^{\infty}{\widehat{\rho}}\bigl(b^{k/n}z\bigr).$$

(3.2)

Define a constant K as

$$K:=\int_{D_1\times\cdots\times D_d}|\det X |^{-1}\prod _{j=1}^d \rho *\bar {\rho}(dx_j).$$

In the following calculation, we use Jensen’s inequality together with (2.8), change variables as 〈z,x _j 〉=u _j for 1≤j≤d, and apply the generalized Hölder’s inequality. Thus, replacing b by b ^1/n, we obtain from (2.3) that

Therefore, the b-decomposable distribution μ satisfying (1.1) has a bounded continuous density for all b in a left neighborhood of 1. Hence, setting μ=ν _b , we see that 1 is not a limit point of $\mathbf{ S}^{*}_{\bot}$. Thus (ii) and (iii) are true. □

Remark 3.2

Peres et al. [20] showed that S _⊥ is a G _δ set. Since $|\widehat{\nu_{b}}(z)|$ is continuous in b∈[1/(n+1),n/(n+1)] for n∈ℕ, $\mathbf{S}^{*}_{\bot}$ is clearly a G _δ set. Note that $\widehat{\nu_{b^{1/n}}}(z) = \prod_{k=0}^{n-1}\widehat{\nu_{b}}(b^{k/n}z)$ for all n∈ℕ. Hence it is clear that b ^1/n∈(2⁻¹,1)∖S _⊥ (resp. $b^{1/n} \in(2^{-1},1)\setminus\mathbf{ S}^{*}_{\bot}$) for all n∈ℕ provided that b∈(2⁻¹,1)∖S _⊥ (resp. $b \in (2^{-1},1)\setminus\mathbf{ S}^{*}_{\bot}$). Thus we see that the assertions of Lemma 3.1 are true for the sets (2⁻¹,1)∖S _⊥ and $(2^{-1},1)\setminus\mathbf{S}^{*}_{\bot}$ instead of E by replacing the interval (0,1) with the interval (2⁻¹,1).

Proposition 3.3

Suppose that B ₃ is of type A. Then the following hold.

• (i) 1 is not a limit point of $\mathbf{ S}^{*}_{\bot}\cup\mathbf{ T}^{*}$.

• (ii) There are N ₂∈ℕ and t ₂∈ℕ such that J _b (t ₂)≤N ₂ for all b in a left neighborhood of 1.

• (iii) Fix a full probability distribution ρ on ℝ^d satisfying (1.2). Fix an integer k≥0. Then, for all b in a left neighborhood of 1, any b-decomposable distribution μ on ℝ^d satisfying (1.1) has a bounded density of class C ^k(ℝ^d).

• (iv) We have

  $$ \liminf_{\theta\downarrow1} D(\theta) > 0.$$

  (3.3)

Proof

Suppose that B ₃ is of type A. Assertion (i) is obvious from (i) and (ii) of Proposition 3.1 and (iii) of Proposition 3.2. The proof of assertion (ii) is similar to that of (i) of Proposition 3.2 by using the Baire-Hausdorff theorem again and is omitted. Thus there are N ₂∈ℕ and t ₂∈ℕ such that J _b (t ₂)≤N ₂ for all b in a left neighborhood I ₂ of 1. Fix a full probability distribution ρ on ℝ^d satisfying (1.2). Let μ _(n) be the b ^1/n-decomposable distribution defined by (3.2) with b∈I ₂. Then we see from (2.1), (2.8), and (2.9) that, for every integer k≥0, there is a sufficiently large n∈ℕ such that

Hence the b-decomposable distribution μ satisfying (1.1) has a bounded density of class C ^k(ℝ^d) for all b in a left neighborhood of 1. Thus (iii) is true. Finally, we prove (iv). Let $\gamma:=t_{2}^{-1}$ and I ₂:=(δ,1) with some δ∈(0,1). We see from (ii) and (2.9) that, for all b∈(δ,1),

Hence we obtain from (2.8) that, for all $b \in(\delta^{2^{-k}},\delta^{2^{-k-1}}]$ with k∈ℤ₊,

$$ \exp\bigl(-H_b(u)\bigr) = O\bigl( u^{-\gamma2^k}\bigr).$$

(3.4)

Let $\epsilon:= \sqrt{2 \gamma|\log\delta|}/(4\pi)$. Suppose that (3.3) fails. Then there exists a sufficiently large k∈ℕ such that, for some $\theta\in[\delta^{-2^{-k-1}},\delta^{-2^{-k}})$ and t∈[1,θ),

Therefore we have

Since γ2^k≥4π ² ϵ ²/logθ, we see from (3.4) that this is a contradiction. Thus we have proved (iv). □

Consequently, we can establish the following facts on the types of B ₂ and B ₃.

Theorem 3.1

• (i) If 1 is a limit point of S _⊥ (resp. $\mathbf{ S}^{*}_{\bot}$), then S _⊥ (resp. $\mathbf{S}^{*}_{\bot}$) is an uncountable dense set in (2⁻¹,1) and B ₂ and B ₃ are of type B.

• (ii) B ₂ is of type A if and only if, for some fixed p≠2⁻¹, ν _b,p has a bounded continuous density for all b in a left neighborhood of 1.

• (iii) B ₃ is of type A if and only if, for some fixed p≠2⁻¹, ν _b,p has a density of class C ¹(ℝ) for all b in a left neighborhood of 1.

Proof

The proof of (i) is clear from Remark 3.2 and Propositions 3.1, 3.2, and 3.3. Fix p∈(0,2⁻¹)∪(2⁻¹,1). Suppose that B ₂ is of type A. Then we see from Proposition 3.2 that ν _b,p has a bounded continuous density for all b in a left neighborhood of 1. Conversely, suppose that B ₂ is of type B. Then we find from (ii) of Lemma 2.1 that ν _b,p does not have an L ²-density for some b arbitrarily close to 1. Thus (ii) is true. Next suppose that B ₃ is of type A. Then we see from Proposition 3.3 that ν _b,p has a density of class C ¹(ℝ) for all b in a left neighborhood of 1. Conversely, suppose that B ₃ is of type B. Note that ν _b,p has a compact support. Thus we find from (ii) of Lemma 2.1 that, for some b arbitrarily close to 1, $\widehat{\nu_{b,p}}(z)$ does not have uniform decay at infinity and hence ν _b,p does not have a density of class C ¹(ℝ). Thus the proof of (iii) is complete. □

We finish this section by posing three open problems which are of interest from the view-point of number theory. It is known that if the answer to Lehmer’s problem in [14] on the minimum Mahler measure for integer polynomials is “no” (that is, so-called Lehmer’s conjecture is true), then 1 is not a limit point of T. It should be noted that if the answer to one of those open problems is “yes”, then 1 is not a limit point of T ^∗(⊃T).

Problem 1

Is it true that lim inf _θ↓1 D(θ)>0?

Problem 2

Is it true that all real numbers in a left neighborhood of 1 are EK numbers?

Problem 3

Is it true that if b ⁻¹∈(1,∞) is not a PV nor Salem number, then b is an EK number?

4 Applications to Refinable Distributions

Let λ∈ℝ with |λ|>1. A Schwartz distribution function f _λ on ℝ is called λ-refinable function if there exist m≥2 and c _j ,d _j ∈ℝ for 1≤j≤m with $\sum_{j=1}^{m} c_{j} =| \lambda|$ such that, in the sense of Schwartz distribution theory,

$$ f_{\lambda}(x)= \sum_{j=1}^mc_j f_{\lambda}(\lambda x -d_j).$$

(4.1)

Note that f _λ always has a compact support. A probability distribution μ _λ on ℝ is called λ-refinable distribution if there exist m≥2 and c _j >0, d _j ∈ℝ for 1≤j≤m with $\sum_{j=1}^{m} c_{j} =| \lambda |$ such that

$$ \mu_{\lambda}(dx)= \sum_{j=1}^m\frac{c_j }{|\lambda|}\mu_{\lambda }(\lambda dx -d_j)$$

(4.2)

with the understanding that ∫ _B μ _λ (λdx−d _j )=μ _λ (λB−d _j ) for Borel sets B in ℝ. We assume that μ _λ is not a delta measure. If the λ-refinable distribution μ _λ is absolutely continuous, then its density f _λ (x) satisfies (4.1) in the usual sense. The Bernoulli convolution ν _b,p is a typical example of b ⁻¹-refinable distributions with m=2,c ₁=b ⁻¹ p,c ₂=b ⁻¹(1−p),d ₁=0, and d ₂=b ⁻¹. Note that λ-refinable distributions μ _λ are λ ⁻²-decomposable because they satisfy (1.1) with b=λ ⁻² and

$$\rho(dx) = \Biggl(\sum_{j=1}^mc_j |\lambda|^{-1} \delta_{d_j}(\lambda dx)\Biggr)*\Biggl(\sum_{j=1}^mc_j |\lambda|^{-1} \delta_{d_j}\bigl(\lambda^2 dx\bigr)\Biggr).$$

Thus they are homogeneous self-similar measures. In particular, if λ>1, then μ _λ is λ ⁻¹-decomposable with $\rho(dx) = \sum_{j=1}^{m} c_{j} \lambda^{-1} \delta_{d_{j}}(\lambda dx)$.

Dai et al. [3] discussed several cases where λ-refinable distributions have uniform decay at infinity. In the first assertion of the following theorem, we solve the third open problem raised by Dai et al. [3] in their appendix.

Theorem 4.1

• (i) Every real number λ in (−∞,−1)∪(1,∞) outside a set of Hausdorff dimension 0 has the property that, for any λ-refinable distribution μ _λ on ℝ, $|\widehat{\mu_{\lambda}}(z)|$ has uniform decay at infinity.

• (ii) Fix $c'_{j} >0$ and $d'_{j} \in\mathbb{R}$ for 1≤j≤m. Then there are positive strictly decreasing sequences $\{a_{k}\}_{k=0}^{\infty }$ and $\{e_{k}\}_{k=0}^{\infty}$ with lim _k→∞ a _k =1 and lim _k→∞ e _k =0 such that the following statement is true for each integer k≥0: For λ∈(1,a _k ) outside a set of Hausdorff dimension less than e _k , any λ-refinable distribution μ _λ on ℝ satisfying (4.2) with $c_{j}=c'_{j}\lambda$ and $d_{j}= d'_{j} \lambda$ has a density of class C ^k(ℝ).

Proof

Theorem 4.1 is a direct consequence of Theorems 1.5 and 2.1. □

We can show the absolute continuity of refinable distributions μ _λ for a.e. λ∈(1,a) with some a>1 without assuming the transversality condition used in Theorem 1.3 of Peres and Solomyak [21]. Moreover, we can give the decay order of the Fourier transforms of μ _λ for a.e. λ∈(1,2).

Proposition 4.1

Let λ>1 and $q:= \sum_{j=1}^{m}c_{j}^{2}\lambda^{-2}$. Then we have the following.

• (i) Let N be the smallest positive integer satisfying 2^−Nlog3+q<1. Then, for a.e. $\lambda\in(1,(9/5)^{2^{-N}})$, μ _λ has an L ²-density. In particular, let m≥3 and c _j =λ/m for 1≤j≤m. Then N=1 and μ _λ has an L ²-density for a.e. $\lambda\in(1,3/{\sqrt{5}})$.

• (ii) Let n∈ℤ₊. We have, for a.e. $\lambda\in (2^{2^{-n-1}},2^{2^{-n}})$,

  $$\bigl|\widehat{\mu_{\lambda}}(z)\bigr|= O\bigl(|z|^{(-0.0013)(1-q)2^n}\bigr).$$

Proof

The distribution μ _λ is λ ⁻¹-decomposable with $\rho(dx) = \sum_{j=1}^{m} c_{j} \lambda^{-1}\delta_{d_{j}}(\lambda dx)$. Note that $\rho*\bar{\rho}(\{0\})= q$. Thus the proof is clear from Theorem 1.4. □

We add a result on the non-smoothness of the distributions in L(b). We say that a probability distribution μ on ℝ^d belongs to the class W ^∞(ℝ^d) if, for every γ>0, $|\widehat{\mu}(z)| = O(|z|^{-\gamma})$ as |z|→∞. If μ∈W ^∞(ℝ^d), then μ has a bounded density of class C ^∞(ℝ^d). The converse is also true provided that the support of μ is compact. Dubickas and Xu [5] proved in their Theorem 1.1 that a λ-refinable distribution μ _λ does not have a density of class C ^∞(ℝ), provided that λ>1 and that all $d_{j}/d_{j_{0}}$ for 1≤j≤m are rational with some $d_{j_{0}} \ne 0$. Thus ν _b,p does not have a density of class C ^∞(ℝ) for every b and p in (0,1). Recently, Wang and Xu [27] proved that all λ-refinable distributions μ _λ do not have densities of class C ^∞(ℝ). We also give another proof. Define the Laplace transform L _ρ (u) on [0,∞) for a probability distribution ρ on [0,∞) as

$$L_{\rho}(u):= \int_{0-}^{\infty}e^{-ux}\rho(dx).$$

Under the assumption that 0<ρ({0})<1, we define a regularly varying function $K_{\lambda_{0}}(x)$ on (0,∞) with the index −λ ₀:=logρ({0})/|logb| for a probability distribution ρ on [0,∞) as

$$K_{\lambda_0}(x):= x^{- \lambda_0} \exp\biggl( \int_1^x\frac{\log\rho (\{0\}) - \log L_{\rho}(u)}{u \log b} du \biggr).$$

For two positive functions f ₁(x) and f ₂(x) on (0,∞), we define the relation f ₁(x)≍f ₂(x) by 0<lim inf _x→∞ f ₁(x)/f ₂(x) and lim sup _x→∞ f ₁(x)/f ₂(x)<∞. The following lemma is due to Proposition 4.1 of Watanabe [31].

Lemma 4.1

Let μ be a b-decomposable distribution on [0,∞) with ρ in (1.1). If 0<ρ({0})<1, then

$$\mu\bigl([0,1/t]\bigr)\asymp K_{\lambda_0}(t) \quad\mbox{\textit{as} } t \to\infty.$$

Proposition 4.2

Let μ∈L(b) on ℝ^d satisfying (1.1).

• (i) Suppose that $\rho*\bar{\rho}(\{0\}) > 2^{-1}$. Then μ∉W ^∞(ℝ^d). Moreover, if the support of ρ is compact, then μ does not have a density of class C ^∞(ℝ^d).

• (ii) Suppose that d=1 and there exists a∈ℝ such that ρ((−∞,a))=0 and 0<ρ({a})<1. Let k ₀:=[logρ({a})/logb]. Then μ does not have a density of class $C^{k_{0}}(\mathbb{R})$. In particular, all λ-refinable distributions μ _λ with λ>1 do not have densities of class C ^∞(ℝ).

Proof

First we prove (i). If $\rho*\bar{\rho}(\{0\}) = 1$, then μ is a delta measure. Thus, let $q := \rho*\bar{\rho}(\{0\})\in (2^{-1},1)$ and set $\eta(dx):= (\rho*\bar{\rho}(dx) -q \delta_{0}(dx))/(1-q)$. Note from (1.2) that η has a finite log-moment. Using the inequality 1−x≥exp(a ⁻¹ xlog(1−a)) for 0≤x≤a<1, we see from (ii) of Lemma 2.3 that

where γ:=2⁻¹ C ₂|log(2q−1)|>0. Thus μ∉W ^∞(ℝ^d). Note that the support of μ is compact if and only if so is that of ρ. Thus the second assertion is clear. Next we prove (ii). Suppose that d=1 and there exists a∈ℝ such that ρ((−∞,a))=0 and 0<ρ({a})<1 and assume that μ has a density of class C ^k(ℝ) with k∈ℤ₊. By translating the support of ρ, we can assume that a=0. Noting that μ((−∞,0))=0, we see that

$$\mu\bigl([0,1/t]\bigr)= o\bigl(t^{-k-1}\bigr)\quad\mbox{as } t \to \infty.$$

Thus we find from Lemma 4.1 that k+1≤λ ₀, but see from the definitions of k ₀ and λ ₀ that λ ₀<k ₀+1. Hence μ does not have a density of class $C^{k_{0}}(\mathbb{R})$. The second assertion is obvious. □

5 Applications to Lévy Processes

Let 0<b<1 and t>0. Let ζ _t be an infinitely divisible distribution on ℝ^d defined by

$$ \widehat{\zeta}_t(z) = \exp\Biggl(t \sum_{n=0}^{\infty}\int_{\mathbb {R}^d}\biggl(\exp\bigl(i \bigl\langle b^nz,x\bigr\rangle\bigr)-1- \frac{i\langle b^nz,x\rangle}{1+|x|^2} \biggr)\nu (dx)+it \langle z,x_0\rangle\Biggr),$$

(5.1)

where x ₀∈ℝ^d and ν is a measure on ℝ^d with ν({0})=0 and

$$ \int_{\mathbb{R}^d}\bigl(|x|^2 \wedge\log\bigl(2 + |x|\bigr)\bigr)\nu(dx) < \infty.$$

(5.2)

Then μ=ζ _t satisfies (1.1) with ρ=ρ _t defined by

$$\widehat{\rho}_t(z) = \exp\biggl(t \int_{\mathbb{R}^d}\biggl(\exp\bigl(i \langle z,x\rangle\bigr)-1- \frac{i\langle z,x\rangle}{1+|x|^2} \biggr)\nu(dx)+it(1-b) \langle z,x_0\rangle\biggr).$$

Moreover (5.2) is equivalent to (1.2). Hence ζ _t ∈L(b) and it is the distribution of X _t for some Lévy process {X _t } on ℝ^d without Gaussian part. See Lemma 2.1 of Watanabe [29] and Sato [25]. Thus {ζ _t } satisfies the convolution semigroup property, that is, ζ _s ∗ζ _t =ζ _s+t for all s,t>0. We assume that the linear span of the support of ν is the whole space ℝ^d. By virtue of Wolfe’s theorem in [36], ζ _t is either singular or absolutely continuous for t>0. Thus there are logically three cases:

• Case A. ζ _t is absolutely continuous for all t>0;

• Case B. ζ _t is singular for all t>0;

• Case C. There is T∈(0,∞) such that ζ _t is singular for 0<t<T and absolutely continuous for t>T.

We show the existence of Cases A, B, and C, respectively, in Propositions 5.1, 5.3, and Theorem 5.1 below.

Proposition 5.1

Let d=1. Suppose that ν in (5.1) satisfies ν(ℝ)=∞. Then we have the following.

• (i) If b is a PS number, then ζ _t is absolutely continuous with a bounded continuous density for all t>0.

• (ii) If b is an EK number, then ζ _t ∈W ^∞(ℝ) for all t>0.

Proof

Let δ>0 and set C _δ :=ν(|x|>δ). Applying Jensen’s inequality, we have

$$\bigl|\widehat{\zeta_t}(z)\bigr| = \exp\biggl(-t\int_{\mathbb{R}}H_b\bigl(|zx|\bigr)\nu (dx)\biggr) \le C_{\delta}^{-1}\int _{|x| >\delta}\exp\bigl(-tC_{\delta} H_b\bigl(|zx|\bigr)\bigr)\nu(dx). $$

(5.3)

Note that as δ→0, C _δ →∞. Thus, if b is a PS number, then we see from Theorem 1.2 that $\int_{\mathbb{R}}|\widehat{\zeta}_{t}(z)| dz < \infty$ for all t>0. Moreover, if b is an EK number, then we find from Theorem 1.3 that ζ _t ∈W ^∞(ℝ) for all t>0. □

Proposition 5.2

Let d=1. Suppose that ν in (5.1) satisfies ν(ℝ)<∞. Then we have the following.

• (i) Let n∈ℤ₊. For a.e. $b \in((5/9)^{2^{-n}},(5/9)^{2^{-n-1}})$, ζ _t is absolutely continuous with a bounded continuous density for all t>2⁻ⁿ(ν(ℝ))⁻¹log3.

• (ii) Let n,k∈ℤ₊. For a.e. $b \in(2^{-2^{-n}},2^{-2^{-n-1}})$, ζ _t is absolutely continuous with a bounded density of class C ^k(ℝ) for all t>371(k+1)2⁻ⁿ(ν(ℝ))⁻¹.

Proof

Note that (5.3) holds. Thus assertions (i) and (ii) follow from (i) of Remark 1.2 and (i) of Remark 1.3, respectively. □

Proposition 5.3

Let d=1. Suppose that ν in (5.1) is a finite discrete measure on ℤ. If b ⁻¹ is a PV number, then ζ _t is singular for all t>0.

Proof

Let $\{\theta_{j}\}_{j=1}^{m}$ be the Galois conjugates of a PV number b ⁻¹. Note that |θ _j |<1 for 1≤j≤m. Thus, as in the proof of (ii) of Lemma 2.3, there is a constant C>0 such that, for x∈ℤ,

$$\sum_{k=1}^{\infty} \Biggl(1- \cos\Biggl(2\pi \sum_{j=1}^m \theta_j^kx\Biggr)\Biggr) \le C \log \bigl(2 + |x|\bigr). $$

(5.4)

Since b ⁻¹ is an algebraic integer, we find that, for n∈ℕ, there is N _n ∈ℤ such that $b^{-n} = N_{n} -\sum_{j=1}^{m} \theta_{j}^{n}$. Thus we see from (5.4) and (ii) of Lemma 2.3 that, for x∈ℤ,

It follows from (5.2) that

Thus, by virtue of the Riemann-Lebesgue theorem, ζ _t is singular for all t>0. □

Proposition 5.4

• (i) If ν in (5.1) satisfies ν(ℝ^d)<∞, then ζ _t ∉W ^∞(ℝ^d) for all t>0.

• (ii) If d=1, ν((−∞,0))=0, and 0<ν((0,∞))<∞, then ζ _t does not have a density of class C ^∞(ℝ) for all t>0.

Proof

Define a finite measure σ on ℝ^d as

$$\sigma(dx):= \nu(dx) + \nu(-dx).$$

Then it holds that, for sufficiently small t>0,

Thus we see from Proposition 4.2 that ζ _t ∉W ^∞(ℝ^d) for sufficiently small t>0, and hence so is for all t>0. We have proved (i). Suppose that d=1, ν((−∞,0))=0, and 0<ν((0,∞))<∞. Since ρ=ρ _t satisfies the assumption of (ii) of Proposition 4.2 with $a =- t \int_{0}^{\infty} x(1+x^{2})^{-1}\nu (dx)+t(1-b) x_{0}$, the assertion (ii) is clear from Proposition 4.2. □

The upper Hausdorff dimension of a probability distribution μ on ℝ^d is denoted by dim^∗ μ, that is,

$$\dim^* \mu:= \inf\bigl\{\dim E : \mu(E) = 1 \bigr\}.$$

The entropy of a discrete probability measure μ on ℝ^d is denoted by H(μ), that is,

$$H(\mu) := - \sum_{a \in A} \mu\bigl(\{a\}\bigr) \log\mu \bigl(\{a\}\bigr),$$

where the set A is given by A={a∈ℝ^d:μ({a})>0}. For a finite discrete measure μ on ℝ^d, define H(μ) by H(μ):=H((μ(ℝ^d))⁻¹ μ). Let η be a finite discrete measure on ℝ^d with η({0})=0. Define a compound Poisson distribution η _t for t≥0 as

$$\eta_t := \exp\bigl(-t\eta\bigl(\mathbb{R}^d\bigr)\bigr)\sum_{n=0}^{\infty }(n!)^{-1}t^n\eta^{n*}.$$

Note that η _t is a discrete distribution for all t≥0. Define a function h _η (t) on ℝ₊ as h _η (t):=H(η _t ). The following is due to Proposition 5.1 of Watanabe [29].

Lemma 5.1

Let η be a finite discrete measure on ℝ^d with η({0})=0. Then we have the following.

• (i) If H(η)=∞, then h _η (t)=∞ for all t>0.

• (ii) If H(η)<∞, then h _η (t)<∞ for all t≥0 and it is positive, continuous, and strictly increasing for t>0 with h _η (0+)=h _η (0)=0 and lim _t→∞ h _η (t)=∞.

In the last theorem, we discover that, as time increases, the distribution of a certain Lévy process on ℝ^d can change from singular with arbitrarily small dimension to absolutely continuous with a density of class C ^k(ℝ^d) of any order k.

Theorem 5.1

Suppose that ν in (5.1) is a finite discrete measure on ℝ^d with H(ν)<∞. Then the following hold.

• (i) We have

  $$\dim^* \zeta_t \leq\frac{h_{\nu}(t )}{| \log b|}$$

  and ζ _t is singular for $0 < t < h_{\nu}^{-1}(d |\log b|) <\infty$.

• (ii) If b is a PS number, then ζ _t is absolutely continuous with a bounded continuous density for all sufficiently large t>0.

• (iii) If b is an EK number, then, for every k∈ℤ₊, ζ _t is absolutely continuous with a bounded density of class C ^k(ℝ^d) for all sufficiently large t>0.

Proof

Assertion (i) is due to Theorem 5.1 of Watanabe [29]. Since ζ _t is full and ζ _t ∈L(b), assertions (ii) and (iii) follow from Theorems 1.2 and 1.3, respectively. □

Finally we raise two open problems on the temporal evolution of continuities of ζ _t .

Problem 4

Is it true that if ζ _t is singular for all t>0, then b ⁻¹ is a PV number?

Problem 5

Is it true that if ζ _t is absolutely continuous for some t>0, then it has a bounded continuous density for all sufficiently large t>0?