Lobachevskii Journal of Mathematics Vol. 12, 2003, 41 – 50 \begin{matrix}c \\ ◯ \end{matrix}K. Igudesman

K. Igudesman

LACUNARY SELF-SIMILAR FRACTAL SETS AND INTERSECTION OF CANTOR SETS

(submitted by M. A. Malakhaltsev)

Abstract. The problem on intersection of Cantor sets was examined in many papers. To solve this problem, we introduce the notion of lacunary self-similar set. The main difference to the standard (Hutchinson) notion of self-similarity is that the set of similarities used in the construction may vary from step to step in a certain way. Using a modification of method described in [3], [4], we find the Hausdorff dimension of a lacunary self-similar set.

In one form or another, the problem on intersection of Cantor sets was examined in many papers. In [5], [6], [8] the intersection of so-called thick Cantor sets was considered. In [2] it was proved that the intersection of two standard Cantor sets can be of any Hausdorff dimension from 0 to \frac{ln 2}{ln 3}. In [7] there was investigated the intersection of standard Cantor sets one of which is translated by an element of Cantor set.

To solve this problem, we introduce the notion of lacunary self-similar set. The main difference to the standard (Hutchinson) notion of self-similarity is that the set of similarities used in the construction may vary from step to step in a certain way. Using a modification of method described in [3], [4], we find a Hausdorff dimension of a lacunary self-similar set.

Let K be the (ordinary) Cantor set in [0, 1]. It is shown that for almost all a ∈ [0, 1] (with respect to Lebesgue’s measure) the set (a + K) ⋂ K is the lacunary self-similar set and its Hausdorff dimension is equal to \frac{ln 2}{3 ln 3}. Then we give some generalization of this fact.

1. Lacunary self-similar sets

Let us recall the definition of Hausdorff measure, dimension and metric. Let U be a nonempty set in ℝ^{n}, |U| = sup{|x − y| : x, y ∈ U}. If E ⊂ ⋃_{i} U_{i}, and 0 < |U_{i}| ≤ δ for each i, then {U_{i}} is called a δ-covering of E. Given E ⊂ ℝ^{n} and s ≥ 0, for any δ > 0 the outer measure H_{δ}^{s}(E) of E is defined by the equality

Supported by Grant of RFFI N 00-01-00308.

H_{δ}^{s}(E) = inf ∑_{i=1}^{∞} |U_{i}|^{s},

where inf is taken over all countable δ-coverings {U_{i}} of E.

The Hausdorff outer s-dimensional measure H^{s} on ℝ^{n} is defined by the equality

H^{s}(E) = lim_{δ→0} H_{δ}^{s}(E).

This limit exists, but can be infinite because H_{δ}^{s} does not decrease as δ → 0.

It is known that the σ-algebra of H^{s}-measurable sets contains the Borel sets. And for any E ⊂ ℝ^{n} there exists a unique number dim(E) ∈ [0, n] such that

H^{t}(E) = \{ \begin{matrix}∞, & t < dim(E) \\ 0, & t > dim(E) \end{matrix} .

dim(E) is called the Hausdorff dimension (the fractal dimension) of E. If E ⊂ ℝ^{n}, and δ ≥ 0, the δ-parallel body of E is the closed set

[E]_{δ} = {x ∈ ℝ^{n} : inf_{y∈E} |x − y| ≤ δ}.

Let Γ be a collection of nonempty compact subset of ℝ^{n}. The Hausdorff metric d on Γ is

d(E, F) = inf{δ : E ⊂ [F]_{δ}, F ⊂ [E]_{δ}}.

It is known that Γ endowed with the Hausdorff metric d is a complete metric space .

Let us construct a lacunary self-similar set and find its Hausdorff dimension. Note that the collection of lacunary self-similar sets contains the well-known self-similar sets.

A mapping ψ : ℝ^{n} → ℝ^{n} is called a similitude if |ψ(x) − ψ(y)| = r|x − y| for all x, y ∈ ℝ^{n}, where r < 1. Clearly, any similitude is a continuous mapping.

Let {ψ_{j}}_{j=1}^{m} be a set of similitudes with ratios {r_{j}}_{j=1}^{m}. We set r = max_{1≤j≤m} r_{j}.

Let N_{k} ⊆ {1, …, m} be a nonempty set, for any positive integer k (if N_{k} = {1, …, m} for all k, then the following construction gives a self-similar set).

We set L^{k} = ∏_{i=1}^{k} N_{i}, L_{p}^{k} = ∏_{i=p}^{k} N_{i} (p ≤ k). For any sequence (j_{1} … j_{k}) ∈ L^{k}

and any set F we set F_{j_{1}...j_{k}} = (ψ_{j_{1}} ∘ … ∘ ψ_{j_{k}})(F). Now, let Γ be a collection of nonempty compact sets, and ψ^{k} : Γ → Γ,

ψ^{k}(F) = ⋃_{L^{k}}(ψ_{j_{1}} ∘ … ∘ ψ_{j_{k}})(F) = ⋃_{L^{k}} F_{j_{1}...j_{k}} .

for any positive integer k.

Theorem 1. There exists a unique set E ∈ Γ such that for all F ∈ Γ

ψ^{k}(F) ⟶^{k→∞} E

with respect to the Hausdorff metric d.

⋄ If F, G ∈ Γ, then, by the definition of d,

d(ψ^{k}(F), ψ^{k}(G)) ≤ sup_{L^{k}} d(F_{j_{1}...j_{k}}, G_{j_{1}...j_{k}}) ≤ r^{k}d(F, G) .

Take any F ∈ Γ and set M = sup_{1≤j≤m} d(F, F_{j}). For any positive integers p and

q (p < q),

d(ψ^{p}(F), ψ^{q}(F)) ≤ d(ψ^{p}(F), ψ^{p}( ⋃_{L_{p+1}^{q}} F_{j_{p+1}...j_{q}})) ≤ r^{p}d(F, ⋃_{L_{p+1}^{q}} F_{j_{p+1}...j_{q}})

≤ r^{p} sup_{L_{p+1}^{q}} d(F, F_{j_{p+1}...j_{q}}) ≤ r^{p} sup_{L_{p+1}^{q}}(d(F, F_{j_{p+1}}) + d(F_{j_{p+1}}, F_{j_{p+1}j_{p+2}}) + ⋯

+d(F_{j_{p+1}...j_{q−1}}, F_{j_{p+1}...j_{q}}) ≤ r^{p}(M + rM + r^{2}M + ⋯ r^{q−p−1}M) < r^{p} \frac{M}{1 − r} .

Hence d(ψ^{p}(F), ψ^{q}(F)) ⟶^{p→∞} 0. As Γ endowed with the Hausdorff metric d is a complete metric space, ψ^{k}(F) converges to a nonempty compact set E.

For any G ∈ Γ

d(ψ^{k}(G), E) ≤ d(ψ^{k}(G), ψ^{k}(F)) + d(ψ^{k}(F), E) ⟶^{k→∞} 0 ,

and so E is unique. ⋄

Let K be the set of finite sequences {(j_{1}, …, j_{k})}_{L^{k},k}. A finite subset P ⊂ K is called a tree if for any (j_{1}, …, j_{k}) ∈ K there exists (i_{1}, …, i_{p}) ∈ P such that j_{q} = i_{q} for all q = 1, …, min{k, p}, moreover, if p ≤ k, then this (i_{1}, …, i_{p}) ∈ P is unique. Note that for any k the set L^{k} is a tree.

Given a tree P, we set p = inf{k : (j_{1}, …, j_{k}) ∈ P} (the length of the shortest branch of the tree) and q = sup{k : (j_{1}, …, j_{k}) ∈ P} (the length of the longest branch of the tree).

Lemma 1. For any tree P and any non-negative numbers {a_{k}}_{k=1}^{m} we have

∑_{P} a_{j_{1}} … a_{j_{k}} ≥ inf_{p≤k≤q} ∑_{L^{k}} a_{j_{1}} … a_{j_{k}} ,

where p is the length of the shortest branch of P, and q is the length of the longest one.

⋄ Let P_{j_{1}...j_{p}} = {(i_{1}, …, i_{k}) ∈ P : (j_{1}, …, j_{p}) = (i_{1}, …, i_{p})}, where (j_{1}, …, j_{p}) ∈ L^{p}. If (l_{1}, …, l_{p}) ∈ L^{p}, then

P′ = ⋃_{L^{p}} ⋃_{P_{l_{1}...lp}}(j_{1}, …, j_{p}, l_{p+1}, …, l_{k})  (1)

is a tree. Note that the first p elements of (j_{1}, …, j_{p}, l_{p+1}, …, l_{k}) run through L^{p} and the remaining elements run through the ends of (l_{1}, …, l_{k}) ∈ P_{l_{1}...l_{p}}.

We set

μ_{j_{1}...j_{p}} = \{ \begin{matrix}∑_{P_{j_{1}...jp}} a_{j_{p+1}} … a_{j_{k}} & if & (j_{1}, …, j_{p}) ∉ P \\ 1 & if & (j_{1}, …, j_{p}) ∈ P , \end{matrix}

so

∑_{P} a_{j_{1}} … a_{j_{k}} = ∑_{L^{p}} a_{j_{1}} … a_{j_{p}}μ_{j_{1}...j_{p}} .  (2)

Let μ = inf_{L^{p}} μ_{j_{1}...j_{p}}. From (2) it follows that

∑_{P} a_{j_{1}} … a_{j_{k}} ≥ ∑_{L^{p}} a_{j_{1}} … a_{j_{k}}

if μ = 1.

If μ < 1, then there exists (l_{1}, …, l_{p}) ∉ P such that μ = μ_{l_{1}...l_{p}}. Thus, using (2), we get

∑_{P} a_{j_{1}} … a_{j_{k}} ≥ μ_{l_{1}...l_{p}} ∑_{L^{p}} a_{j_{1}} … a_{j_{p}}

= ∑_{L^{p}} a_{j_{1}} … a_{j_{p}} ∑_{P_{l_{1}...lp}} a_{l_{p+1}} … a_{l_{k}} = ∑_{P′} a_{j_{1}} … a_{j_{k}} .

Thus it follows from (1) that P′ is a tree. Clearly, the length of the shortest branch of P′ is equal to p′ > p, and the length of the longest branch of P′ is equal to q′ ≤ q.

We proceed in this way until, after a finite number of steps, we reach a tree L^{k}, where p ≤ k ≤ q. This completes the proof. ⋄

It is clear that lim inf_{k→∞} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} does not increase as t increases from 0

to ∞. Furthermore, if t < t′, then

inf_{l≥k} ∑_{L^{l}}(r_{j_{1}} … r_{j_{l}})^{t} = inf_{l≥k} ∑_{L^{l}}(r_{j_{1}} … r_{j_{l}})^{t′+(t−t′)} ≥ (r^{k})^{(t−t′)} inf_{l≥k} ∑_{L^{l}}(r_{j_{1}} … r_{j_{l}})^{t′} .

Hence there exists a unique number s ≥ 0 such that

lim inf_{k→∞} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = \{ \begin{matrix}∞, & t < s \\ 0, & t > s \end{matrix} .  (3)

Lemma 2. Let {V_{i}} be a collection of disjoint open subset of ℝ^{n} such that each V_{i} contains a ball of radius c_{1}ρ and is contained in a ball of radius c_{2}ρ. Then any ball B of radius ρ intersects, at most, (1 + 2c_{2})^{n}c_{1}^{−n} of the sets V̄_{i} (the bar denotes closure).

⋄ If V̄_{i} ⋂ B ≠ ∅, V̄_{i} is contained in a ball concentric with B and of radius (1 + 2c_{2})ρ. Let h elements of the collection {V̄_{i}} intersect B, then summing up the volumes of the corresponding interior balls, we get h(c_{1}ρ)^{n} ≤ (1 + 2c_{2})^{n}ρ^{n}, which proves the Lemma. ⋄

We say that the set of similitudes {ψ_{j}}_{j=1}^{m} satisfies the open set condition if there exists a nonempty bounded open set V ⊂ ℝ^{n} such that

ψ_{j}(V) ⊂ V and ψ_{j}(V) ⋂ ψ_{i}(V) = ∅  (4)

for any i, j = \overline{1, m}, j ≠ i. Thus the sets \{V_{j_{1}...j_{k}}\}_{L^{k},k} form a net in the sense that any two of sets from the collection are either disjoint, or one set is included into the other one. The collection {V_{j_{1}...j_{k}}}_{P} is disjoint for any tree P.

If (4) holds, then {ψ^{k}(V̄)}_{k} is a decreasing sequence of compact sets, which converges to E with respect to the Hausdorff metric by Theorem 1. It follows

from the definition of Hausdorff metric that E = ⋂_{k=1}^{∞} ψ^{k}(V̄).

The set E from Theorem 1 is called lacunary self-similar, if the open set conditions holds true.

Theorem 2. The Hausdorff dimension of the lacunary self-similar set E is equal to dim(E) = s, where s is determined by (3).

⋄ For any L^{k} (moreover, for any tree P) the collection {V̄_{j_{1}...j_{k}}}_{L^{k}} is a cover of E. As

lim inf_{k→∞} ∑_{L^{k}} |V̄_{j_{1}...j_{k}}|^{t} = |V̄ | lim inf_{k→∞} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = \{ \begin{matrix}∞, & t < s \\ 0, & t > s \end{matrix}

and |V̄_{j_{1}...j_{k}}| ≤ r^{k}|V̄ | ⟶^{k→∞} 0, we get dim(E) ≤ s.

To prove the opposite inequality we show that, if t ∈ [0, s), then H^{t}(E) = ∞. Since E is compact, it is sufficient to prove that

lim inf_{δ→0} ∑ |U_{i}|^{t} = ∞ ,

where the infimum is taken over all finite δ-covers {U_{i}} of E. Given any δ-cover {U_{i}}_{i=1}^{N} of E, we can cover E by balls {B_{i}}_{i=1}^{N} with |B_{i}| ≤ 2|U_{i}|, then

∑_{i=1}^{N} |U_{i}|^{t} ≥ 2^{−t} ∑_{i=1}^{N} |B_{i}|^{t} .  (5)

Suppose that an open set V such that (4) holds true contains a ball of radius c_{1} and is contained in a ball of radius c_{2}. Take any ρ ∈ (0, 1). For each infinite sequence (j_{1}, j_{2}, …) with (j_{1}, …, j_{k}) ∈ L^{k} for all k, curtail the sequence at the least value of k such that

( min_{1≤j≤m} r_{j})ρ ≤ r_{j_{1}} … r_{j_{k}} ≤ ρ ,  (6)

and let us denote by P the set of finite sequences obtained in this way. It is clear that P is a tree.

Each V_{j_{1}...j_{k}} contains a ball of radius c_{1}r_{j_{1}} … r_{j_{k}} and hence a ball of radius c_{1}ρ( min_{1≤j≤m} r_{j}), by (6), and is contained in a ball of radius c_{2}r_{j_{1}} … r_{j_{k}} and there-

fore of radius c_{2}ρ. By Lemma 2, any ball B of radius ρ intersects, at most, h = (c_{1} + 2c_{2})^{n}(c_{1} min_{1≤j≤m} r_{j})^{−n} sets of collection {V̄_{j_{1}...j_{k}}}_{P}. Note that h does

not depend on ρ.

Let B be a ball of radius ρ. We denote

D = {(j_{1}, …, j_{k}) ∈ P : V̄_{j_{1}...j_{k}} ⋂ B ≠ ∅} .

The set D contains, at most, h elements. Hence

∑_{D}(r_{j_{1}} … r_{j_{k}})^{t} ≤ hρ^{t} = h|B|^{t} ,

hence

|B|^{t} ≥ h^{−1} ∑_{D}(r_{j_{1}} … r_{j_{k}})^{t} .  (7)

Suppose that for any ball B_{i} from (5) we have constructed a tree P_{i} and a subset D_{i} ⊂ P_{i} . It follows from (7) that

∑_{i=1}^{N} |B_{i}|^{t} ≥ h^{−1} ∑_{i=1}^{N} ∑_{D_{i}}(r_{j_{1}} … r_{j_{k}})^{t} .  (8)

Let S = ⋃_{i=1}^{N} D_{i}. From the construction of the S we see that the collection

{V̄_{j_{1}...j_{k}}}_{S} is a (2c_{2}δ)-cover of E. We set

P = {(j_{1}, …, j_{k}) ∈ S : for any l < k (j_{1}, …, j_{l}) ∉ S} .

Clearly, P ⊂ S is a tree, hence the collection {V̄_{j_{1}...j_{k}}}_{P} is a (2c_{2}δ)-cover of E. It follows from Lemma 1 that

∑_{i=1}^{N} ∑_{D_{i}}(r_{j_{1}} … r_{j_{k}})^{t} ≥ ∑_{P}(r_{j_{1}} … r_{j_{k}})^{t} ≥ inf_{p≤k≤q} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} ,  (9)

where p and q are the lengths of the shortest and the longest branches of the tree P, respectively.

Thus by (5), (8), (9),

∑_{i=1}^{N} |U_{i}|^{t} ≥ 2^{−t}h^{−1} inf_{p≤k≤q} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} .

As |B_{i}| → 0 as δ → 0, we have p → ∞. It follows from the equality

lim inf_{k→∞} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = ∞, t < s, that H^{t}(E) = ∞. ⋄

Let {M_{l}}_{l=1}^{2^{m}−1} be the collection of all nonempty subsets of {1, …, m}. We will denote by χ_{M_{l}} the characteristic function of M_{l}

χ_{M_{l}}(N_{k}) = \{ \begin{matrix}1 & if & M_{l} = N_{k} \\ 0 & if & M_{l} ≠ N_{k} \end{matrix}

for any positive integer k and l = 1, …, (2^{m}−1). Set f_{l}(k) = ∑_{j=1}^{k} χ_{M_{l}}(N_{j}).

Corollary 1. Suppose that lim_{k→∞} \frac{f_{l}(k)}{k} = P_{l} for any l = 1, …, (2^{m}−1) and s is the number such that

∏_{l=1}^{2^{m}−1} (∑_{M_{l}} r_{j}^{s})^{P_{l}} = 1 .  (10)

Then the Hausdorff dimension of the lacunary self-similar set E is equal to dim(E) = s.

⋄ We have

∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = ∏_{i=1}^{k} (∑_{N_{i}} r_{j}^{t}) = ∏_{l=1}^{2^{m}−1} (∑_{M_{l}} r_{j}^{t})^{f_{l}(k)}

= ( ∏_{l=1}^{2^{m}−1} (∑_{M_{l}} r_{j}^{t})^{( \frac{f_{l}(k)}{k} −P_{l})} ∏_{l=1}^{2^{m}−1} (∑_{M_{l}} r_{j}^{t})^{P_{l}} )^{k}

for any positive integer k. Hence

lim inf_{k→∞} ∑_{L^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = \{ \begin{matrix}∞, & t < s \\ 0, & t > s \end{matrix} . ⋄

For instance, let Ω = {ω = (N_{1}, N_{2}, N_{3}, …)} be the set of sequences of independent random nonempty subsets of {1, …, m}, where N_{k} = M_{l} with probability P_{l}, l = 1, …, (2^{m}−1). We will denote by P the probability measure on Ω. Using Theorem 1 we construct the lacunary self-similar set E_{ω} for any ω ∈ Ω, if the open set conditions holds true. From the strong law of large numbers it may be concluded that

P \{ω ∈ Ω : lim_{k→∞} \frac{f_{l}(k)}{k} = P_{l}\} = 1 ,

and so

P {ω ∈ Ω : dim(E_{ω}) = s} = 1 ,

where s is determined by (10).

2. The intersection of Cantor’s sets

Let us set K_{0} = [0, 1], K_{1} = [0, \frac{1}{3}] ⋃[\frac{2}{3}, 1], K_{2} = [0, \frac{1}{9}] ⋃[\frac{2}{9}, \frac{1}{3}] ⋃[\frac{2}{3}, \frac{7}{9}] ⋃[\frac{8}{9}, 1], etc., K_{i+1} is obtained from K_{i} by cutting out the open middle part of each interval in K_{i}. Thus, K_{i} consists of 2^{i} closed intervals of length 3^{−i}. Taking

the intersection of all sets K_{i}, we obtain the Cantor set K = ⋂_{i=0}^{∞} K_{i}.

Consider the similitudes of the real line ψ_{1}(x) = \frac{x}{3}, ψ_{2}(x) = \frac{x}{3} + \frac{2}{3}. Since

K_{i} = ⋃_{j_{1}...j_{i}}(ψ_{j_{1}} ∘ … ∘ ψ_{j_{i}})([0, 1]) ,

where the sum is taken over all i-tuples {j_{1} … j_{i}}, the Cantor set is the selfsimilar set (it is the particular case of a lacunary self-similar set). It is known that the Hausdorff dimension of the Cantor set is dim(K) = \frac{ln 2}{ln 3}.

For a ∈ [0, 1], we set K + a = {x : x − a ∈ K}, E_{a} = K ⋂(K + a).

Theorem 3. For almost all a ∈ [0, 1] with respect to Lebesgue’s measure, E_{a} is a lacunary self-similar set and its Hausdorff dimension is equal to \frac{ln 2}{3 ln 3}.

⋄ For any non-negative integer i the set S_{a,i} = K_{i} ⋂(K_{i} + a) is a covering of

E_{a} . It is clear that {S_{a,i}} is a decreasing sequence of sets, and E_{a} = ⋂_{i=0}^{∞} S_{a,i}.

From the definition of δ-parallel body we conclude that S_{a,i} ⊂ [S_{a,i+1}]_{3^{−i}}. Hence

d(S_{a,i}, E_{a}) ⟶^{i→∞} 0 .  (11)

Let a = 0, a_{1}a_{2}a_{3} … be the triadic expansion of a ∈ [0, 1]. We can exclude from the consideration the set of a for which the triadic expansion is ambiguously determined, because Lebesgue’s measure of this set, is equal to zero (for these a the set E_{a} is either finite or similar to Cantor’s set). We set

λ_{a}(k) = \{ \begin{matrix}0 & if & ∑_{i=0}^{k−1} a_{i} = 0 (mod 2), \\ 1 & if & ∑_{i=0}^{k−1} a_{i} = 1 (mod 2), \end{matrix}

for any positive integer k (we state a_{0} = 0). Let us consider the nonempty sets N_{a,k} ⊂ {1, 2}

N_{a,k} = \{ \begin{matrix}\{1\} & if & a_{k} ≠ 2, λ_{a}(k) = 1, \\ \{2\} & if & a_{k} ≠ 0, λ_{a}(k) = 0, \\ \{1, 2\} & if & a_{k} = 2, λ_{a}(k) = 1 & or a_{k} = 0, λ_{a}(k) = 0 . \end{matrix}

The similitudes ψ_{1}(x) = \frac{x}{3}, ψ_{2}(x) = \frac{x}{3} + \frac{2}{3} with ratios r_{1} = r_{2} = \frac{1}{3} and sets N_{a,i} generate (see. Section 1) the lacunary self-similar set G_{a} (the open set condition holds for the open interval (0, 1)). As in Section 1 we set

L_{a}^{k} = ∏_{i=1}^{k} N_{a,i} , ψ_{a}^{k}(F) = ⋃_{L_{a}^{k}}(ψ_{j_{1}} ∘ … ∘ ψ_{j_{k}})(F) = ⋃_{L_{a}^{k}} F_{j_{1}...j_{k}} ,

where F is a nonempty compact set. By Theorem 1,

d(G_{a}, ψ_{a}^{k}([0, 1])) ⟶^{k→∞} 0 .  (12)

It follows from the construction that S_{a,k} ⊂ ψ_{a}^{k}([0, 1]) and ψ_{a}^{k}([0, 1]) ⊂ [S_{a,k}]_{3^{−k}}, hence

d(ψ_{a}^{k}([0, 1]), S_{a,k}) ⟶^{k→∞} 0 .  (13)

Thus by (11), (12), (13),

d(G_{a}, E_{a}) ≤ d(G_{a}, ψ_{a}^{k}([0, 1])) + d(ψ_{a}^{k}([0, 1]), S_{a,k}) + d(S_{a,k}, E_{a}) ⟶^{k→∞} 0 ,

so E_{a} = G_{a}.

To find the Hausdorff dimension of E_{a} it is sufficient to calculate s_{a} for which

lim inf_{k→∞} ∑_{L_{a}^{k}}(r_{j_{1}} … r_{j_{k}})^{t} = lim inf_{k→∞} 3^{−tk}M_{a}(k) = \{ \begin{matrix}∞, & t < s_{a} \\ 0, & t > s_{a} \end{matrix}  (14)

where M_{a}(k) is the number of elements of L_{a}^{k}.

Let us denote by χ_{Z} the characteristic function of a set Z. If Z is an integer, we consider Z as the subset of the set of integers. By construction we have

M_{a}(k) = 2^{f_{a}(k)}, where f_{a}(k) = ∑_{i=1}^{k}(χ_{0}(a_{i})χ_{0}(λ_{a}(i)) + χ_{2}(a_{i})χ_{1}(λ_{a}(i))) .

(15)

It is clear that, given a triadic expansion of a, we can calculate the Hausdorff dimension of E_{a} with the use of (14) and (15). However, for an arbitrary irrational number we cannot even calculate explicitly number characteristics of its triadic expansion, e.g. the proportion of digits 0, 1, 2 in its triadic expansion. We only known [1] that almost all numbers in [0, 1] (with respect to Lebesgue’s measure) have the property that the proportion of digits 0, 1, 2 in their triadic expansion equals \frac{1}{3}.

Let Ω = {ω = (a_{1}, a_{2}, a_{3}, …)}, where a_{k} are independent random variables taking values 0, 1, 2 with probability \frac{1}{3}. Let us consider the mapping a : Ω →

[0, 1], a = a(ω) = ∑_{k=1}^{∞} \frac{a_{k}}{3^{k}}, then ω is the triadic expansion of a. It is known [1]

that for any Borel set B ⊂ [0, 1] we have P {ω : a(ω) ∈ B} = mes(B), where P is a probability measure on Ω and ”mes” is the Lebesgue measure.

Now let us consider the mapping γ : Ω → Ω̂, γ(ω) = ω̂ = (â_{1}, â_{2}, â_{3}, …), where

â_{k} = 0 if a_{k} = 0, λ_{a}(k) = 0;
â_{k} = 1 if a_{k} = 1, λ_{a}(k) = 0;
â_{k} = 2 if a_{k} = 2, λ_{a}(k) = 0;
â_{k} = 3 if a_{k} = 0, λ_{a}(k) = 1;
â_{k} = 4 if a_{k} = 1, λ_{a}(k) = 1;
â_{k} = 5 if a_{k} = 2, λ_{a}(k) = 1 .

The probability P on Ω induces the probability P̂ = γ_{#}P on Ω̂. With respect to P the sequence of random variables â_{1}, â_{2}, â_{3}, … is a homogeneous Markov chain with the starting distribution p = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 0, 0, 0) and the matrix of transition probabilities

∥p_{i,j}∥ = ( \begin{matrix}\frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{matrix} )_{.}

It follows from the strong law of large numbers that

P \{ω : lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k} χ_{0}(a_{i}) = \frac{1}{3}\} = 1 .

Hence

P̂ \{ω̂ : lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k}(χ_{0}(â_{i}) + χ_{3}(â_{i})) = \frac{1}{3}\} = 1 .  (16)

Since the starting distribution and the matrix of transition probability do not change if we interchange the events a_{k} = 3 and a_{k} = 5, and the set in the braces in (16) depends only on the starting distribution and the matrix of transition probability, we have

P̂ \{ω̂ : lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k}(χ_{0}(â_{i}) + χ_{5}(â_{i})) = \frac{1}{3}\} = 1 .

It follows from χ_{0}(â_{i}) = χ_{0}(a_{i})χ_{0}(λ_{a}(i)) and χ_{5}(â_{i}) = χ_{2}(a_{i})χ_{1}(λ_{a}(i)) that

mes \{a ∈ [0, 1] : lim_{k→∞} \frac{f_{a}(k)}{k} = \frac{1}{3}\}

= P \{ω : lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k}(χ_{0}(a_{i})χ_{0}(λ_{a}(i)) + χ_{2}(a_{i})χ_{1}(λ_{a}(i))) = \frac{1}{3}\}

50 K. IGUDESMAN

= P̂ \{ω̂ : lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k}(χ_{0}(â_{i}) + χ_{5}(â_{i})) = \frac{1}{3}\} = 1 .

From this and Corollary 1 we conclude that for almost all a ∈ [0, 1] the Hausdorff dimension of E_{a} equals s, where s is determined by

3^{− \frac{2}{3} s}(3^{−s} + 3^{−s})^{\frac{1}{3}} = 1\frac{ln 2}{3 ln 3} . ⋄

The next Theorem gives a generalization of the previous one. Let us denote by K^{p} the set of numbers in [0, 1] which admit base p-expansion without odd digits. Note that K^{2} = {0} and K^{3} is the middle third Cantor set. Given a ∈ [0, 1], we set K^{p} + a = {x : x − a ∈ K^{p}}, E_{a}^{p} = K^{p} ⋂(K^{p} + a).

Theorem 4. If p is an even number, then for almost all a ∈ [0, 1] with respect to Lebesgue’s measure, E_{a}^{p} = ∅.

If p is an odd number, then for almost all a ∈ [0, 1] with respect to Lebesgue’s measure, E_{a}^{p} is a lacunary self-similar set and

dim(E_{a}^{p}) = \frac{2 ln((\frac{p+1}{2})!) − ln(\frac{p+1}{2})}{p ln p} .

⋄ The proof of the first statement is trivial. The second statement can be proved just in the same way as Theorem 3. ⋄

References

[1] Borovkov A.A. Probability theory. / M.: Science, 1986. – 431 P. (Russian).

[2] G.Davis, Hu Tian-You On the structure of the intersection of two middle third Cantor sets. / Pabl. Mat., Barc. 39, No.1, 43-60 (1995).

[3] Falconer K.J. The Geometry of Fractal Sets / Cambridge University Press. – 1985. – 162 P.

[4] Hutchinson J.E. Fractals and self-similarity // Indiana University Mathematics Journal. – 1981. – 30, 713-47.

[5] Kraft R. Random intersections of thick Cantor sets. // Trans. Am. Math. Soc. 352, No.3, 1315-1328 (2000).

[6] Kraft R. A golden Cantor set. // Am. Math. Mon. 105, No.8, 718-725 (1998).

[7] Li Wenxia, Hiao Dongmei Intersection of translations of Cantor triadic set. // Acta. Math. Sci (Engl. Ed) 19, No.2, 214-219 (1999).

[8] Williams R.F. How big is the intersection of two thick Cantor sets? // Continuum theory and dynamical systems, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Arcata/CA (USA) 1989, Contemp. Math. 117, 163-175 (1991).

Kazan State University, Faculty of Mathematics and Mechanics, Kremlevskaya, 18, Kazan: 420008, Russia E-mail address: Konstantin.Igudesman@ksu.ru

Received December 24, 2002