Non-differentiability of devil’s staircases

and dimensions of subsets of Moran sets

Wenxia Li, ^{∗} Dongmei Xiao

Department of Mathematics, Central China Normal University,

Wuhan 430079, P.R. China F.M. Dekking ^{†} Thomas Stieltjes Institute of Mathematics Delft University of Technology ITS (CROSS) Mekelweg 4, 2628 CD Delft, The Netherlands

Abstract

Let C be the homogeneous Cantor set invariant for x → ax and x → 1 − a + ax. It has been shown by Darst that the Hausdorff dimension of the set of non-differentiability points of the distribution function of uniform measure on C equals (dim_{H} C)^{2} = (log 2/ log a)^{2}. In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, ⋯, r}. Let F be a Moran set associated with {0 < a_{i} < 1, i ∈ Ω} and Ω^{ω} = Ω × Ω × · · · . Let ϕ be the associated coding map from Ω^{ω} onto F. Fix a nonempty set Γ ⊆ Ω with Γ^{c} ≠ ∅ and let z(σ, n) denote the position of the n-th occurrence of the elements of Γ in σ ∈ Ω^{ω}. For given 0 ≤ ξ ≤ 1, let

∗Supported by the National Science Foundation of China 10071027. Email:

wxli@mail.ccnu.edu.cn, dmxiao@mail.ccnu.edu.cn

†Email: f.m.dekking@its.tudelft.nl

Λ = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{z(σ, n + 1)}{z(σ, n)} = ξ^{−1}\} , F_{ξ} = ϕ(Λ),

and

Λ^{∗} = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{z(σ, n + 1)}{z(σ, n)} ≥ ξ^{−1}\} , F_{ξ}^{∗} = ϕ(Λ^{∗}).

We show that dim_{P} F_{ξ} = dim_{P} F_{ξ}^{∗} = dim_{B} F_{ξ} = dim_{B} F_{ξ}^{∗} = s with ∑_{j∈Ω} a_{i}^{s} = 1, and dim_{H} F_{ξ} = dim_{H} F_{ξ}^{∗} = η where η is such that

ξ log ∑_{j∈Ω} a_{j}^{η} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{η} = 0.

Key Words: Hausdorff dimension; Packing dimension; Moran set; AMS Subject Classifications: 28A80, 28A78.

1 Introduction

Let h_{i}(x) = ax + i(1 − a), i = 0, 1 with x ∈ [0, 1] and 0 < a < \frac{1}{2}. Then there exists a unique non-empty compact set C ⊂ [0, 1] such that

C = h_{0}(C) ⋃ h_{1}(C).

It is well-known that the Hausdorff dimension of C equals dim_{H} C = −\frac{log 2}{log a}. Let μ be the uniform probability measure on C. Consider the distribution function which is often referred to as the Devil’s staircase (for a = \frac{1}{3}):

F (x) = μ([0, x]), x ∈ [0, 1].

It is easy to check that the derivative of F (x) is zero for all x ∈ [0, 1] \ C and the upper derivative of F (x) is infinite on C. Let S be the set of points at which F (x) is not differentiable, i.e., the set of points in C at which the lower derivative of F (x) is finite. S can be decomposed into

S = N^{+} ⋃ N^{−} ⋃{t : t is an endpoint of C},  (1)

where N^{+}(N^{−}) is the set of non-end points of C at which the lower right (left) derivative of F (x) is finite. Each t ∈ C can be encoded in the usual way by a 0−1 sequence, denoted by {t}. Now let z(t, n) denote the position of the n-th zero in {t}. The set N^{+} (symmetrically for N^{−}) is characterized by R.Darst [1] as follows:

[a] if t ∈ N^{+}, then lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} ≥ −\frac{log a}{log 2};

[b] if lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} > −\frac{log a}{log 2}, then t ∈ N^{+}.

By means of the above [a] and [b] R. Darst [1] proves that

dim_{H} S = dim_{H} N^{+} = [\frac{log 2}{log a}]^{2} = (dim_{H} C)^{2}.

One of the results of this paper is that S is not a regular set: we will prove that the packing dimension of S, dim_{P} S = dim_{H} C. For non-homogeneous Cantor sets C it is much more difficult to determine dim_{H} S. The most studied non-homogeneous Cantor set in R is the set C satisfying C = ⋃_{j=0}^{r} h_{j}(C) with a disjoint union, where the h_{j}’s are similitude mappings with ratios 0 < a_{j} < 1. It is well known that dim_{H} C = s with ∑_{j=0}^{r} a_{j}^{s} = 1. Consider the self-similar probability measure μ on C corresponding to the probability vector (a_{0}^{s}, a_{1}^{s}, ⋯, a_{r}^{s}). The direct motivation of this paper is trying to determine dimensions of the set S of non-differentiability of points in C. The dimension formula given in this paper can be employed to obtain the same results as for the homogeneous Cantor set case, i.e., dim_{H} S = (dim_{H} C)^{2} = s^{2} and dim_{P} S = dim_{B} S = dim_{H} C = s (see [5]). On the other hand, one can also consider the problem of non-differentiability for higher dimensional Cantor sets. For example, in the two-dimensional case let the non-empty compact set C be defined by C = ⋃_{j=0}^{3} h_{j}(C), where h_{j}(x, y) = (x, y)A + b_{j} with the matrix A = diag(a, a), 0 < a < \frac{1}{2}, b_{0} = (0, 0), b_{1} = (1−a, 0), b_{2} = (1−a, 1−a) and b_{3} = (0, 1−a). Let F (x, y) be the distribution function of the uniform probability measure on C. Consider the set S of points in C which are not partially differentiable in the x-direction. A similar decomposition as in (1) can be made for S. Here each t ∈ C can be encoded by a sequence with symbols from {0, 1, 2, 3}. Now we let z(t, n) denote the position of the n-th occurrence of the elements of the set {0, 3}. One can check that N^{+} can also be characterized by the above [a] and [b]. In this special case our formula (8) yields dim_{P} N^{+} = dim_{B} N^{+} = −\frac{log 4}{log a} and dim_{H} N^{+} = (\frac{log 2}{log a})^{2} − \frac{log 2}{log a}. We remark that even although N^{+} is not a regular set, Marstrand’s product theorem does apply, and so this result might also be obtained in that way. The sets considered above are special cases of a class of subsets of general Moran fractals. We will focus on calculating the Hausdorff, packing and box dimension of subsets of these Moran fractals which are characterized by location codes like that in [a] and [b] above. Denote Ω = {0, 1, ⋯, r}, where r is a positive integer. We use the following notations.

(i) Ω^{ω} = {σ = (σ(1), σ(2), ⋯) : 0 ≤ σ(j) ≤ r};

(ii) Ω^{k} = {σ = (σ(1), σ(2), ⋯, σ(k)) : 0 ≤ σ(j) ≤ r} for k ∈ N and Ω^{∗} = ⋃_{k=1}^{∞}Ω^{k}; (iii) |·| is used to denote the length of word. 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};

(iv) σ|k = (σ(1), σ(2), ⋯, σ(k)) for σ ∈ Ω^{ω} and k ∈ N;

(v) For σ ∈ Ω^{k}, the cylinder set C(σ) with base σ is defined as C(σ) = {τ ∈ Ω^{ω} : τ|k = σ} for k ∈ N.

Fixing a nonempty compact set J ⊂ R^{n} with \overline{intJ} = J and positive real numbers 0 < a_{i} < 1, i = 0, 1, ⋯, r, the related Moran set (or Moran fractal) is defined in the following way. Let also a number c > 0 be given. Step 1. For each σ ∈ Ω^{k}, k ∈ N, construct a compact set J_{σ} ⊂ R^{n} by induction:

• A family {J_{j} : j = 0, 1, ⋯, r} of nonoverlapping nonempty compact subsets of J is chosen for k = 1 such that \overline{intJ_{j}}=J_{j}, |J_{j}| = a_{j}|J| where | · | denotes the diameter of a set.

• Suppose that J_{σ} is given for some σ ∈ Ω^{k}. Take a family {J_{σ∗i} : i = 0, 1, ⋯, r} of nonoverlapping nonempty compact subsets of J_{σ} such that \overline{intJ_{σ∗i}} = J_{σ∗i}, |J_{σ∗i}| = a_{i}|J_{σ}| and J_{σ∗i} contains an open ball of diameter c|J_{σ∗i}|. Step 2. The Moran fractal F associated with {0 < a_{i} < 1, i = 0, 1, ⋯, r} and the J_{σ}, σ ∈ Ω^{∗} is defined as the nonempty compact set

F = ⋂_{k=1}^{∞} ⋃_{σ∈Ω^{k}} J_{σ}.  (2)

It is well known that dim_{H} F = dim_{P} F = dim_{B} F = s and F is an s-set, where

∑_{j∈Ω} a_{j}^{s} = 1.  (3)

Define ϕ: Ω^{ω} ⟶ R^{n} by

{ϕ(σ)} = ⋂_{k=1}^{∞} J_{σ|k}.  (4)

It is easy to see that ϕ(Ω^{ω}) = F and ϕ(C(σ)) = F ⋂ J_{σ} by (2) and ϕ is a continuous surjection. Each x ∈ F can be encoded via ϕ. An infinite sequence σ is called a location code of x ∈ F if ϕ(σ) = x. Here we would like to point out that there may be multiple location codes for some x ∈ F. However the number of location codes for any x ∈ F is bounded by a positive constant independent of x ∈ F. For convenience we often use x(k) to denote the k-th component of a location code of x ∈ F. Let h_{i} : R^{n} ⟶ R^{n}, 0 ≤ i ≤ r. Denote h_{σ}(x) = h_{σ(1)} ∘ ⋯ ∘ h_{σ(k)}(x) for σ ∈ Ω^{k} and x ∈ R^{n}. A Moran fractal is termed as map-specified if there exist similitude contractions h_{i}, i = 0, 1, ⋯, r, such that J_{σ} = h_{σ}(J) for any σ ∈ Ω^{∗}. In this case F is actually the self-similar set determined by {h_{i}, 0 ≤ i ≤ r}, which satisfies the open set condition with respect to the open set O = intJ (i.e. ⋃_{i=0}^{r} h_{i}(O) ⊂ O with a disjoint union on the left) and the coding map ϕ in (4) can be changed into

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

Now let Γ ⊆ Ω = {0, 1, ⋯, r} be non-empty such that Γ^{c} ≠ ∅. Let z(σ, n) denote the position of the n-th occurrence of elements of Γ in σ ∈ Ω^{ω}:

z(σ, n) = \{ \begin{matrix}k, & if σ(k) ∈ Γ and #\{1 ≤ i < k : σ(i) ∈ Γ\} = n − 1 \\ +∞, & if #\{1 ≤ i < +∞ : σ(i) ∈ Γ\} < n. \end{matrix} (5)

For given 0 ≤ ξ ≤ 1, let

Λ = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{z(σ, n + 1)}{z(σ, n)} = ξ^{−1}\} and F_{ξ} = ϕ(Λ),  (6)

and

Λ^{∗} = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{z(σ, n + 1)}{z(σ, n)} ≥ ξ^{−1}\} and F_{ξ}^{∗} = ϕ(Λ^{∗}),  (7)

where we adopt the convention that 0^{−1} = +∞ and \frac{+∞}{+∞} = +∞. It is easy to check that if σ ∈ Λ (or Λ^{∗}) then for any j ∈ Ω we have j ∗ σ ∈ Λ (or Λ^{∗}). This means that F_{ξ} and F_{ξ}^{∗} are both dense in F. For any fixed 0 ≤ ξ ≤ 1, define the function

T (x) = ξ log ∑_{j∈Ω} a_{j}^{x} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{x}.

It is easy to verify that T (x) is strictly decreasing in [0, s]. Since T (0) > 0 and T (s) ≤ 0, there exists a unique 0 < η ≤ s such that T (η) = 0, i.e.,

ξ log ∑_{j∈Ω} a_{j}^{η} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{η} = 0.  (8)

η is a function η(ξ) of ξ in [0, 1]. It is easy to verify that η(ξ) is strictly increasing and continuous and η(0) ≤ η(ξ) ≤ η(1) = s with ∑_{j∈Γ^{c}} a_{j}^{η(0)} = 1. In the present paper, we shall prove: dim_{H} F_{ξ} = dim_{H} F_{ξ}^{∗} = η and dim_{P} F_{ξ} = dim_{P} F_{ξ}^{∗} = dim_{B} F_{ξ} = dim_{B} F_{ξ}^{∗} = s where F_{ξ}, F_{ξ}^{∗}, η and s are defined in (6),(7) , (8) and (3) respectively. Obviously this result can be employed to obtain Darst’s result for dim_{H} N^{+}. In fact, for any ϵ > 0 and ξ = −\frac{log 2}{log a} we have F_{ξ+ϵ}^{∗} ⊆ N^{+} ⊆ F_{ξ}^{∗} if we take r = 1, Γ = {0} and a_{i} = a for i = 0, 1. Therefore from the continuity of η we obtain letting ϵ ↓ 0 that dim_{H} N^{+} = dim_{H} F_{ξ}^{∗} = (\frac{log 2}{log a})^{2}. In addition we have dim_{B} N^{+} = dim_{P} N^{+} = −\frac{log 2}{log a}.

2 Dimensions of F_{ξ} and F_{ξ}^{∗}

In this section the dimensions of F_{ξ} and F_{ξ}^{∗} are obtained. The following proposition will be employed. Part (A) can be found in [2] for the more general Moran fractal structure and a simplified proof is given for this special case in [4].

Proposition 2.1 Let M = ϕ(∏_{i=1}^{∞} Ω_{i}) where Ω_{i} ⊆ Ω, i ∈ N. Let d(k) be such that

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

Then (A) dim_{H} M = lim inf_{k→∞} d(k) (see [2]), and (B) \overline{dim}_{B}M = dim_{P} M = lim sup_{k→∞} d(k) (see [3]).

Theorem 2.2 We have dim_{H} F_{ξ} = dim_{H} F_{ξ}^{∗} = η and dim_{P} F_{ξ} = dim_{P} F_{ξ}^{∗} = dim_{B} F_{ξ} = dim_{B} F_{ξ}^{∗} = s where F_{ξ}, F_{ξ}^{∗}, η and s are defined in (6),(7), (8) and (3) respectively.

Proof Let H^{d}(·) denote d-dimensional Hausdorff measure. The case ξ = 1 is simple, and left to the reader. We now consider the case 0 < ξ < 1. We shall first give upper bounds for the Hausdorff dimension of F_{ξ}^{∗}. This part of the proof is similar to that in [1]. Note that 0 < η < s when 0 < ξ < 1. At first we will show dim_{H} F_{ξ}^{∗} ≤ η. Fix an arbitrary d with s ≥ d > η. Note that T (d) < 0, i.e.,

ξ log ∑_{j∈Ω} a_{j}^{d} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{d} < 0.  (9)

Note that from (9) and d ≤ s it follows that

− log ∑_{j∈Γ^{c}} a_{j}^{d} > 0.  (10)

Thus we get

\frac{log ∑_{j∈Ω} a_{j}^{d}}{− log ∑_{j∈Γ^{c}} a_{j}^{d}} < ξ^{−1} − 1,

by (9) and (10). Let t > 0 be such that

\frac{log ∑_{j∈Ω} a_{j}^{d}}{− log ∑_{j∈Γ^{c}} a_{j}^{d}} = ξ^{−1} − 1 − t.  (11)

We will define a positive integer n^{∗} (depending on d); for k ≥ n^{∗}, we will specify u_{k} so that (I) the limsup of the sets

E_{k} = {x : x(i) ∈ Γ^{c} for k < i ≤ u_{k}}, k ≥ n^{∗}

satisfies the formula

F_{ξ}^{∗} ⊆ lim sup_{k→∞} E_{k} = ⋂_{m=n^{∗}}^{∞} ⋃_{k≥m} E_{k} =^{Δ} E^{∞},  (12)

(II) the inequality (∑_{j∈Ω} a_{j}^{d})^{k}(∑_{j∈Γ^{c}} a_{j}^{d})^{u_{k}−k} ≤ k^{−2} is satisfied. Taking logs in (II) and using (10), we obtain the equivalent inequality:

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

Choose n^{∗} large enough to assure that when k ≥ n^{∗}

\frac{2 log k}{−k log ∑_{j∈Γ^{c}} a_{j}^{d}} < \frac{t}{2} and \frac{1}{k} < \frac{t}{8}.  (14)

Now for each k ≥ n^{∗} we can choose u_{k} such that

ξ^{−1} − \frac{t}{2} < \frac{u_{k}}{k} and \frac{u_{k} − 1}{k} ≤ ξ^{−1} − \frac{t}{2}.  (15)

By (15) and the second inequality of (14), we obtain

ξ^{−1} − \frac{t}{2} < \frac{u_{k}}{k} < ξ^{−1} − \frac{t}{4}.  (16)

According to (11), (14) and the first inequality in (16), for k ≥ n^{∗}

\frac{log ∑_{j∈Ω} a_{j}^{d}}{− log ∑_{j∈Γ^{c}} a_{j}^{d}} + \frac{2 log k}{−k log ∑_{j∈Γ^{c}} a_{j}^{d}} + 1 < ξ^{−1} − 1 − t + \frac{t}{2} + 1 < \frac{u_{k}}{k} .

So (13), i.e., the inequality in (II) is satisfied for k ≥ n^{∗}. To verify (12), we need to show that for each point x ∈ F_{ξ}^{∗} with location code σ there exists a strictly increasing sequence {k_{i} : i ∈ N} of positive integers such that x ∈ E_{k_{i}}. By the definitions (5) and (7) of z(σ, n) and F_{ξ}^{∗}, for each x ∈ F_{ξ}^{∗} with a location code σ ∈ Λ^{∗} there are two different cases, i.e., #{1 ≤ i < +∞ : σ(i) ∈ Γ} = +∞ or #{1 ≤ i < +∞ : σ(i) ∈ Γ} < +∞. In the former case, there exists a strictly increasing sequence {n_{i} : i ∈ N} of positive integers such that z(σ, n_{1}) ≥ n^{∗} and

\frac{z(σ, n_{i} + 1)}{z(σ, n_{i})} > ξ^{−1} − \frac{t}{4}.  (17)

Taking k_{i} = z(σ, n_{i}) and using (17) as well as the second inequality in (16), we have z(σ, n_{i} + 1) > u_{k_{i}}, which implies that x ∈ E_{k_{i}}. In the latter case, it is easy to see that x ∈ E_{k} for all k ≥ max{1 ≤ i < +∞ : σ(i) ∈ Γ}. On the other hand, since each E_{k} can be covered with {J_{σ∗τ} : σ ∈ Ω^{k} and τ = (τ(1), τ(2), ⋯, τ(u_{k} − k)) with τ(j) ∈ Γ^{c} for j = 1, ⋯, u_{k} − k}, for any m ≥ n^{∗} we have

H^{d}(E^{∞}) ≤ H^{d}( ⋃_{k≥m} E_{k}) ≤ |J|^{d} ∑_{k≥m}[(∑_{j∈Ω} a_{j}^{d})^{k}( ∑_{j∈Γ^{c}} a_{j}^{d})^{u_{k}−k}] ≤ |J|^{d} ∑_{k≥m} k^{−2},

by (I) and (II). So we have H^{d}(E^{∞}) = 0 by letting m ⟶ ∞. Consequently, H^{d}(F_{ξ}^{∗}) = 0 which implies dim_{H} F_{ξ}^{∗} ≤ η. We now turn to the second part of the proof of the case 0 < ξ < 1, where we shall show dim_{H} F_{ξ} ≥ η and dim_{P} F_{ξ} ≥ s by proving that for any fixed 0 < d < η and 0 < d^{∗} < s there exists a subset E = E(d, d^{∗}) of F_{ξ} such that dim_{H} E ≥ d and dim_{P} E ≥ d^{∗}. Let the constant c be defined by

c = max{log #Γ, log #Γ^{c}, | log ∑_{i∈Γ} a_{j}^{s}|, | log ∑_{j∈Γ^{c}} a_{j}^{s}|}.  (18)

Note that both functions T (x) and G(x) =^{Δ} log ∑_{j∈Ω} a_{j}^{x} are strictly decreasing with T (η) = 0 and G(s) = 0. Therefore we can choose a 0 < ϵ < 1 which satisfies:

[ϵ1]. the solutions x of the following inequality will be in [d, +∞)

T (x) = ξ log ∑_{j∈Ω} a_{j}^{x} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{x} ≤ \frac{6cϵ}{1 − ϵ};  (19)

[ϵ2]. the solutions x of the following inequality will be in [d^{∗}, +∞)

log ∑_{j∈Ω} a_{j}^{x} ≤ \frac{5cϵ}{1 − ϵ};  (20)

[ϵ3]. with t being such that ∑_{j∈Γ^{c}} a_{j}^{t} = 1,

\frac{4cϵ}{1 − ϵ} < log(1 + ∑_{j∈Γ} a_{j}^{t}).  (21)

Given any 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} < ⋯, we construct a set E as follows:

E = {x : x(k_{i}), x(u_{i}), x(u_{i,j}) ∈ Γ and x(k) ∈ Γ^{c} for k_{i} < k < u_{i}, i ≥ 1, 1 ≤ j ≤ n_{i}}.

The set E is a closed subset of F. In the definition of E, when k_{i} < k < u_{i}, x(k) is a restricted Γ^{c}-choice; when k = k_{i}, u_{i} and u_{i,j}, x(k) is a restricted Γ-choice; and the rest of x(k) is a free choice, i.e., Ω-choice. In the following we specify k_{i}, u_{i}, u_{i,1}, ⋯, u_{i,n_{i}} depending on ϵ to construct the set E = E_{ϵ}.

By N_{Ω}(k), N_{Γ}(k) and N_{Γ^{c}}(k) we denote the number of Ω-, Γ- and Γ^{c}-choices among the first k entries in a location code of a point in E respectively. Thus we have N_{Ω}(k) + N_{Γ}(k) + N_{Γ^{c}}(k) = k for k ∈ N. Note that for k_{i} − 1 ≤ j ≤ u_{i}, N_{Ω}(j) = N_{Ω}(k_{i}). Now define a sequence of positive integers b_{i}, i ∈ N, by

b_{i+1} = b_{i}ξ^{−1} + θ_{i}, where i ∈ N and 0 ≤ θ_{i} < 1,  (22)

with b_{1} = 6. Thus the b_{i} increase strictly and tend to +∞. We take the sequence of positive integers k_{1}, u_{1}, u_{1,1}, ⋯, u_{1,n_{1}}, k_{2}, u_{2}, ⋯ in the definition of E as the sequence b_{1}, b_{2}, · · ·. From the definition of E and (22) it follows that E ⊆ F_{ξ} since

ξ^{−1} = lim_{i→∞} \frac{u_{i}}{k_{i}} ≤ lim sup_{n→∞} \frac{z(σ, n + 1)}{z(σ, n)} ≤ lim sup_{i→∞} \frac{b_{i+1}}{b_{i}} = ξ^{−1}.

We have not yet specified indices n_{i}. Suppose that the n_{j} are defined for j = 1, 2, ⋯, i − 1, then also k_{i} and u_{i} are determined. Letting n_{i} vary, we have (using Stolz’s theorem in the second line)

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}{(ξ^{−1} − 1)u_{i,n_{i}−1}} = 1.

Therefore noting that we can identify k_{i+1} with u_{i,n_{i}+1}, we can choose n_{i} such that

N_{Ω}(k_{i+1}) ≥ (1 − ϵ)k_{i+1}.  (23)

Now take E =^{Δ} E_{ϵ} corresponding to the choice of n_{i} satisfying (23). According to Proposition 2.1 we have dim_{H} E = lim inf_{k→∞} d(k) where d(k) satisfies

(∑_{j∈Ω} a_{j}^{d(k)})^{N_{Ω}(k)}(∑_{j∈Γ} a_{j}^{d(k)})^{N_{Γ}(k)}( ∑_{j∈Γ^{c}} a_{j}^{d(k)})^{k−N_{Ω}(k)−N_{Γ}(k)} = 1. (24)

Taking logs in (24) we get after some algebra

ξ log ∑_{j∈Ω} a_{j}^{d(k)} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{d(k)} = (1 − \frac{ξk}{N_{Ω}(k)}) log ∑_{j∈Γ^{c}} a_{j}^{d(k)}

+\frac{ξN_{Γ}(k)}{N_{Ω}(k)} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)}) .  (25)

We shall show that there exist an i^{∗} such that d(k) ≥ d when i ≥ i^{∗} and k_{i} ≤ k < k_{i+1}. Then dim_{H} E ≥ d by Proposition 2.1. At first note that since 0 ≤ d(k) ≤ s, we have ∑_{j∈Ω} a_{j}^{d(k)} ≥ 1 and therefore by (24) we have

d(u_{i,j}) ≤ d(k) when u_{i,j} < k < u_{i,j+1}, 0 ≤ j ≤ n_{i}, with u_{i,0} =^{Δ} u_{i} and

u_{i,n_{i}+1} =^{Δ} k_{i+1}. So we only need to consider d(k) for k_{i} ≤ k ≤ u_{i} and k = u_{i,j}. When k_{i} ≤ k ≤ u_{i}, then N_{Ω}(k) = N_{Ω}(k_{i}) and hence the equality (25) can be written as

T (d(k)) = ξ log ∑_{j∈Ω} a_{j}^{d(k)} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{d(k)}
= (1 − \frac{ξk}{N_{Ω}(k_{i})}) log ∑_{j∈Γ^{c}} a_{j}^{d(k)}
+ \frac{ξN_{Γ}(k)}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)}) .  (26)

Note that by (22) and (23)

(1 − ϵ)k_{i} ≤ N_{Ω}(k_{i}) ≤ k_{i} and u_{i} = ξ^{−1}k_{i} + θ for some 0 ≤ θ < 1. (27)

So we have

\frac{N_{Γ}(k)}{N_{Ω}(k_{i})} = \frac{k − N_{Ω}(k_{i}) − N_{Γ^{c}}(k)}{N_{Ω}(k_{i})} ≤ \frac{k_{i} − N_{Ω}(k_{i}) + 1}{N_{Ω}(k_{i})} ≤ \frac{ϵ}{1 − ϵ} + \frac{1}{N_{Ω}(k_{i})}, (28)

and

1 − \frac{kξ}{N_{Ω}(k_{i})} ≥ 1 − \frac{u_{i}ξ}{N_{Ω}(k_{i})} = 1 − \frac{k_{i} + ξθ}{N_{Ω}(k_{i})} ≥ 1 − \frac{1}{1 − ϵ} − \frac{1}{N_{Ω}(k_{i})}
= −\frac{ϵ}{1 − ϵ} − \frac{1}{N_{Ω}(k_{i})},  (29)

by (27). Now because u_{i} → ∞, also k_{i} → ∞ and by (23), we can take i^{∗} by requiring that when i ≥ i^{∗},

[i^{∗}1]. for N_{Ω}(k_{i}) we have

\frac{1}{N_{Ω}(k_{i})} ≤ \frac{ϵ}{1 − ϵ};  (30)

[i^{∗}2]. for u_{i} we have

(ξ^{−1} − 1)u_{i} > 1 + \frac{max\{| log ∑_{j∈Γ} a_{j}^{d}|, log(#Γ)\}}{log ∑_{j∈Ω} a_{j}^{d}}.  (31)

Note that when i ≥ i^{∗} and k_{i} ≤ k ≤ u_{i}, we have log ∑_{j∈Γ^{c}} a_{j}^{d(k)} < 0. Otherwise suppose that log ∑_{j∈Γ^{c}} a_{j}^{d(k)} ≥ 0, i.e., d(k) ≤ t with ∑_{i∈Γ^{c}} a_{i}^{t} = 1. Thus from (26),(28), (18) and (30) it follows that

log (1 + ∑_{j∈Γ} a_{j}^{t}) = log ∑_{j∈Ω} a_{j}^{t} ≤ log ∑_{j∈Ω} a_{j}^{d(k)}

= (1 − \frac{k}{N_{Ω}(k_{i})}) log ∑_{j∈Γ^{c}} a_{j}^{d(k)} + \frac{N_{Γ}(k)}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)})
≤ \frac{N_{Γ}(k)}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)}) ≤ \frac{4cϵ}{1 − ϵ},

which contradicts (21). Therefore when i ≥ i^{∗} and k_{i} ≤ k ≤ u_{i}, from (26), (29), (18), (28) and (30) it follows that

T (d(k)) = ξ log ∑_{j∈Ω} a_{j}^{d(k)} + (1 − ξ) log ∑_{j∈Γ^{c}} a_{j}^{d(k)}

= (1 − \frac{ξk}{N_{Ω}(k_{i})}) log ∑_{j∈Γ^{c}} a_{j}^{d(k)} + \frac{ξN_{Γ}(k)}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)})
≤ (−\frac{ϵ}{1 − ϵ} − \frac{1}{N_{Ω}(k_{i})}) log ∑_{j∈Γ^{c}} a_{j}^{d(k)}
+ \frac{ξN_{Γ}(k)}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)}) ≤ \frac{6cϵ}{1 − ϵ},

i.e., d(k) ≥ d by (19). This takes care of k with k_{i} ≤ k ≤ u_{i}. But actually, when i ≥ i^{∗}, d(u_{i,j}) ≥ d also holds for all j = 1, 2, ⋯ n_{i}. Otherwise, suppose d(u_{i,j^{∗}}) < d for some 1 ≤ j^{∗} ≤ n_{i}. Taking k = u_{i} and u_{i,j^{∗}} in (24), we get

(∑_{j∈Ω} a_{j}^{d(u_{i})})^{N_{Ω}(u_{i})}(∑_{j∈Γ} a_{j}^{d(u_{i})})^{N_{Γ}(u_{i})}( ∑_{j∈Γ^{c}} a_{j}^{d(u_{i})})^{N_{Γ^{c}}(u_{i})} = 1, (32)

and

(∑_{j∈Ω} a_{j}^{d(u_{i,j^{∗}})})^{N_{Ω}(u_{i,j^{∗}})}(∑_{j∈Γ} a_{j}^{d(u_{i,j^{∗}})})^{N_{Γ}(u_{i,j^{∗}})}( ∑_{j∈Γ^{c}} a_{j}^{d(u_{i,j^{∗}})})^{N_{Γ^{c}}(u_{i,j^{∗}})}
= 1.  (33)

since d(u_{i}) ≥ d > d(u_{i,j^{∗}}), N_{Γ^{c}}(u_{i,j^{∗}}) = N_{Γ^{c}}(u_{i}), N_{Γ}(u_{i,j^{∗}}) = N_{Γ}(u_{i})+j^{∗} and

N_{Ω}(u_{i,j^{∗}}) = N_{Ω}(u_{i})+∑_{k=0}^{j^{∗}−1}((ξ^{−1} −1)u_{i,k} +θ_{i,k} −1) with u_{i,0} =^{Δ} u_{i}, it follows

from (32) and (33) that

(∑_{j∈Ω} a_{j}^{d(u_{i,j^{∗}})})^{∑_{k=0}^{j^{∗}−1}((ξ^{−1}−1)u_{i,k}+θ_{i,k}−1)}(∑_{j∈Γ} a_{j}^{d(u_{i,j^{∗}})})^{j^{∗}} < 1,  (34)

and also since d(u_{i,j^{∗}}) < d < s we have

∑_{j∈Ω} a_{j}^{d(u_{i,j^{∗}})} > 1.  (35)

Therefore (34), (35) and the monotone increasing of u_{i,k}, k = 0, ⋯, j^{∗} − 1, imply

(ξ^{−1} − 1)u_{i} − 1 ≤ \frac{∑_{k=0}^{j^{∗}−1}((ξ^{−1} − 1)u_{i,k} + θ_{i,k} − 1)}{j^{∗}} < \frac{− log ∑_{j∈Γ} a_{j}^{d(u_{i,j^{∗}})}}{log ∑_{j∈Ω} a_{j}^{d(u_{i,j^{∗}})}}
≤ \frac{| log ∑_{j∈Γ} a_{j}^{d(u_{i,j^{∗}})}|}{log ∑_{j∈Ω} a_{j}^{d(u_{i,j^{∗}})}} ≤ \frac{max\{| log ∑_{j∈Γ} a_{j}^{d}|, log(#Γ)\}}{log ∑_{j∈Ω} a_{j}^{d}},

which is impossible by (31). Thus we complete the proof of dim_{H} E ≥ d for the subset E of F_{ξ}. Since d < η was arbitrary, and since F_{ξ} ⊆ F_{ξ}^{∗} it now follows by combining this result with the first part of the proof that dim_{H} F_{ξ} = dim_{H} F_{ξ}^{∗} = η. Finally taking k = k_{i} in (25) and noting that −\frac{ϵ}{1−ϵ} ≤ 1 − \frac{k_{i}}{N_{Ω}(k_{i})} ≤ 0, then when i ≥ i^{∗} we have

log ∑_{j∈Ω} a_{j}^{d(k_{i})} = (1 − \frac{k_{i}}{N_{Ω}(k_{i})}) log ∑_{j∈Γ^{c}} a_{j}^{d(k_{i})}
+ \frac{N_{Γ}(k_{i})}{N_{Ω}(k_{i})} (log ∑_{j∈Γ^{c}} a_{j}^{d(k_{i})} − log ∑_{j∈Γ} a_{j}^{d(k_{i})}) ≤ \frac{5cϵ}{1 − ϵ},

by (28), (30) and (18). It follows from (20) that d(k_{i}) ≥ d^{∗} if i ≥ i^{∗}. By Proposition 2.1 it follows that dim_{P} E = \overline{dim}_{B}E ≥ d^{∗}. Consequently we get dim_{P} F_{ξ}^{∗} = \overline{dim}_{B}F_{ξ}^{∗} = dim_{P} F_{ξ} = \overline{dim}_{B}F_{ξ} = s. The denseness of Λ and Λ^{∗} in Ω^{ω} implies that \underline{dim}_{B}F_{ξ}^{∗} = \underline{dim}_{B}F_{ξ} = \underline{dim}_{B}F = s. This ends the proof of the case 0 < ξ < 1. We now consider ξ = 0. In this case F_{0} = F_{0}^{∗}. That dim_{H} F_{0} ≥ η, where η satisfies ∑_{j∈Γ^{c}} a_{j}^{η} = 1, follows again from Proposition 2.1 by constructing an M contained in F_{0} of type M = ϕ(∏_{i=1}^{∞} Ω_{i}) that has Hausdorff dimension η. In fact define M by requiring that Ω_{i} = Γ if i = 2^{k^{2}} and Ω_{i} = Γ^{c} for i ≠ 2^{k^{2}}. That dim_{H} F_{0} ≤ η follows from the case 0 < ξ < 1 by using continuity of T and the fact that F_{0} ⊆ F_{1/k}^{∗} for k = 1, 2, · · · . To prove that dim_{P} F_{0} = s one can proceed similarly to the case 0 < ξ < 1, but now constructing E = E_{ϵ} = {x : x(k_{i}) ∈ Γ and x(k) ∈ Γ^{c} for k_{i} < k ≤ u_{i}, i ≥ 1}, which has u_{i} = (i + 1)k_{i} and N_{Ω}(k_{i+1}) ≥ (1 − ϵ)k_{i+1} for all i. □

References

[1] Richard Darst, Hausdorff dimension of sets of non-differentiability points of Cantor functions. Math. Proc. Camb. Phi. Soc., 117(1995), 185–191.

[2] Su Hua, On the Hausdorff dimension of generalized self-similar sets. Acta Mathematical Applicata Sinica (in chinese), 17(1994), 551–558.

[3] Su Hua & Wenxia Li, Packing Dimension Of Generalized Moran Sets. Progress In Natural Science, 2(1996), 148-152.

[4] Wenxia Li & F.M. Dekking, The dimension of subsets of Moran sets determined by the success run behaviour of their codings. Monatshefte fur Mathematik,131(2000), 309–320.

[5] F.M. Dekking & Wenxia Li , How smooth is a devil’s staircase?. In preparation.