THE DIMENSION OF SETS DETERMINED BY THEIR CODE BEHAVIOR

WENXIA LI

Abstract. By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile an interesting decomposition of Moran fractals are given.

AMS Subject Classifications: 28A80, 28A78.

1. Introduction

Fractal dimensions such as Hausdorff, box and packing dimensions are introduced to measure the sizes of fractal sets and are employed in many different disciplines. Unfortunately, it is very difficult to determine the exact fractal dimensions of general fractal sets. Some results on fractal dimensions are obtained for those fractal sets with a special structure. Among them is a typical fractal structure termed as Moran set or Moran fractal. The Moran fractal is an extension of the self-similar set with separation property. The latter is in fact a map-specified Moran fractal. As a fractal set the various dimensions of the Moran fractal have been determined. One turns to studying its various subsets. These subsets generally play an important part in the study of fractals, e.g., they act as the multifractals (ref. [1]), the set of normal numbers and the set of the real numbers in [0, 1] for which the digits in their decimal expansion has a prescribed frequency (ref. [4]), the set of non-differentiability points of Cantor function (ref. [2], [3]), etc. Moran fractal is constructed in a good way so that it can be well encoded. Let Ω = {0, 1, ⋯, r} where r is a positive integer. We write Ω^{ω} = {σ = (σ(1), σ(2), ⋯) : σ(j) ∈ Ω} and Ω^{∗} = ⋃_{k=1}^{∞}Ω^{k} with Ω^{k} = {σ = (σ(1), σ(2), ⋯, σ(k)) : σ(j) ∈ Ω} for k ∈ N. |σ| is used to denote the length of word σ ∈ Ω^{∗}. For any σ, τ ∈ Ω^{∗} write σ ∗ τ = (σ(1), ⋯, σ(|σ|), τ(1), ⋯, τ(|τ|)), and write τ ∗ σ = (τ(1), ⋯, τ(|τ|), σ(1), σ(2), ⋯) for any τ ∈ Ω^{∗}, σ ∈ Ω^{ω}. For σ ∈ Ω^{k}, let C(σ) =

Supported by the National Natural Science Foundations of China, #10071027 .

2 WENXIA LI

{τ ∈ Ω^{ω} : τ|k = σ} where τ|k = (τ(1), τ(2), ⋯, τ(k)). C(σ) is termed as the cylinder set with base σ. Fixing a nonempty compact set J ⊂ R^{n} with \overline{intJ} = J and positive real numbers 0 < a_{j} < 1, j = 0, 1, ⋯, r. First choose a family \{J_{j} : j ∈ Ω\} of nonoverlapping nonempty compact subsets of J such that \overline{intJ_{j}}=J_{j} and |J_{j}| = a_{j}|J| where | · | denotes the diameter of set. Suppose that J_{σ} is given for some σ ∈ Ω^{k}. Take a family {J_{σ∗j} : j ∈ Ω} of nonoverlapping nonempty compact subsets of J_{σ} such that \overline{intJ_{σ∗j}} = J_{σ∗j}, |J_{σ∗j}| = a_{j}|J_{σ}|. We assume that there is a constant 0 < c < 1 such that each J_{σ} contains an open ball of diameter c|J_{σ}|. The Moran fractal F associated with {0 < a_{j} < 1, j ∈ Ω} and the J_{σ}, σ ∈ Ω^{∗} is defined as the nonempty compact set

F = ⋂_{k=1}^{∞} ⋃_{σ∈Ω^{k}} J_{σ}. (1)

We shall refer to J_{σ} as a k-th level component set of F if σ ∈ Ω^{k}. Define ϕ: Ω^{ω} ⟶ R^{n} by

{ϕ(σ)} = ⋂_{k=1}^{∞} J_{σ|k}. (2)

It is easy to see that ϕ(Ω^{ω}) = F and ϕ(C(σ)) = F ⋂ J_{σ} by (1) and (2). But ϕ may not be an injection. An important property of ϕ is that there is positive constant c_{1}, independent of x ∈ F, such that

sup_{x∈F} #{ϕ^{−1}(x)} < c_{1}, (3)

by means of Lemma 9.2 in [5]. Thus each x ∈ F can be encoded via ϕ: σ ∈ Ω^{ω} is called a location code of x ∈ F if ϕ(σ) = x. Note that each x ∈ F has at most c_{1} codes by (3). In this paper sometimes we use (x(1), x(2), ⋯) ∈ Ω^{ω} to denote a specified location code of x ∈ F if no confusion arises. A Moran fractal is termed map-specified if there exist similitude contractions h_{j}(x) : R^{n} ⟶ R^{n}, j = 0, 1, ⋯, r, such that J_{σ} = h_{σ}(J) for any σ ∈ Ω^{∗} where h_{σ}(x) = h_{σ(1)} ∘ ⋯ ∘ h_{σ(|σ|)}(x). In this case F is actually the self-similar set determined by {h_{j} : j ∈ Ω}, which satisfies the open set condition with respect to the open set O = intJ, i.e., ⋃_{j=0}^{r} h_{j}(O) ⊆ O with a disjoint union on the left, and the coding map ϕ in (2) can be rewritten as

\{ϕ(σ)\} = ⋂_{k=1}^{∞} h_{σ|k}(\overline{O}) = \{ lim_{k→∞} h_{σ|k}(0)\}. (4)

Now let Γ ⊆ Ω = {0, 1, ⋯, r} be non-empty such that Γ^{c} ≠ ∅. For n ∈ N and σ ∈ Ω^{ω}, we define N_{n}(σ), the length of the Γ-run starting at n, by

N_{n}(σ) = \{ \begin{matrix}0, & if σ(n) ∉ Γ \\ k, & if σ(j) ∈ Γ for j = n, n + 1, ⋯, n + k − 1, and σ(n + k) ∉ Γ \\ +∞, & if σ(k) ∈ Γ for k ≥ n. \end{matrix}

Hence for each σ ∈ Ω^{ω} the sequence {N_{n}(σ)} characterize its behavior. Let {Q(n)} be a sequence of real numbers such that Q(n) ↑ +∞ as n ⟶ +∞ and lim_{n→+∞} \frac{Q(n)}{n} = α ∈ [0, +∞]. Let

(5) Λ = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{N_{n}(σ)}{Q(n)} = 1\} and M = ϕ(Λ).

It is easy to see that M is dense in F since Λ is dense in Ω^{ω}. The Hausdorff dimension of M is expected depending on α. For the extreme case when α = 0, it was proved in [8] that

dim_{H} M = dim_{P} M = dim_{B} M = dim_{H} F = s,

where s is given by

∑_{j∈Ω} a_{j}^{s} = 1. (6)

Throughout this paper the positive real number s is always assumed giving in this equation. While for the general case when α ∈ (0, +∞), we have

dim_{H} M = η̃ and dim_{P} M = dim_{B} M = dim_{H} F = s,

where η̃ is determined by

(7) \frac{1}{1 + α} log ∑_{j∈Ω} aη̃ ; j + \frac{α}{1 + α} log ∑_{j∈Γ} aη̃ ; j = 0.

This can be easily derived from the result in [9]. In this paper we discuss the fractal dimensions of M for the another extreme case when α = +∞. We obtain Theorem. Let {Q(n)} be a sequence of real numbers such that Q(n) ↑ +∞ as n ⟶ +∞ and lim_{n→+∞} \frac{Q(n)}{n} = +∞. Let M be defined by (5). Let η be given by

∑_{j∈Γ} a_{j}^{η} = 1. (8)

Then dim_{H} M = η with H^{η}(M) = +∞, dim_{B} M = dim_{P} M = dim_{H} F = s with s given in (6), and M is dense in F.

Summarizing above, the dim_{H} M for all α ∈ [0, +∞] can be given by the unified formula (7) if we adopt the convention that \frac{1}{∞} = 0 and \frac{∞}{∞} = 1. Taking Q(n) = tn^{2} in (5), we can get F_{t} ⊆ F for t ∈ (0, +∞) by

(9) Λ_{t} = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{N_{n}(σ)}{n^{2}} = t\} and F_{t} = ϕ(Λ_{t}).

Then we can obtain a decomposition of F, i.e., F = ⋃_{t∈[0,+∞]} F_{t} by allowing t = 0, +∞ in (9). These F_{t} have the following properties and will be verified at the end of next section:

(P1) F = ⋃_{t∈[0,+∞]} F_{t} with sup_{x∈F} #{t : x ∈ F_{t}} < c_{1} by (3), i.e., each x ∈ F

belongs to at most c_{1} distinct F_{t}; (P2) Each F_{t} is dense in F; (P3) dim_{H} F_{t} = η for t ∈ (0, ∞], dim_{H} F_{0} = s. dim_{P} F_{t} = dim_{B} F_{t} = s for

t ∈ [0, ∞]; (P4) F_{0} is an s-set, while H^{η}(F_{t}) = ∞ for t ∈ (0, ∞]; (P5) Moreover, if F is map-specified with similitudes h_{j}, j = 0, 1, ⋯, r, i.e., a

self-similar set, then each F_{t} is invariant under {h_{j} : j = 0, 1, ⋯, r}, i.e.,

F_{t} = ⋃_{j=0}^{r} h_{j}(F_{t}).

On the other hand, the F_{t} can be considered as the level sets of the measurable function Y (σ) = lim sup_{n→∞} \frac{N_{n}(σ)}{n^{2}}: Ω^{ω} ⟶ [0, +∞]. It can be employed to construct some very interesting functions, for example, for any 0 < η < 1 a measurable function g(x) : [0, 1] ⟶ [0, +∞] can be made such that for any t ∈ (0, +∞] the t-level set of g(x) is dense in [0, 1] and has Hausdorff dimension η.

DIMENSION OF SETS DETERMINED BY CODE 5

2. Proofs

In this section, a general dimension result on a class of subsets of Moran sets is first obtained. Then this result will be applied to give a decomposition of Moran sets. The following proposition will be employed. Proposition. Let A = ϕ(∏_{i=1}^{∞} Ω_{i}) where the Ω_{i} are non-empty subsets of Ω = {0, 1, ⋯, r}, i ∈ N. Let d(k) be such that

∏_{i=1}^{k}( ∑_{j∈Ω_{i}} a_{j}^{d(k)}) = 1.

Then dim_{H} A = lim inf_{k→∞} d(k) =^{Δ} \underline{d} (see [6]). Proof This result can be found in [6] for the more general Moran fractal structure. Here a simplified proof is given for this special case. For any d > \underline{d}, there exists a sequence {n_{k} : k = 1, 2, ⋯ } such that d(n_{k}) < d. For any δ > 0, we can take k large enough such that {J_{σ} : σ ∈ ∏_{i=1}^{n_{k}} Ω_{i}} is a δ-covering of A. Then

H_{δ}^{d}(A) ≤ ∑ ; σ∈∏_{i=1}^{n_{k}} Ω_{i} |J_{σ}|^{d} = |J|^{d} ∏_{i=1}^{n_{k}}( ∑_{j∈Ω_{i}} a_{j}^{d}) ≤ |J|^{d} ∏_{i=1}^{n_{k}}( ∑_{j∈Ω_{i}} a_{j}^{d(n_{k})}) = |J|^{d},

which implies that dim_{H} A ≤ d. So we have dim_{H} A ≤ \underline{d}. Now we turn to prove that dim_{H} A ≥ \underline{d}. Let us suppose without loss of generality that \underline{d} > 0. Then we only need to prove that for any fixed 0 < d < \underline{d}, dim_{H} A ≥ d. Let us construct a probability measure μ̃ on S^{ω} =^{Δ} ∏_{i=1}^{∞} Ω_{i} such that for any

σ ∈ S^{k} =^{Δ} ∏_{i=1}^{k} Ω_{i}, k = 1, 2, ⋯

μ̃(C^{∗}(σ)) = \frac{∏_{i=1}^{k} a_{σ(i)}^{d}}{∏_{i=1}^{k}(∑_{j∈Ω_{i}} a_{j}^{d})},

where C^{∗}(σ) =^{Δ} {θ ∈ S^{ω} : θ|k = σ} is a cylinder in S^{ω} with base σ. Let μ be the image measure under ϕ defined in (2) and restricted to S^{ω} here. For ϵ > 0,

write S_{ϵ} = {σ ∈ ⋃_{k=1}^{∞} S^{k} : a_{σ} =^{Δ} ∏_{i=1}^{|σ|} a_{σ(i)} ≤ ϵ and a_{σ|(|σ|−1)} > ϵ}. For any

x ∈ A, ϵ > 0, by B(x, ϵ) we denote the closed ball with center at x and radius ϵ.

Let L_{x,ϵ} =^{Δ} {σ ∈ S_{ϵ} : J_{σ} ⋂ B(x, ϵ) ≠ ∅}. Since A = ⋃_{σ∈S_{ϵ}}(A ⋂ J_{σ}) and J_{σ}, σ ∈ S_{ϵ}

are pairwise nonoverlapping, there exists a finite constant c̄ independent of x, ϵ such that 1 ≤ #L_{x,ϵ} ≤ c̄. Thus

μ(B(x, ϵ)) ≤ ∑_{σ∈L_{x,ϵ}} μ̃(C^{∗}(σ)) = ∑_{σ∈L_{x,ϵ}} \frac{∏_{k=1}^{|σ|} a_{σ(k)}^{d}}{∏_{i=1}^{|σ|}(∑_{j∈Ω_{i}} a_{j}^{d})} ≤ c̄ϵ^{d},

6 WENXIA LI

where for the last inequality we use that ∏_{i=1}^{|σ|}(∑_{j∈Ω_{i}} a_{j}^{d}) ≥ 1 when ϵ is small enough, since d < \underline{d}. Then by Frostman’s lemma (ref. [5]) we obtain H^{d}(A) > 0 which implies that dim_{H} A ≥ d. QED Remark. Some further results on the set A defined in Proposition are: \overline{dim}_{B}A = dim_{P} A = lim sup_{k→∞} d(k) (see [7]), and that 0 < H\underline{d}(A) < +∞ if and only if 0 < lim inf_{k→∞} ∏_{i=1}^{k}(∑_{j∈Ω_{i}} a\underline{d} ; j) < +∞ (see [6] & [10]).

Proof of Theorem. (I) We prove dim_{H} M = η. At first we show dim_{H} M ≤ η. To do this, we prove below that dim_{H} M ≤ d for any η < d < s. Now fix arbitrarily d with η < d < s. Then we have

∑_{j∈Γ} a_{j}^{d} < 1 and ∑_{j∈Ω} a_{j}^{d} > 1.

Since lim_{k→∞} \frac{Q(k)}{k} = ∞, we can take k^{∗} such that when k ≥ k^{∗}

(10) \frac{[0.5Q(k)]}{k} ≥ \frac{−2 log k}{k log(∑_{j∈Γ} a_{j}^{d})} − (1 − \frac{1}{k} ) \frac{log(∑_{j∈Ω} a_{j}^{d})}{log(∑_{j∈Γ} a_{j}^{d})},

where for a real number a, [a] is the maximal integer not more than a throughout this paper. For k ≥ k^{∗}, take

u_{k} = k − 1 + [0.5Q(k)], (11)

and

E_{k} = {x ∈ F : x(i) ∈ Γ for k ≤ i ≤ u_{k}}. (12)

Thus from the definitions of M and E_{k} in (5) and (12) it follows

M ⊆ lim sup_{k→∞} E_{k} = ⋂_{m=k^{∗}}^{∞} ⋃_{k≥m} E_{k} =^{Δ} E^{∞}.

Since each E_{k} can be covered with {J_{σ∗τ} : σ ∈ Ω^{k−1} and τ ∈ Γ^{u_{k}−k+1}}, for any m ≥ k^{∗} we have

H^{d}(E^{∞}) ≤ H^{d}( ⋃_{k≥m} E_{k}) ≤ |J|^{d} ∑_{k≥m} (∑_{j∈Ω} a_{j}^{d})^{k−1} (∑_{j∈Γ} a_{j}^{d})^{u_{k}−k+1} .

Here we declare that for k ≥ k^{∗} we have

(∑_{j∈Ω} a_{j}^{d})^{k−1} (∑_{j∈Γ} a_{j}^{d})^{u_{k}−k+1} ≤ k^{−2},

which is equivalent to

(13) \frac{u_{k}}{k} ≥ \frac{−2 log k}{k log(∑_{j∈Γ} a_{j}^{d})} + (1 − \frac{1}{k} ) − (1 − \frac{1}{k} ) \frac{log(∑_{j∈Ω} a_{j}^{d})}{log(∑_{j∈Γ} a_{j}^{d})}.

But (13) holds by (10) and (11). Therefore we have H^{d}(E^{∞}) ≤ |J|^{d} ∑_{k≥m} k^{−2} ⟶ 0 as m ⟶ ∞. Without loss of generality we assume η > 0. To show dim_{H} M ≥ η, we prove below that for arbitrarily fixed 0 < d < η, there exists a subset E of M such that dim_{H} E ≥ d. Let the positive constant c_{2} be defined by

c_{2} =^{Δ} − log ∑_{j∈Γ^{c}} a_{j}^{s}. (14)

Consider strictly decreasing continuous function

G(x) = log ∑_{j∈Γ} a_{j}^{x}, 0 ≤ x ≤ s.

Let 0 < ϵ < 1 be defined by \frac{c_{2}ϵ}{1−ϵ} = G(d). Now define a sequence of positive integers b_{i}, i ∈ N, by

b_{i+1} = b_{i} + [Q(b_{i} + 1)]. (15)

Here we take b_{1} large enough to ensure Q(k) ≥ 2 whenever k ≥ b_{1}. Thus b_{i} > i. So the b_{i} increase strictly and tend to +∞. Construct a set E as follows:

E = {x ∈ F : all x(k) ∈ Γ but x(b_{i}) ∈ Γ^{c}, i = 1, 2, ⋯, }.

Now for any k ∈ N there is an i with b_{i} ≤ k < b_{i+1}. Thus for any σ ∈ Ω^{ω} with ϕ(σ) ∈ E, because N_{k}(σ) ≤ N_{b_{i}+1}(σ) = b_{i+1} − b_{i} − 1, N_{b_{i}}(σ) = 0 and Q is non-decreasing we have

lim sup_{k→∞} \frac{N_{k}(σ)}{Q(k)} ≤ lim sup_{i→∞} \frac{N_{b_{i}+1}(σ)}{Q(b_{i} + 1)} = lim sup_{i→∞} \frac{b_{i+1} − b_{i} − 1}{Q(b_{i} + 1)} = 1

by (15). Therefore E ⊆ M and by Proposition dim_{H} E = lim inf_{k→∞} d(k) where d(k) with b_{i} ≤ k < b_{i+1} is determined by

( ∑_{j∈Γ^{c}} a_{j}^{d(k)})^{i} (∑_{j∈Γ} a_{j}^{d(k)})^{k−i} = 1. (16)

Thus 0 ≤ d(k) ≤ s Note that

lim_{i→∞} \frac{i}{b_{i}} = lim_{i→∞} \frac{i − (i − 1)}{b_{i} − b_{i−1}} = lim_{i→∞} \frac{1}{Q(b_{i−1} + 1)} = 0.

Thus we can take i^{∗} such that when i ≥ i^{∗} we have \frac{i}{b_{i}} < ϵ. As a result, when k ≥ b_{i^{∗}} from (16) and (14) it follows

G(d(k)) = log ∑_{j∈Γ} a_{j}^{d(k)} = \frac{−i}{k − i} log ∑_{j∈Γ^{c}} a_{j}^{d(k)} ≤ \frac{c_{2}i}{k − i} ≤ \frac{c_{2}ϵ}{1 − ϵ} = G(d).

So we have d(k) ≥ d, implying dim_{H} E = lim inf_{k→∞} d(k) ≥ d. (II) We prove dim_{B} M = dim_{P} M = s. The dim_{B} M = dim_{B} F = s can be derived from the density result proved later. Thus it suffices to prove for any given 0 < d < s, dim_{P} M ≥ d. Let the positive constant c_{3} be defined by

(17) c_{3} =^{Δ} max{log #Γ, log #Γ^{c}, | log ∑_{j∈Γ} a_{j}^{s}|, | log ∑_{j∈Γ^{c}} a_{j}^{s}|}.

Consider the non-negative strictly decreasing function

G^{∗}(x) = log ∑_{j∈Ω} a_{j}^{x}, 0 ≤ x ≤ s.

Let 0 < ϵ < 1 be defined by \frac{3c_{3}ϵ}{1−ϵ} = G^{∗}(d). Take the sequence of integers 0 < k_{1} < u_{1} < u_{1,1} < ⋯ < u_{1,n_{1}} < k_{2} < u_{2} < ⋯ < k_{i} < u_{i} < u_{i,1} < u_{i,2} < ⋯ < u_{i,n_{i}} < k_{i+1} < u_{i+1} < ⋯ as {b_{i}, i = 1, 2, ⋯ } defined in (15) and construct a set E as follows:

(18) E = {x ∈ F : x(b_{i}) ∈ Γ^{c} and x(k) ∈ Γ for k_{i} < k < u_{i}, i ≥ 1}.

Obviously for any choice of {n_{i}, i = 1, 2, ⋯ }, the set E is a closed set with E ⊆ M ⊆ F and is a generalized Moran fractal described in Proposition since E = ϕ(∏_{i=1}^{∞} Ω_{i}) with Ω_{b_{i}} = Γ^{c}, Ω_{k} = Γ when k_{i} < k < u_{i} and Ω_{i} = Ω for the rest. Denote

N_{Ω}(k) =^{Δ} #{i : Ω_{i} = Ω, 1 ≤ i ≤ k}, N_{Γ}(k) =^{Δ} #{i : Ω_{i} = Γ, 1 ≤ i ≤ k}

DIMENSION OF SETS DETERMINED BY CODE 9

and

N_{Γ^{c}}(k) =^{Δ} #{i : Ω_{i} = Γ^{c}, 1 ≤ i ≤ k}.

Thus we have that N_{Ω}(k) + N_{Γ}(k) + N_{Γ^{c}}(k) = k for k ∈ N. Note that for k_{i} − 1 ≤ k ≤ u_{i}, N_{Ω}(k) = N_{Ω}(k_{i} − 1). For convenience we put

f_{i} = N_{Ω}(k_{i} − 1).

We shall now make a choice for {n_{i}, i = 1, 2, ⋯ }, based on the previously defined ϵ. Suppose that the n_{ℓ} are defined for ℓ = 1, 2, ⋯, i − 1, then also k_{i} and u_{i} are determined. Letting n_{i} vary, we have

lim_{n_{i}→∞} \frac{N_{Ω}(u_{i,n_{i}})}{u_{i,n_{i}}} = lim_{n_{i}→∞} \frac{N_{Ω}(u_{i}) + u_{i,n_{i}} − u_{i} − n_{i}}{u_{i,n_{i}}}
= 1 − lim_{n_{i}→∞} \frac{n_{i}}{u_{i,n_{i}}} = 1 − lim_{n_{i}→∞} \frac{n_{i} − (n_{i} − 1)}{u_{i,n_{i}} − u_{i,n_{i}−1}}
= 1 − lim_{n_{i}→∞} \frac{1}{[Q(u_{i,n_{i}−1} + 1)]} = 1.

Therefore we can choose n_{i} such that

f_{i+1} = N_{Ω}(k_{i+1} − 1) ≥ (1 − ϵ)k_{i+1}. (19)

According to Proposition we have \overline{dim}_{B}E = dim_{P} E = lim sup_{k→∞} d(k) where d(k) satisfies

(∑_{j∈Ω} a_{j}^{d(k)})^{N_{Ω}(k)}( ∑_{j∈Γ^{c}} a_{j}^{d(k)})^{N_{Γ^{c}} (k)}(∑_{j∈Γ} a_{j}^{d(k)})^{N_{Γ}(k)} = 1. (20)

Also we have 0 ≤ d(k) ≤ s. Taking logs in (20), and using N_{Ω}(k)+N_{Γ}(k)+N_{Γ^{c}}(k) = k we get

log ∑_{j∈Ω} a_{j}^{d(k)} = −\frac{N_{Γ^{c}}(k)}{N_{Ω}(k)} log ∑_{j∈Γ^{c}} a_{j}^{d(k)}
(21)  − \frac{k − N_{Ω}(k) − N_{Γ^{c}}(k)}{N_{Ω}(k)} log ∑_{j∈Γ} a_{j}^{d(k)}.

Taking k = k_{i} in (21), we have

log ∑_{j∈Ω} a_{j}^{d(k_{i})} = −\frac{N_{Γ^{c}}(k_{i})}{f_{i}} (log ∑_{j∈Γ^{c}} a_{j}^{d(k_{i})} − log ∑_{j∈Γ} a_{j}^{d(k_{i})})
(22)  − \frac{k_{i} − f_{i}}{f_{i}} log ∑_{j∈Γ} a_{j}^{d(k_{i})}.

10 WENXIA LI

Note that

0 ≤ \frac{N_{Γ^{c}}(k_{i})}{f_{i}} = \frac{k_{i} − N_{Ω}(k_{i}) − N_{Γ}(k_{i})}{f_{i}} ≤ \frac{k_{i} − f_{i}}{f_{i}} = \frac{k_{i}}{f_{i}} − 1 ≤ \frac{ϵ}{1 − ϵ} (23)

by (19). Note that | log ∑_{j∈Γ^{c}} a_{j}^{d(k_{i})}| ≤ c_{3}, | log ∑_{j∈Γ} a_{j}^{d(k_{i})}| ≤ c_{3} by (17). Therefore

G^{∗}(d(k_{i})) = log ∑_{j∈Ω} a_{j}^{d(k_{i})} ≤ \frac{ϵ}{1 − ϵ} · 2c_{3} + \frac{ϵ}{1 − ϵ} · c_{3} = \frac{3c_{3}ϵ}{1 − ϵ} = G^{∗}(d),

by (22) and (23), which means that d(k_{i}) ≥ d. So we get \overline{dim}_{B}E = dim_{P} E ≥ d. (III) We prove H^{η}(M) = +∞. Let E be defined in (18). We show that H^{η}(E) = +∞. Denote G =^{Δ} ∏_{i=1}^{∞} Ω_{i} and

G^{∗} =^{Δ} {σ : σ ∈ G_{k}, k ≥ 1} with G_{k} =^{Δ} ∏_{i=1}^{k} Ω_{i}. Let Q = {J_{σ} : σ ∈ G^{∗}}. For α ≥ 0

define

H_{Q}^{α}(E) =^{Δ} lim_{δ→0^{+}} inf{ ∑_{σ} |J_{σ}|^{α} : {J_{σ}} is a finite nonoverlapping

(24) δ − covering ofE and J_{σ} ∈ Q}.

Then we have (see [6] [10])

c_{α}H_{Q}^{α}(E) ≤ H^{α}(E) ≤ H_{Q}^{α}(E), (25)

where c_{α} is a positive number depending on α. For σ ∈ G^{∗} denote

C_{G}(σ) =^{Δ} {τ ∈ G : τ∥σ| = σ}.

A finite nonoverlapping δ-covering {J_{σ} : σ ∈ T ⊆ G^{∗}} of E is called full if ⋃_{σ∈T} C_{G}(σ) = G. Note that a finite nonoverlapping covering of E may not be full. Let {J_{σ} : σ ∈ T ⊆ G^{∗}} be a finite nonoverlapping δ-covering of E. If it is not full, then for each σ ∈ T let

G_{σ}^{∗} = {τ ∈ G^{∗} : |J_{τ} | ≤ |J_{σ}| but |J_{τ|(|τ|−1)}| > |J_{σ}|, J_{τ} ⋂ J_{σ} ≠ ∅}.

Then there is a positive constant q independent of σ such that #G_{σ}^{∗} < q by means of Lemma 9.2 in [5]. Take D = (⋃_{σ∈T} G_{σ}^{∗}). By T̃ we denote the subset of D by deleting those τ ∈ D for which there is a γ ∈ D with |γ| < |τ| and τ||γ| = γ. Thus we get a full finite nonoverlapping δ-covering {J_{σ} : σ ∈ T̃ ⊆ G^{∗}} of E satisfying

∑_{σ∈T̃} |J_{σ}|^{η} ≤ ∑_{σ∈T} ∑_{τ∈G_{σ}^{∗}} |J_{τ} |^{η} ≤ q ∑_{σ∈T} |J_{σ}|^{η},

DIMENSION OF SETS DETERMINED BY CODE 11

i.e., for each a finite nonoverlapping δ− covering {J_{σ} : σ ∈ T ⊆ G^{∗}} of E there is a full finite nonoverlapping δ− covering {J_{σ} : σ ∈ T̃ ⊆ G^{∗}} of E such that

∑_{σ∈T} |J_{σ}|^{η} ≥ q^{−1} ∑_{σ∈T̃} |J_{σ}|^{η}, (26)

where the positive real number q is independent of the choice of the finite nonoverlapping δ− covering of E. On the other hand, for each full finite nonoverlapping δ-covering {J_{σ} : σ ∈ T̃ ⊆ G^{∗}} of E, let k_{1} = max_{σ∈T̃} |σ| and k_{2} = min_{σ∈T̃} |σ|. In [6] and [7], it states that there is k_{2} ≤ k ≤ k_{1} such that

∑_{σ∈T̃} |J_{σ}|^{η} ≥ ∑_{σ∈G_{k}} |J_{σ}|^{η} = |J|^{η} ∏_{i=1}^{k} ∑_{j∈Ω_{i}} a_{j}^{η}. (27)

From (24),(25),(26) and (27) it follows that we only need to prove

lim inf_{k→∞} ∏_{i=1}^{k} ∑_{j∈Ω_{i}} a_{j}^{η} = +∞. (28)

By the definitions of E and η in (18) and (8) we have

log ∏_{i=1}^{k} ∑_{j∈Ω_{i}} a_{j}^{η} = log ((∑_{j∈Ω} a_{j}^{η})^{N_{Ω}(k)}( ∑_{j∈Γ^{c}} a_{j}^{η})^{N_{Γ^{c}} (k)})
(29)  = N_{Ω}(k) (log ∑_{j∈Ω} a_{j}^{η} + \frac{N_{Γ^{c}}(k)}{N_{Ω}(k)} log ∑_{j∈Γ^{c}} a_{j}^{η}) .

Noting that lim_{k→∞} N_{Ω}(k) = +∞, ∑_{j∈Ω} a_{j}^{η} > 1 and lim_{k→∞} \frac{N_{Γ^{c}} (k)}{N_{Ω}(k)} = 0, thus (28) holds by (29). For the density result, it is derived directly from the fact that if σ ∈ Λ then for any k ∈ N those τ ∈ Ω^{ω} with τ(i) = σ(i), i ≥ k will lie in Λ. QED

As an application of the Theorem we can now prove (P1), (P2), (P3), (P4) and (P5) of the previous section. Proof of (P1)–(P5). By (3) and the definition (9) of Λ_{t} we have ⋃_{t∈[0,+∞]} Λ_{t} = Ω^{ω} which leads to (P1). (P2) can be got directly from the definition (9) of F_{t}. For t ∈ (0, ∞), we have dim_{H} F_{t} = η with H^{η}(F_{t}) = ∞ and dim_{P} F_{t} = dim_{B} F_{t} = s by taking Q(n) = tn^{2} in Theorem. If taking Q(n) = n^{3} in Theorem, we have F_{+∞} ⊇ M which implies dim_{H} F_{+∞} ≥ η with H^{η}(F_{+∞}) = ∞ and dim_{P} F_{+∞} = dim_{B} F_{+∞} = s. If taking Q(n) = n log n in Theorem, we have F_{0} ⊇ M, which

12 WENXIA LI

implies dim_{P} F_{0} = dim_{B} F_{0} = s. dim_{H} F_{+∞} ≤ η can be got from the fact F_{+∞} ⊆ lim sup_{k→∞} E_{k} with E_{k} being defined in (12) and Q(k) = k^{2} in (11). Finally some results in [8] imply that F_{0} is an s-set. Thus we get (P3) and (P4). Note that if σ ∈ Λ_{t} then for any j ∈ Ω we have j ∗ σ ∈ Λ_{t} since

lim sup_{n→∞} \frac{N_{n}(j ∗ σ)}{n^{2}} = lim sup_{n→∞} [\frac{N_{n−1}(σ)}{(n − 1)^{2}} · \frac{(n − 1)^{2}}{n^{2}} ] = t.

Thus Λ_{t} = ⋃_{j=0}^{r} j∗Λ_{t} where j∗Λ_{t} = {j∗σ : σ ∈ Λ_{t}}, which get (P5) by (4). QED

In the following example we give a measurable function g(x) : [0, 1] ⟶ [0, +∞] such that for any given 0 < η < 1 the t-level set, t ∈ (0, +∞], of g(x) is dense in [0, 1] and has Hausdorff dimension η. Example. Take r = 2, Γ = {0, 2} and J = [0, 1]. Take positive real numbers a_{0}, a_{1} and a_{2} such that ∑_{i=0}^{2} a_{i} = 1 and a_{0}^{η} + a_{2}^{η} = 1. Consider the map-specified Moran fractal F with h_{0}(x) = a_{0}x, h_{1}(x) = a_{1}x + a_{0} and h_{2}(x) = a_{2}x + a_{0} + a_{1}, x ∈ R^{1}. Then we have F = [0, 1] and s = 1. Note that each x ∈ F either has unique location code or has only two location codes in Ω^{ω} = {0, 1, 2} × {0, 1, 2} × · · ·. In the former case, the corresponding x only lies in one of the sets F_{t}. In the latter case, one of the two location codes only has components 0 except for finitely many components, the other only has components 2 except for finitely many components. So the corresponding x only lies in F_{∞}. Then (F_{t}) with t ∈ [0, +∞], defined in (9), is a partition of F (= [0, 1]), satisfying the properties of (P1)-(P5). Define function g : [0, 1] ⟶ [0, ∞] by

g(x) = t if x ∈ F_{t}.

Thus we have g^{−1}(t) = F_{t}. Then by Theorem we get a measurable function g : [0, 1] ⟶ [0, +∞] satisfying:

• each t-level set g^{−1}(t) ⊆ [0, 1] is dense in [0, 1], t ∈ [0, +∞];

• the t-level set g^{−1}(t) has Hausdorff dimension η and infinite Hausdorff measure in η for t ∈ (0, +∞], g^{−1}(0) is a 1-set and g^{−1}(t) have packing and box dimensions 1 for all t ∈ [0, +∞];

• g^{−1}(t) = ⋃_{i=0}^{2} h_{i}(g^{−1}(t)).

References

[1] R. Cawley & R. D. Mauldin, Multifractal decomposition of Moran fractals. Adv. in Math., 92(1992), 196–236.

[2] DARST R (1995) Hausdorff dimension of sets of non-differentiability points of Cantor functions. Math. Proc. Camb. Phi. Soc., bf 117: 185–191.

[3] DEKKING FM & LI WX (2001) How smooth is a devil’s staircase?. To appear in Fractals.

[4] H. G. Eggleston, The fractional dimension of a set defined by decimal properties. Quart. Journ. of Math. (Oxford), Vol. 20, March 1949,31–36.

[5] FALCONER KJ (1990) Fractal geometry–Mathematical foundations and applications. Chichester: John Wiley & Sons Ltd.

[6] HUA S (1994) On the Hausdorff dimension of generalized self-similar sets. Acta Mathematica Applicata Sinica (in chinese) 17: 551–558.

[7] HUA S & LI WX (1996) Packing Dimension Of Generalized Moran Sets. Progress In Natural Science 2: 148-152.

[8] LI WX & DEKKING FM (2000) The dimension of subsets of Moran sets determined by the success run behaviour of their codings. Monatshefte fur Mathematik, 131, 309–320..

[9] LI WX, XIAO DM & DEKKING FM (2002), Non-differentiability of devil’s staircases and dimensions of subsets of Moran fractals. Math. Proc. Camb. Phi. Soc., 133: 345–355.

[10] LI WX & XIAO DM (1998) A note on generalized Moran set. Acta Mathematica Scientia. 18(supp.): 88–93.

Department of Mathematics, East China Normal University, Shanghai 200062, P.R. China

E-mail address: wxli@math.ecnu.edu.cn