ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 108–111 (July 20, 1999) S 1079-6762(99)00068-2

THICKNESS MEASURES FOR CANTOR SETS

S. ASTELS

(Communicated by Yitzhak Katznelson)

Abstract. For a fixed k ≥ 1 let C_{1}, …, C_{k} be generalized Cantor sets. We examine various criteria under which C_{1} +⋯+C_{k} contains an interval. When these criteria do not hold, we give a lower bound for the Hausdorff dimension of C_{1} + ⋯ + C_{k}. Our work will involve the development of two different types of thickness measures.

1. Introduction

We define a generalized Cantor set (henceforth known as a Cantor set) to be any set C of real numbers of the form

C = I \ ⋃_{i≥1} O_{i}

where I is a finite closed interval and {O_{i} ; i ≥ 1} is a countable (finite or infinite) collection of disjoint open intervals contained in I. We may inductively define a tree D that will represent C. Let the root of the tree be the interval I. We say that {I} is the zeroth level of the tree. Now suppose we have defined our tree up to the n^{th} level. We define the (n + 1)^{th} level of the tree as follows. Let I^{w} be an n^{th} level vertex of our tree. Assume first that

I^{w} ∩ ( ⋃_{i≥1} O_{i}) ≠ ∅.

Let O_{I^{w}} be the interval in the set {O_{i} ; i ≥ 1} of least index which is contained in I^{w}, and let I^{w0} and I^{w1} be closed intervals with

I^{w} = I^{w0} ∪ O_{I^{w}} ∪ I^{w1}.

We let I^{w0} and I^{w1} be subvertices of I^{w} in D. If

I^{w} ∩ ( ⋃_{i≥1} O_{i}) = ∅,

then we set I^{w0} = I^{w} and let I^{w0} be a subvertex of I^{w} in D. We repeat this process for every vertex I^{w} in the n^{th} level of D. The (n + 1)^{th} level of the tree is the set

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors March 15, 1999 and, in revised form, May 12, 1999.

1991 Mathematics Subject Classification. Primary 58F12; Secondary 28A78.

Key words and phrases. Cantor sets, sums of sets, Hausdorff dimension.

Research supported in part by the Natural Sciences and Engineering Research Council of Canada.

of vertices I^{v} in D with |v| = n + 1, where |v| denotes the length of the word v. We continue this process inductively, creating the infinite tree D. Note that

{O_{I^{w}} ; I^{w} is a bridge of D} = {O_{i} ; i ≥ 1},

hence

C = ⋂_{n=0}^{∞} ( ⋃_{|w|=n} I^{w}).

Any tree with this property is said to be a derivation of the Cantor set C from I. The intervals I, I^{0}, … are called bridges of the derivation, while the open intervals O_{I}, O_{I^{0}}, … are called gaps of C.

Cantor sets arise naturally in many areas of mathematical inquiry, including the examination of the Markoff spectrum and the study of the chaotic behavior of certain families of functions (see, for example, [2] and [4]). Of interest to us here is the following problem. Define the sum of sets E_{1}, …, E_{n} to be the set

E_{1} + ⋯ + E_{n} = {e_{1} + ⋯ + e_{n} ; e_{i} ∈ E_{i} for 1 ≤ i ≤ n}.

For k ≥ 2 let C_{1}, …, C_{k} be Cantor sets derived from I_{1}, …, I_{k} respectively. In this paper we discuss conditions under which C_{1} + ⋯ + C_{k} contains an interval. We also give bounds for the Hausdorff dimension of C_{1} + ⋯ + C_{k}.

2. Thickness

Let C be a Cantor set. We define the thickness of C, τ(C), to be infinity if {O_{i} ; i ≥ 1} is empty. Otherwise we put

τ(C) = sup_{D} inf_{A∈D} min \{\frac{|A^{0}|}{|O_{A}|}, \frac{|A^{1}|}{|O_{A}|}\}

where the supremum is over all derivations D of C and the infimum is over all bridges A of D. It is not difficult to show that the supremum is always attained (see, for example, [1], Lemma 3.1).

In 1979 Sheldon Newhouse [3] proved the following result.

Theorem 2.1. Let C_{1} and C_{2} be Cantor sets derived from I_{1} and I_{2} respectively, with τ(C_{1})τ(C_{2}) > 1. Then either I_{1} ∩ I_{2} = ∅, C_{1} is contained in a gap of C_{2}, C_{2} is contained in a gap of C_{1} or C_{1} ∩ C_{2} ≠ ∅.

In fact, if Newhouse’s proof is slightly altered, then we may replace the condition “τ(C_{1})τ(C_{2}) > 1” in Theorem 2.1 with the weaker condition “τ(C_{1})τ(C_{2}) ≥ 1”. This strengthened version of Theorem 2.1 has the following corollary.

Theorem 2.2. For j = 1 or j = 2 let C_{j} be a Cantor set derived from I_{j}, with O_{j} a gap of maximal size in C_{j}. Assume that

|O_{1}| ≤ |I_{2}| and |O_{2}| ≤ |I_{1}|.

If τ(C_{1})τ(C_{2}) ≥ 1, then C_{1} + C_{2} = I_{1} + I_{2}.

If τ(C_{1})τ(C_{2}) < 1, then the work of Newhouse does not yield any non-trivial results. This case was the main focus of the author in [1], where a best-possible lower bound for the thickness of a finite sum of Cantor sets was found.

For a Cantor set C we define the normalized thickness of C, γ(C), to be

γ(C) = \frac{τ(C)}{τ(C) + 1}.

Theorem 2.3. Let k be a positive integer and for j = 1, 2, …, k let C_{j} be a Cantor set derived from I_{j}, with O_{j} a gap of maximal size in C_{j}. Let S_{γ} = γ(C_{1}) + ⋯ + γ(C_{k}).

1. If S_{γ} ≥ 1, then C_{1} + ⋯ + C_{k} contains an interval. Otherwise C_{1} + ⋯ + C_{k} contains a Cantor set of thickness at least

\frac{S_{γ}}{1 − S_{γ}} .

2. If

(1)  |I_{r+1}| ≥ |O_{j}|  for r = 1, …, k − 1 and j = 1, …, r,
(2)  |I_{1}| + ⋯ + |I_{r}| ≥ |O_{r+1}|  for r = 1, …, k − 1

and S_{γ} ≥ 1, then

C_{1} + ⋯ + C_{k} = I_{1} + ⋯ + I_{k}.

3. If (1) and (2) hold and S_{γ} < 1, then

τ(C_{1} + ⋯ + C_{k}) ≥ \frac{S_{γ}}{1 − S_{γ}} .

Proof. See [1], Theorem 2.4.

Note that in the case k = 2 the condition “γ(C_{1}) + γ(C_{2}) ≥ 1” is equivalent to the condition “τ(C_{1})τ(C_{2}) ≥ 1”, hence Theorem 2.3 implies Theorem 2.2.

Let dim_{H}(E) denote the Hausdorff dimension of the set E. There is a connection between thickness and Hausdorff dimension, as illustrated by the next theorem.

Theorem 2.4. If C is a Cantor set, then

dim_{H}(C) ≥ \frac{log 2}{log (2 + \frac{1}{τ(C)})}.

Proof. See [4], p. 77.

Using Theorem 2.3 and Theorem 2.4 we can establish the following lower bound for the Hausdorff dimension of a sum of Cantor sets.

Theorem 2.5. For k ∈ ℤ^{+} let C_{1}, …, C_{k} be Cantor sets. Then

dim_{H}(C_{1} + ⋯ + C_{k}) ≥ \frac{log 2}{log (1 + \frac{1}{min\{γ(C_{1})+···+γ(C_{k}),1\}})}.

It may be the case that C_{1}+⋯+C_{k} contains an interval yet γ(C_{1})+⋯+γ(C_{k}) < 1. In this case better results may be gained by employing a concept known as maximal thickness.

3. Maximal thickness

We define the maximal thickness and normalized maximal thickness of a Cantor set C to be

τ_{M}(C) = sup_{C′⊆C} τ(C′) and γ_{M}(C) = \frac{τ_{M}(C)}{τ_{M}(C) + 1}

respectively, where the supremum is taken over all Cantor sets C′ contained in C. Note that for any Cantor set C it follows trivially that τ(C) ≤ τ_{M}(C). Using Theorems 2.3 and 2.5 we may establish the following result.

Theorem 3.1. Let C_{1}, …, C_{k} be Cantor sets. If

γ_{M}(C_{1}) + ⋯ + γ_{M}(C_{k}) > 1,

then C_{1} + ⋯ + C_{k} contains an interval. Otherwise

γ_{M}(C_{1} + ⋯ + C_{k}) ≥ γ_{M}(C_{1}) + ⋯ + γ_{M}(C_{k})

and

dim_{H}(C_{1} + ⋯ + C_{k}) ≥ \frac{log 2}{log (1 + \frac{1}{γ_{M} (C_{1})+···+γ_{M} (C_{k})})}.

References

[1] S. Astels. Cantor sets and numbers with restricted partial quotients, Trans. Amer. Math. Soc. (to appear).

[2] Thomas W. Cusick and Mary E. Flahive. The Markoff and Lagrange spectra, Amer. Math. Soc., Providence, RI, 1989. MR 90i:11069

[3] Sheldon E. Newhouse. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. MR 82e:58067

[4] J. Palis and F. Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge University Press, Cambridge, 1993. MR 94h:58129

Department of Pure Mathematics, The University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

   E-mail address: sastels@barrow.uwaterloo.ca