and the packing dimension of the intersection E ∩ G, where G is an arbitrary Borel set on the circle

Bing Li, Narn-Rueih Shieh, and Yimin Xiao

Abstract. Let E be the Dvoretzky random covering sets on the circle. By applying the method of limsup type random fractals, as illustrated in Khosh- nevisan, Peres and Xiao [24], we determine the hitting probability P(E∩G 6= ∅) and the packing dimension of the intersection E ∩ G, where G is an arbitrary Borel set on the circle.

Introduction

We begin with a brief review on random coverings. Let {ω_n}_n≥1be a sequence of independent random variables on (Ω, B, P) which are uniformly distributed over the unit interval I = [0, 1). Let {ln}n≥1 be a sequence of positive real numbers which is decreasing to zero. For every n ≥ 1, denote by In := (ωn, ωn+ ln)(mod 1) the random interval whose starting point and lengthe are determined by ωn and ln

respectively. Define the random covering set as E := lim sup

n→∞

In= {t ∈ I : t ∈ In for infinitely many n ≥ 1}.

The set E consists of the points which are covered by {I_n} infinitely often (i.o. for short). The Borel-Cantelli Lemma implies that the Lebesgue measure of the random set E is either 1 or 0 almost surely according to the divergence or convergence of the seriesP∞

n=1l_n.

It was Dvoretzky [5] who called the attention on study of such random sets; he raised the question that under what condition on {ln} one can have

(0.1) [0, 1) = lim sup

n→∞

I_n a.s.

In the literature this is referred to as the Dvoretzky covering problem and had attracted the attention of P. Billard, J.-P. Kahane, B. Mandelbrot, among others, before it was completely solved by L. A. Shepp in 1972. Shepp [31] provided a necessary and sufficient condition for (0.1) to hold, namely

(0.2)

∞

X

n=1

1 n²exp

l1+ · · · + ln



= ∞.

1991 Mathematics Subject Classification. 60D05, 52A22, 28A78, 28A80.

Key words and phrases. Random covering sets, Dvoretzky random covering, hitting proba- bility, packing dimension, Hausdorff dimension, limsup random fractals.

Research supported in part by NSF of China Grant #11201155.

Research supported in part by NSF grant DMS-1006903.

1

Since then the topic has been under active development and there have been many extensions and refinements. We refer to [20, Chapter 11] for a systematic account on the Dvoretzky covering problem and its higher dimensional extensions, to the survey articles [8, 22, 23] for historical accounts and connections to multiplicative processes, and to [1, 4, 7, 8, 9, 10, 14, 21] and the references therein for further information. It should also be mentioned that Jonasson and Steif [16] (see also [15]) have recently extended the Dvoretzky covering model by including time dynamics (In the first variant, they identify I = [0, 1) with the unit circle C and allow the centers of In(n ≥ 1) perform independent Brownian motions on C, each with variance 1. In the second variant, they associate independent Poisson processes with the different intervals.) The work of Jonasson and Steif [16] has revealed rich structures in dynamical random coverings and raised more interesting questions about properties of the random covering sets, including their fractal dimensions and hitting probabilities.

This paper is concerned with the geometric and potential-theoretic properties of the Dvoretzky covering set E = lim sup

n→∞

In. It is known that the set E is a.s. dense in I and is of second category ([20, Chapter 5, Proposition 11]). Thus, the upper box dimension of the set E is 1 almost surely. Several authors have investigated the Hausdorff dimension and other fractal properties of E and/or its complement F∞ = I\E (which is called the uncovered set). For example, Fan and Wu [10]

considered the Hausdorff dimension of the set E for the special case l_n =_n^a_γ, where a > 0 and γ > 1 are constants, they proved that dim_H(E) = ¹_γ a.s., where dim_H denotes Hausdorff dimension. Durand [4] considered a general sequence {ln} with P∞

n=1ln< ∞ and proved, among other things,

(0.3) dim_HE = α and dim_PE = 1 a.s., where α is defined by

(0.4) α := inf

 s > 0 :

∞

X

n=1

l^s_n< ∞



= sup

 s > 0 :

∞

X

n=1

l^s_n= ∞



with the convention that sup ∅ = 0 and inf ∅ = 1.

The index α defined in (0.4) is known as the exponent of convergence of the sequence {ln} and can be calculated by using the following formula

(0.5) α = lim sup

n→∞

log n

− log ln

(see [27] p.285 or [30] p.26). Besicovitch and Taylor [3] applied the index α (they also introduced another index–the lower index for {ln}) to characterize the Haus- dorff measure and Hausdorff dimension of a linear compact set K whose complement forms a sequence of open intervals of lengths {ln}. Hawkes [13] showed that α is the upper box dimension of K, and the lower index of {ln} is the lower box dimension of K. Kahane [18, 19] called α the upper Besicovitch-Taylor index of {ln}. Some related indices for {ln} were also discussed in [1] for studying the Carleson problem and covering numbers for the Dvoretzky covering set E.

The following intersection problem is of intrinsic importance in the study of random coverings and other random fractals. For any given set A ⊂ [0, 1), we can ask whether or not it is a.s. covered infinitely often by {I_n}. That is, when

does P(A ⊂ E) = 1 hold? In the case

∞

P

n=1

ln = ∞, which is opposite to what we are considering in this paper, the analogous problem for the uncovered set F_∞has been investigated by several authors. For example, Kahane [20] considered the case l_n =^β_n, 0 < β < 1 and showed that P(A ⊂ E) = 1 (equivalently, P A∩F∞6= ∅
= 0) if dim_H(A) < β, whilst P(A ⊂ E) = 0 (equivalently, P A ∩ F∞ 6= ∅

= 1) if dim_H(A) > β. For a more general case, Hawkes [12] proved that, if the set A satisfies a regularity condition (which in particular requires dim_H(A) = dim_P(A)), then P(A ⊂ E) = 1 or 0 according as dimH(A) < τ and dim_H(A) > τ , where τ is the index of {l_n} defined by

τ = lim sup

n→∞

Pn i=1li

log n .

More precise hitting probability results for the Poisson covering (see Mandelbrot [26]) have been established by using the connection between F_∞and the range of a subordinator; see Fitzsimmons, et al. [11]. However, the problems for determining the hitting probabilities of the Dvoretzky random covering set E had never been studied.

The purpose of this paper is to study the hitting probabilities of the Dvoretzky covering set E, as well as fractal dimensions of the intersection E ∩ G, when it is nonempty. Our main result (Theorem 2.1 below) shows that the hitting probability P(E ∩ G 6= ∅) is determined by dimP(G), the packing dimension of G (see (0.7) below). This is in contrast with the hitting probability results for the random set F_∞, where Hausdorff dimension plays the natural role. Theorem 2.1 will allow us to determine the packing dimension of E ∩ G for any analytic set G ⊆ [0, 1) and provide a refinement (under an extra condition) of (0.3) obtained by Durand [4].

Recall that packing dimension was introduced in the early 1980s by Tricot [33]

as follows. For any ε > 0 and any bounded set G ⊂ R, let N (G, ε) be the smallest number of balls of radius ε needed to cover G. The upper box dimension of G is defined as

(0.6) dim_M(G) = lim sup

ε→0

log N (G, ε)

− log ε and the packing dimension of G is defined as

(0.7) dim_P(G) = inf

 sup

n

dim_MG_n : G ⊂

∞

[

n=1

G_n

 ,

where the infimum is taken over all countable coverings {Gn} of G. It is well known that 0 ≤ dim_H(G) ≤ dim_P(G) ≤ dim_M(G) ≤ 1 for every set G ⊂ R. Similarly to Hausdorff dimension, packing dimension has been shown to be a useful tool for characterizing fractal sets and for studying “roughness” of stochastic processes. We refer to Falconer [6] and Mattila [28] for further properties of packing dimension and to Taylor [32] and Xiao [34] for extensive surveys on its applications to random fractals.

The rest of this paper is organized as follows. In Section 2 we state the main results and provide some discussions and examples. The proofs of the theorems are given in Section 3 and they rely on the general method on limsup random fractals in Khoshnevisan, Peres and Xiao [24]. We remark that our argument extends that in [24] and shows that their Theorems 3.1, 3.3 and corollaries still hold if their

Condition 4 is replaced by a weaker condition. Finally Section 4 contains some technical results on the upper Besicovitch-Taylor index. In particular we apply the results in Lapidus and van Frankenhuysen [25] to show that every sequence {ln} associated to a self-similar string has its upper Besicovitch-Taylor index equal to the self-similarity dimension and satisfies the condition (C) in this paper.

1. Main results and examples Throughout this paper we assumeP∞

n=1l_n < ∞. Thus the Lebesgue measure of the Dvoretzky covering set E is 0 almost surely.

Let α be the upper Besicovitch-Taylor index of {ln}. Then by Proposition 3.1 below, we have

(1.1) α = lim sup

k→∞

log₂nk

k , where log₂is the logarithm in base 2 and n_k is defined as

nk= #n

n ∈ N : lⁿ ∈ [2^−k+1, 2^−k+2)o

(k ≥ 2).

Here #A denotes the cardinality of the set A.

To state the main results of this paper we will make use of the following con- dition (C):

(C) There exists an increasing sequence of positive integers {k_i} such that

(1.2) lim

i→∞

ki+1

k_i = 1 and

(1.3) lim

i→∞

log₂nk_i

ki

= α < 1.

Theorem 1.1. Let E be the Dvoretzky covering set associated with the sequence {ln} whose upper Besicovitch-Taylor index is α. If the condition (C) holds, then for every analytic set G ⊂ [0, 1), we have

P E ∩ G 6= ∅
=

(0 if dim_P(G) < 1 − α, 1 if dim_P(G) > 1 − α.

Remark 1.2. Some remarks are in order.

(i) It is clear that if

(1.4) lim

k→∞

log₂n_k

k = α < 1,

then condition (C) holds. We will give several interesting examples of sequences {l_n} that satisfy (1.4).

(ii) If G is regular in the sense that dim_M(G) = dim_M(G), where dim_M(G) is the lower box dimension of G, which is defined by replacing limsup in (0.6) by liminf, then condition (C) is surplus. This follows from the proof of Theorem 1.1 below, in which we can take N = {ki₀, ki₀+1, . . . } for some i0≥ 1.

(iii) From the first part of the proof of Theorem 1.1, we see that the conclusion dim_P(G) < 1 − α implies P(E ∩ G 6= ∅) = 0, even without the condition (C).

(iv) By Proposition 3.3 in Section 4, we see that (1.4) can be replaced by the following: there exists a constant b ∈ (1, 2] such that

(1.5) lim

k→∞

log_bmk

k = α < 1, where m_k = #n ∈ N : ln∈ [b^−k+1, b^−k+2) . (v) When P∞

n=1ln = ∞, but Shepp’s condition (0.2) is not satisfied, then E 6= [0, 1). One can consider the random set F_∞ = [0, 1)\E of the un- covered points. The fractal dimension and hitting probabilities have been studied by Hawkes [12] (see also Kahane [20, Chapter 11]) and have been shown to be very different from Theorem 1.1.

We can extend Theorem 1.1 to the following, which describes the intersection of two independent random covering sets of indices α, α⁰< 1.

Theorem 1.3. Let E and E⁰ be two independent Dvoretzky covering sets on the same probability space, associated to the sequences {ln} and {l⁰_n} respectively.

Suppose both {ln} and {l_n⁰} satisfy the condition (C) with the corresponding upper Besicovitch-Taylor indices α, α⁰ < 1 and possibly different subsequences {ki} and {k_i⁰}. Then for any analytic set G ⊂ [0, 1) satisfying dim_P(G) > 1 − min{α, α⁰}, we have

P E ∩ E⁰∩ G 6= ∅
= 1.

In particular, if dim_P(G) > 1 − α, then

dim_P(E ∩ G) = dim_P(G) a.s.

In the following we provide an estimate on the Hausdorff dimension of the intersection E ∩ G for a given set G.

Theorem 1.4. Let E be the Dvoretzky covering set associated with the sequence {ln} which satisfies the condition (C). Then for any analytic set G ⊂ [0, 1), we have (1.6) dim_H(G) − (1 − α) ≤ dim_H(E ∩ G) ≤ dim_P(G) − (1 − α) a.s.

By taking G = [0, 1) in Theorems 1.3 and 1.4, we obtain dim_H(E) = α and dim_P(E) = 1 almost surely. This recovers the result (0.3) of Durand [4], under the extra condition (C). We remark that our method is different from that of Durand [4].

Corollary 1.5. Assume the conditions of Theorem 1.4 hold. For any analytic set G ⊂ [0, 1) satisfying dim_H(G) = dim_P(G), we have

dim_H(E ∩ G) = dim_H(G) − (1 − α) a.s.

In particular dim_H(E) = α almost surely.

We end this section with some examples.

Example 1.6. 1. If lⁿ ∼ c n^−γ, where c > 0 and γ > 1 are constants and ln ∼ jnmeans lim

n→∞

l_n

j_n = 1, then {ln} satisfies (1.4) with α = _γ¹. Hence Theorem 2.1 provides results on hitting probabilities for the associated Dvoretzky covering set E.

In particular, we have dim_P(E) = 1. This complements the results in Fan and Wu (2004) on the Hausdorff dimension of E. More generally, we can take l_n ∼ c₁n^−γ for even integers n, while l_n ∼ c2n^−γ0 for odd integers n, where both constants γ and γ⁰ are larger than 1. Such sequence satisfies (1.4) with α = max{γ⁻¹, γ⁰⁻¹}.

Thus, by Durand [4] or by our Corollary 1.5, dim_H(E) = α < 1. On the other hand, by [4] or by our Theorem 1.3, dim_P(E) = 1.

2. Let {ln} be the sequence corresponding to the complimentary open intervals of the tertiary Cantor set. That is, l_n = 3^−m when

m−1

P

i=0

2ⁱ < n ≤

m

P

i=0

2ⁱ (m = 1, 2, . . .). Then it can be verified directly that the upper Besicovitch-Taylor index α = log 2/ log 3 and, moreover, (1.4) holds. Hence our results are applicable to the corresponding random covering set E. Consequently, dim_H(E) = log 2/ log 3 and dim_P(E) = 1 almost surely. Similar results hold for the random covering sets associated with more general self-similar sets (or self-similar strings, in the terminology of Lapidus and van Frankenhuysen [25]). See Proposition 3.2.

3. If ln= a⁻ⁿ, where a > 1 is a constant, then {ln} satisfies the condition (1.5) with α = 0, by Remark 1.2 (iv), more generally, Proposition 3.3. Hence Theorem 1.1 holds for such {ln}. In particular, we have dim_HE = 0 and dim_P(E) = 1 almost surely.

4. Finally we provide a simple example of {ln} that satisfies condition (C), but not (1.4). Let β > log 3/ log 2 be a constant. We define

l_n=











3^−m if

m−1

P

i=0

2ⁱ< n ≤

m−1

P

i=0

2ⁱ+ 2^m−1,

n^−β if

m−1

P

i=0

2ⁱ+ 2^m−1< n ≤

m

P

i=0

2ⁱ.

Then we can verify that condition (C) is satisfied with α = log 2/ log 3 and the sub- sequence ki= b(log₂3)ic, where bxc denotes the largest integer ≤ x. However, (1.4) fails. Nevertheless, the theorems in this section are applicable to the corresponding Dvoretzky covering set E.

2. Proofs of the theorems

In this section we prove Theorems 1.1, 1.3 and Theorem 1.4. It will be clear that the method for studying limsup random fractals in Khoshnevisan, Peres and Xiao [24] plays an essential role in our proofs. We remark that, even though the second half of the proof of Theorem 1.1 is a modification of the proof of Theorem 3.1 in [24], our argument is more general and proves that Theorems 3.1 and 3.2 and their corollaries in [24] still hold if their Condition 4 is replaced by

Condition 4⁰: For some constant γ > 0, lim sup

k→∞

log₂pk

k = −γ

and there exists an increasing sequence of positive integers {k_i} satisfying (1.2) such that

i→∞lim

log₂p_ki ki

= −γ.

For proving Theorem 1.1 (and for extending the results in [24]) we will use the following elementary lemma on upper box dimension.

Lemma 2.1. Let {kⁱ} be an increasing sequence of positive integers which sat- isfies (1.2). Then for any bounded set G ⊂ R,

(2.1) dim_M(G) = lim sup

i→∞

log₂N (G, 2^−ki) ki

.

Proof. For any ε > 0, there is an integer i such that 2^−ki+1< ε ≤ 2^−ki. Thus for any bounded set G ⊂ R we have

N (G, 2^−ki) ≤ N (G, ε) ≤ N (G, 2^−ki+1).

This implies that

log N (G, 2^−ki) k_ilog 2

ki

k_i+1 ≤log N (G, ε)

− log ε ≤log N (G, 2^−ki+1) k_i+1log 2

ki+1

k_i .

It is clear that (2.1) follows from the above and (1.2).  Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Firstly we show that dimP(G) < 1−α implies P E∩

G 6= ∅
= 0. By (0.7), it suffices to show that whenever dim_M(G) < 1 − α, then E ∩ G = ∅, a.s.

Fix an arbitrary but small η > 0 such that dim_M(G) < 1 − α − η. For any r > 0, denote by Cr = Cr(G) a collection of the smallest number of the intervals with length r that cover the set G. Let Nr(G) = #Cr. Since

lim sup

n→∞

log Nl_n(G)

− log l_n ≤ lim sup

r→0

log Nr(G)

− log r = dim_M(G) < 1 − α − η, there exists an integer n₀∈ N such that

(2.2) Nln(G) < l^{−(1−α−η)}_n

for all n ≥ n₀. For any interval J in [0, 1) with length l_n, since ω_n is uniformly distributed on [0, 1), we have

PIn∩ J 6= ∅ ≤ 3ln. Note that

In∩ G 6= ∅ ⊂ [

J ∈C_ln

In∩ J 6= ∅ , we derive from this and (2.2) that

PIn∩ G 6= ∅ ≤ X

J ∈C_ln

PIn∩ J 6= ∅ ≤ Nl_n(G) · 3ln< 3l^α+η_n for all n ≥ n₀. Hence the seriesP∞

n=1P{In∩ G 6= ∅} converges by the definition of α and η > 0. By the Borel-Cantelli Lemma, we have

PIn∩ G 6= ∅ i.o. = 0.

That is, P∃n0, s.t. ∀n ≥ n₀, I_n∩ G = ∅ = 1. Therefore, E ∩ G = ∅ a.s.

In the following, we prove that if dim_P(G) > 1 − α, then P E ∩ G 6= ∅
= 1.

For this purpose, we construct a random subset E_∗⊂ E and show that P E∗∩ G 6=

∅
= 1. The random subset E∗ is a discrete limsup random fractal as in [24]. Our proof below is a modification and extension of the method in their Section 3 and is divided into two steps.

(i) Construction. For any k ≥ 2, letDkbe the collection of dyadic intervals of the form (₂ⁱ_k,ⁱ⁺¹_2k ), i = 3, 4, . . . , 2^k− 1. Denote by Tk = {n ∈ N : ln ∈ [2^−k+1, 2^−k+2)}

and let n_k= #T_k.

For every J ∈Dk, define Z_k(J ) =

(1 if ∃ n ∈ Tk such that J ⊂ In= (ωn, ωn+ ln), 0 otherwise.

Let

A(k) = [

J ∈Dk Zk(J)=1

J

be the union of open dyadic intervals of order k that are contained in some In with length ln∈ [2^−k+1, 2^−k+2). Observe that

A(k) ⊂ [

n∈Tk

In.

We define E∗:= lim sup_k→∞A(k). From the above, we have E∗⊂ E.

(ii) Hitting probability. Now let G ⊂ [0, 1) be an analytic set such that dim_P(G) >

1 − α. Then by Joyce and Preiss [17], we can find a closed set G_∗⊂ G, such that for all open set V , we have dim_M(G_∗∩ V ) > 1 − α, whenever G_∗∩ V 6= ∅.

In the following, we show P E∗∩ G_∗ 6= ∅
= 1. Our method is a modification and extension of that in Section 3 of [24].

For every J ∈Dk, the probability

P Zk(J ) = 1
= P∃ n ∈ Tk such that J ⊂ (ωn, ωn+ ln)

does not depend on J due to our assumption on {ωn} and our definition of Dk. Denote the above probability by Pk. Then

(2.3) Pk ≤ nk(ln− 2^−k) ≤ 3 nk2^−k. On the other hand,

Pk = P [

n∈T_k

{J ⊂ In}

!

≥ X

n∈T_k

P(Iⁿ ⊃ J ) − X

m∈T_k

X

n∈Tk n6=m

P I^m⊃ J, In ⊃ J

≥ nk ln− 2^−k
− 9n²_k2^−2k

≥ n_k2^−k(1 − 9n_k2^−k).

(2.4)

In the above, we have used the independence of Im and In (m 6= n) to derive the second inequality. Combining (2.3) and (2.4), together with (1.1) and Condition (C), we derive that

(2.5) lim sup

k→∞

log₂Pk

k = −(1 − α)

and there is an increasing sequence of integers {ki} that satisfies (1.2) such that

(2.6) lim

i→∞

log₂P_ki ki

= −(1 − α).

Hence E∗ is a limsup random fractals which satisfies Condition 4⁰ with γ = 1 − α (which is weaker than Condition 4 in Khoshnevisan, Peres and Xiao [24, p.11]).

Still using their terminology, we call E_∗ a limsup random fractal of index 1 − α.

Next we verify their Condition 5 regarding the correlation of Zk(J ) and Zk(J⁰) in [24]. Given J and J⁰ ∈Dk such that the distance d(J, J⁰) ≥ 2^−k+2. Since

Cov Zk(J ), Zk(J⁰)

= E Zk(J )Zk(J⁰)
− E Zk(J )

E Zk(J⁰)

= E Zk(J )Zk(J⁰)
− P_k², (2.7)

we estimate E(Z^k(J )Zk(J⁰)) first,

E(Zk(J )Zk(J⁰)) = P Z^k(J ) = 1, Zk(J⁰) = 1

= P∃ m, n ∈ Tk such that Im⊃ J and In⊃ J⁰

≤ X

m∈Tk

X

n∈Tk, n6=m

P Im⊃ J, In⊃ J⁰

=

 X

m∈T_k

P Im⊃ J



X

n∈Tk, n6=m

P In ⊃ J⁰

 . (2.8)

By (2.7), (2.8) and the first inequality in (2.4) we derive Cov(Z_k(J ), Z_k(J⁰)) ≤ 2

 X

m∈T_k

P Im⊃ J



· X

m∈T_k

X

n∈Tk n6=m

P Im⊃ J, I_n⊃ J⁰

≤ C nk2^−k

E Zk(J )

E Zk(J⁰)
, (2.9)

where the last inequality follows from (2.3) and C > 0 is a finite constant. It follows from (2.9) and (1.1) that for any ε > 0

Cov Zk(J ), Zk(J⁰)
< ε E Zk(J )

E Zk(J⁰)
for all k large enough. This implies that f (k, ε) ≤ 8, where

f (k, ε) = max

J ∈Dk

#J⁰∈Dk: Cov(Zk(J ), Zk(J⁰)) ≥ εE(Z^k(J ))E(Z^k(J⁰)) . In particular,

lim

k→∞

log f (k, ε)

k = 0.

Thus we have shown that Condition 5 in [24] is satisfied with δ = 0.

The rest of the proof follows a similar line as in the proof of Theorem 3.1 in [24]. For convenience of the reader, we give it below. Notice that our set N is determined by Condition (C) and may be different from that in [24].

Fix an open set V ⊂ [0, 1) such that G_∗∩ V 6= ∅. Let Nk be the number of dyadic intervals J ∈Dk such that

(2.10) J ∩ G_∗∩ V 6= ∅.

Since dim_M(G∗ ∩ V ) > 1 − α, we use Lemma 2.1 to derive that, for any β ∈ 1 − α, dim_M(G_∗∩ V )
, Nki ≥ 2^kiβ for infinitely many integers i. This implies the set N defined as

(2.11) N:= {i ≥ 1 : N_ki ≥ 2^kiβ} satisfies #N = ∞. Similarly to [24], we define

S_i= X

J ∈Dki J ∩G∗∩V 6=∅

Z_ki(J ).

Namely, Si is the total number of intervals J ∈Dk_i such that J ∩ G_∗∩ V ∩ A(ki) 6= ∅.

We now show P Sⁱ> 0 i.o.
= 1.

To this end, we estimate Var(S_i) = X

J ∈Dki J ∩G∗∩V 6=∅

X

J 0 ∈Dki J 0 ∩G∗∩V 6=∅

Cov Z_ki(J ), Z_ki(J⁰)
.

Fix ε > 0, for each J ∈Dki which satisfies (2.10), letGki(J ) be the collection of all J⁰∈Dki such that

(i) J⁰∩ G_∗∩ V 6= ∅, and

(ii) Cov Z_ki(J ), Z_ki(J⁰)
≤ εP_k²_i.

If J⁰∈Dk_i satisfies (i), but not (ii), then we say J⁰∈Bk_i(J ). Thus Var(Si) ≤ εN_k²_iP_k²_i+ X

J ∈Dki J 0 ∈Bki(J )

Cov(Zk_i(J ), Zk_i(J⁰))

≤ εN_k²_iP_k²_i+ Nk_i max

J ∈D_ki#Bk_i(J )Pk_i,

where the last term comes from the fact that Cov(Zk(J ), Zk(J⁰)) ≤ E Z^k(J )
= Pk. Since we have shown maxJ ∈Dk#Bk(J ) ≤ 8 for all k large enough, the above implies

lim sup

i→∞

i∈N

Var(Si)

[E(Sⁱ)]² ≤ ε + lim sup

i→∞

i∈N

maxJ ∈D_ki#Bk_i(J ) Nk_iPk_i

= ε.

In the above, we have used that facts that E(Sⁱ) = Nk_iPk_i and Nk_iPk_i → ∞ if i ∈ N and i → ∞ (recall (2.6) and (2.11)). Since ε > 0 is arbitrary, we have

(2.12) lim sup

i→∞

i∈N

Var(Si) [E(Sⁱ)]² = 0.

It follows from the Paley-Zygmund inequality ([20, p.8]) that P Si> 0
≥ (E(Sⁱ))²

E(Si²)

= 1 −Var(S_i)

E(Si²) ≥ 1 − Var(S_i)

E(Si)2.

Combining the above inequality, (2.12) and Fatou’s Lemma, we derive (2.13) P Si> 0 i.o.
≥ lim sup

i→∞ P Si > 0
= 1.

It follows from (2.13) that P

( ^∞ [

k=n

A(k)



∩ G∗∩ V 6= ∅, ∀n ≥ 1 )

= 1

for every open set V with G_∗∩ V 6= ∅. Letting V run over all open interval with rational endpoints, we obtain that (∪^∞_k=nA(k))∩G_∗is a.s. dense in G_∗for all n ≥ 1.

Since

 ^∞ [

k=n

A(k)



∩ G_∗

is an open set in G∗and G∗is a complete metric space, by Baire’s category theorem (see Munkres [29]), we know ∩^∞_n=1(∪^∞_k=nA(k)) ∩ G∗ is a.s. dense in G∗, that is, E∗∩ G∗is a.s. dense in G∗. In particular, E∗∩ G∗6= ∅ a.s. This finishes the proof

of Theorem 1.1. 

Proof of Theorem 1.3. We use the same method as in the proof of Theorem 3.2 in [24]. Let G_∗ be the closed subset of G described in the proof of Theorem 1.1. Suppose dim_P(G) > 1 − min{α, α⁰}, the proof of Theorem 1.1 shows that for any open set V such that V ∩ G_∗6= ∅ we have

P

 [^∞

k=n

Ik

∩ V ∩ G_∗6= ∅, ∀n ≥ 1



= P

 [^∞

k=n

I_k⁰

∩ V ∩ G_∗6= ∅, ∀n ≥ 1



= 1.

By independence, there exists a single null probability event outside which for all open intervals V with rational endpoints satisfying V ∩ G_∗6= ∅, we have

 ^∞ [

k=n

Ik



∩ V ∩ G∗6= ∅ and

 ^∞ [

k=n

I_k⁰



∩ V ∩ G∗6= ∅ for all n ≥ 1.

That is, 

∪^∞_k=nI_k
∩ G∗}n≥1∪

∪^∞_k=nI_k⁰) ∩ G_∗

n≥1 is a countable collection of open, dense subsets of the complete metric space G_∗. Again, Baire’s theorem implies that

P E ∩ E⁰∩ G_∗ is dense in G_∗
= 1.

In particular, E ∩ E⁰∩ G_∗ 6= ∅ a.s. That is, P(E ∩ E⁰∩ G 6= ∅) = 1. This proves the first part of Theorem 1.3.

In order to prove the second half, we regard the set E ∩ G as the target set with respect to the random covering set E⁰. By Theorem 1.1, we know that P(E⁰∩E∩G 6=

∅) = 1 implies dim_P(E ∩ G) ≥ 1 − α⁰ a.s. Therefore, from the above we see that dim_P(G) > 1 − min{α, α⁰} implies dim_P(E ∩ G) ≥ 1 − α⁰ a.s.

Now we assume dim_P(G) > 1 − α. For any α⁰with 1 − dim_P(G) < α⁰ < α, that is, dim_P(G) > 1 − min{α, α⁰}, we have dim_P(E ∩ G) ≥ 1 − α⁰ a.s. Letting α⁰ tend to 1 − dim_P(G) along rational numbers, we obtain

dim_P(E ∩ G) ≥ dim_P(G) a.s.

Therefore, dim_P(E ∩ G) = dim_P(G) a.s. 

Proof of Theorem 1.4. Firstly, we prove the right-hand inequality in (1.6).

By (0.7), it suffices to prove that

(2.14) dim_H(E ∩ G) ≤ dim_M(G) − (1 − α) a.s.

Denote by Cl_n a collection of the smallest number of the intervals with length ln, the union of such intervals covers the set G. Let Nl_n(G) = #Cl_n. Since ξ :=

dim_M(G) ≥ lim sup

n→∞

log N_ln(G)

− log ln , we have

Nln(G) < l^−(ξ+ε)_n

as n large enough, say n ≥ n₁(ε), where ε > 0 is an arbitrary small real number.

LetGn be the collection of the intervals J ∈ C_ln such that J ∩ I_n 6= ∅ and denote T_n= #Gn. For any J ∈Gn, P(In∩ J 6= ∅) ≤ 3ln. Thus

E(Tn) ≤ X

J ∈Gn

P(In∩ J 6= ∅) ≤ 3Nln(G)l_n ≤ 3l_n^1−ξ−ε.

For any θ > ξ − (1 − α), we choose ε > 0 such that 2ε < θ − ξ + (1 − α), then

E

 ^∞

X

n=n₁(ε)

Tnl_n^θ



< 3

∞

X

n=n₁(ε)

l^1−ξ−ε_n l^{ξ−(1−α)+2ε}_n = 3

∞

X

n=n₁(ε)

l^α+ε_n < ∞.

Thus E(P∞

n=1T_nl^θ_n) < ∞. It follows thatP∞

n=1T_nl^θ_n< ∞ a.s.

For any m ≥ 1, the collection {J ∈Gn}_n≥m is a covering of the set E ∩ G, then H^θ E ∩ G
≤

∞

X

n=m

Tnl^θ_n< ∞ a.s.,

which implies dim_H(E ∩ G) ≤ θ a.s. Since θ > dim_M(G) − (1 − α) is arbitrary, this proves that (2.14) holds.

The left-hand inequality in (1.6) can be derived from Theorem 1.1 and the following Lemma, due to Khoshnevisan, Peres, and Xiao [24] (Lemma 3.4 with N = 1 and γ = 1 − α). The proof of Theorem 1.3 completed.  Lemma 2.2. Equip [0, 1] with the Borel σ-field. Suppose E = E(ω) is a random set in [0, 1] (i.e., the indicator function χ_E(ω)(x) is jointly measurable) such that for any compact set F ⊂ [0, 1] with dim_H(F ) > γ, we have P(E ∩ F 6= ∅) = 1.

Then, for any analytic set F ⊂ [0, 1],

dim_H(F ) − γ ≤ dim_H(E ∩ F ) a.s.

3. Technical results

The upper Besicovitch-Taylor index (or the convergence exponent) of {ln} plays an essential role in this paper. In this section we provide some equivalent charac- terizations for this index and elaborate more on the condition (C) and (1.4).

First we show that

Proposition 3.1. For any constant a > 1, let nk = #{n ∈ N : ln∈ [a^−k+1, a^−k+2)}.

Then

(3.1) α = lim sup

k→∞

log_an_k k . Proof. For any γ > α, we have P∞

n=1l^γ_n < ∞ orP∞

k=1n_ka^−γ(k−1) < ∞.

Thus

nka^−γ(k−1) ≤ 1 for all k large, which implies

lim sup

k→∞

log_an_k k ≤ γ.

Hence we have

lim sup

k→∞

log_ank

k ≤ α.

On the other hand, if γ > lim sup_k→∞ ^loga_k^nk, we choose γ⁰ such that lim sup

k→∞

log_ank

k < γ⁰ < γ.

This implies nk≤ a^kγ0 for all k large enough, say k ≥ k0. Hence

∞

X

k=k₀

n_ka^−kγ ≤

∞

X

k=k₀

a^{−k(γ−γ0)}< ∞.

This implies α ≤ γ, which proves α ≤ lim sup_k→∞^loga_k^nk. Therefore (3.1) holds.  For any decreasing sequence {ln} of positive numbers such that

∞

P

n=1

ln < ∞, one can associate the following Dirichlet series

ζ(s) =

∞

X

n=1

l_n^s =

∞

X

n=1

e^{−s ln(l−1n )},

which is called the geometric zeta function in Lapidus and van Franhenhuysen [25]. Then the upper Besicovitch-Taylor index α defined by (0.4) is the abscissa of convergence of the above Dirichlet series. On the other hand, denote by N (x) the counting function defined by

N (x) = #n

n : l⁻¹_n ≤ xo ,

see [25, p.8]. Then n_kin Proposition 3.1 can be written as n_k= N (a^k−1)−N (a^k−2) for a > 1, hence the index α can also be determined by N (x) (we take a = 2):

(3.2) α = lim sup

k→∞

log₂ N (2^k−1) − N (2^k−2)

k .

Thanks to the above we can also apply the results in [25] to calculate the upper Besicovitch-Taylor index of a sequence {l_n}. In the following we focus on sequences which are associated to self-similar sets (or self-similar strings in [25]).

Given an integer M ≥ 2 and constants r₁, . . . , r_M ∈ (0, 1) such that 1 ≥ r₁≥ r2≥ · · · ≥ rM > 0 and R =

M

X

i=1

r_i< 1,

one can construct self-similar sets in [0, 1] with scaling ratios r1, . . . , rM (cf. [6, 25, 28]). Similarly to the tertiary Cantor set in Section 2, we denote the corresponding sequence by {ln}, where each ln is of the form

r₁^k1· · · r^k_M^M, where k1, . . . , kM ∈ N.

It can be verified that the multiplicity of the length r^k₁¹· · · r_M^kMin {ln} is the multino- mial coefficient _k ^q

1··· k_M
, where q = P^M

i=1

ki; see [25, p.24].

By the proof of Theorem 2.3 in [25] we see that the geometric zeta function of {ln} is

(3.3) ζ(s) =

∞

X

q=0

 ^M X

i=1

r^s_i

^q

, ∀s ∈ C.

This, together with (0.4), implies the first assertion of Proposition 3.2 below.

The asymptotic behavior of the counting function N (x) for a sequence {ln} associated to a self-similar set has been studied in [25] (see also the references therein for further information). We notice that the zeta function ζ(s) in (3.3)

satisfies conditions (H1) and (H⁰₂) in [25, p.80] with κ = 0 and A = rM (see [25, pp.121–122]). Hence we can apply Theorem 4.8 in [25, p.88] to obtain that

(3.4) N (x) = X

ω∈D(C)

res x^sζ(s) s ; ω



+ constant

for all x > rM. In the above D(C) denotes the set of complex dimensions of {lⁿ} (i.e., the set of poles of ζ(s) or equivalently the set of solutions of the equation PM

i=1r_i^ω= 1) and res(g(s); ω) denotes the residue of a meromorphic function g(s) at s = ω.

To obtain more explicit information about the terms on the right hand side of (3.4), we distinguish two cases:

Nonlattice case: The additive group generated by log r1, . . . , log rM is dense in R.

Lattice case: There exists some number δ > 0 such that log r1, . . . , log rM ∈ δZ. The largest such δ is called the additive generator and is denoted by r [25, p.34]. The positive constant p = (2π)/(log r⁻¹) is called the oscil- latory period.

In the nonlattice case, it follows from (5.44) in [25, p.126] that (3.5) N (x) = res(ζ; α)x^α

α + o(x^α), as x → ∞.

The lattice case is much simpler since the complex dimension of {ln} are located on finitely many vertical lines [25, Theorem 2.13]. It follows from (5.33) and (5.34) in [25, pp.122-123] that

(3.6) N (x) = res(ζ; α)b^1−{u}

b − 1 2π

p x^α+ o(x^α), as x → ∞,

where log b = 2πα/p, u = p log x/2π, {x} = x − bxc is the fractional part of x.

By (3.5), (3.6) and (3.2) we derive

(3.7) lim

k→∞

log₂ N (2^k−1) − N (2^k−2)

k = α.

In other words, (1.4) always holds for a self-similar sequence {ln}.

Hence we have proved the following proposition.

Proposition 3.2. Let {lⁿ} be the sequence associated to a self-similar set with scaling ratios r1, . . . , rM. Then the upper Besicovitch-Taylor index α of {ln} coin- cides with the self-similarity dimension D, which is the unique constant satisfying

M

X

i=1

r_i^D= 1.

Moreover, (1.4) holds.

As an example, we mention the Fibonacci sequence, which is obtained by taking M = 2, r₁ = 1/2 and r₂= 1/4. Then it can be verified directly that α = log₂φ, where φ = ¹⁺

√5

2 is the golden ratio, and its geometric counting function is given by

NFib(x) = 3 + 4φ

5 φ^−{log2x}x^α− 1 + 7 − 4φ

5 φ^{log2x}x^−α(−1)^blog2xc, see [25, p.124]. It can be verified directly that (1.4) holds.

Finally we show that condition (1.4) can be replaced by (1.5), as stated in Remark 1.2 (iv).

Proposition 3.3. For any constants a > b > 1, let mk = #n : ln ∈ [b^−k+1, b^−k+2)

and let nk = #n : ln ∈ [a^−k+1, a^−k+2) . If lim

k→∞

log_bmk

k = α, then

lim

k→∞

log_ank

k = α.

Proof. We state the elementary fact that if lim

k→∞

log_bmk

k = α, then for any fixed integer τ₀≥ 1, we have

(3.8) lim

k→∞

log_b(m_k+ m_k+1+ · · · + m_k+τ0)

k = α.

This can be verified by the fact that b^(α−)k< mk< b^(α+)k for all k large implies b^(α−)k< mk+ mk+1+ · · · + mk+τ0 < (τ0+ 1)b^{(α+)(k+τ0)}

for all k large.

To prove the lemma, observe that

ln ∈ [a^−k+1, a^−k+2) ⇐⇒ ln ∈ [b^{−(logba)(k−1)}, b^{−(logba)(k−2)}).

Hence

n_k ≤ #n : ln∈ [b^{−b(logba)(k−1)c−1}, b^{−b(logba)(k−2)c})

= m_b(logba)(k−1)c+2+ · · · + m_b(logba)(k−2)c+2, (3.9)

where bxc denotes the largest integer ≤ x, and note that a > b, we have nk ≥ #n : ln∈ [b^{−b(logba)(k−1)c}, b^{−b(logba)(k−2)c−1})

= m_b(log

ba)(k−1)c+1+ · · · + m_b(log

ba)(k−2)c+3. (3.10)

Since limk→∞

log_bmb(logb a)kc

(log_ba)k = α, we derive from (3.8), (3.9) and (3.10) that lim

k→∞

log_an_k

k = lim

k→∞

log_bn_k (log_ba)k = α.

This proves the lemma. 

Acknowledgement. This paper was developed and finished when Bing Li did his post-doc research at National Taiwan University, under a grant from NCTS Taipei Office, and during his visit to Michigan State University. The hospitality of the hosts is appreciated. The authors thank Prof. Ai Hua Fan for his helpful comments.

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Department of Mathematics, South China University of Technology. 510640, Guangzhou, P. R. China.

E-mail address: libing0826@gmail.com

Mathematics Department, Honorary Faculty, National Taiwan University. Taipei 10617, Taiwan.

E-mail address: shiehnr@ntu.edu.tw

Department of Statistics and Probability, 619 Red Cedar Road, Michigan State University, East Lansing, MI 48824, U.S.A.

E-mail address: xiao@stt.msu.edu