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\newcommand\myD{{\mathcal D}}
\newcommand\Z{{\mathbb Z}}

\title{Thickness measures for Cantor sets}
\author{S. Astels}
\address{Department of Pure Mathematics, The University of Waterloo, 
Waterloo, Ontario, Canada N2L 3G1}
\thanks{Research supported in part by the Natural Sciences and
Engineering Research Council of Canada.}

\dateposted{July 20, 1999}
\PII{S 1079-6762(99)00068-2}
\copyrightinfo{1999}{American Mathematical Society}

\keywords{Cantor sets, sums of sets, Hausdorff dimension}
\subjclass{Primary 58F12; Secondary 28A78}

\date{March 15, 1999}
\revdate{May 12, 1999}

\commby{Yitzhak Katznelson}

For a fixed $k\ge 1$ let $C_1,\dots,C_k$ be generalized Cantor sets.  
We examine 
various criteria under which $C_1+\dots + C_k$ contains an interval.  
When these 
criteria do not hold, we give a lower bound for the Hausdorff dimension 
of $C_1+\dots+C_k$.
Our work will involve the development of two different types of 
thickness measures.


We define a {\em generalized Cantor set} (henceforth known as a {\em 
Cantor set})
 to be any set $C$ of real numbers of the
C = I\;\backslash \bigcup_{i\ge 1} O_i
where $I$ is a finite closed interval and $\{O_i\;;\;i\ge 1\}$ is a 
countable (finite or infinite)
collection of disjoint open intervals contained in $I$.  We may 
inductively define a tree $\myD$
that will represent $C$.  Let the root of the tree
be the interval $I$.  We say that $\{I\}$ is the {\em 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\cap \Big(\bigcup_{i\ge 1}O_i\Big) \neq \emptyset.
Let $O_{I^w}$ be the interval in the set $\{O_i\;;\;i\ge 1\}$ of least 
index which is contained
in $I^w$, and let $I^{w0}$  and $I^{w1}$ be closed intervals with
I^w = I^{w0}\cup O_{I^w}\cup I^{w1}.
We let $I^{w0}$ and $I^{w1}$ be subvertices of $I^w$ in $\myD$.
I^w\cap \Big(\bigcup_{i\ge 1}O_i\Big)=\emptyset,
then we set $I^{w0}=I^w$ and let $I^{w0}$ be a subvertex of $I^w$ in 
We repeat this process for every vertex $I^w$ in the $n^{th}$ level of 
The $(n+1)^{th}$ level of the tree is the set of vertices $I^v$ in $\myD$ 
with $|v|=n+1$,
where $|v|$ denotes the length of the word $v$.
We continue this process inductively, creating the infinite tree $\myD$.
Note that
\{O_{I^w}\;;\; \text{$I^w$ is a bridge of $\myD$}\} = \{O_i\;;\; i\ge 1\},
C = \bigcap_{n=0}^\infty\Big(\bigcup_{|w|=n} I^w\Big).
Any tree with this property is said to be a {\em derivation} of the 
Cantor set $C$ from $I$.
The intervals $I, I^0,\dots$ are called {\em bridges} of the derivation,
while the open intervals $O_I, O_{I^0},\dots$ are called {\em gaps} of 

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,
 \cite{cusfl} and \cite{palis}).  Of interest to us here is the 
problem.  Define the sum of sets $E_1,\dots, E_n$ to be the set
E_1+\dots+E_n = \{e_1+\dots+e_n\;;\;\text{$e_i\in E_i$ for $1\le i\le 
For $k\ge 2$ let $C_1, \dots, C_k$ be Cantor sets derived
from $I_1,\dots,I_k$ respectively.  In this paper we discuss conditions
under which
$C_1+\dots+C_k$ contains an interval.  We also give bounds for the 
dimension of $C_1+\dots+C_k$.


Let $C$ be a Cantor set.  We define the {\em thickness} of $C$,
$\tau(C)$, to be infinity if $\{O_i\;;\; i\ge 1\}$ is empty.  Otherwise 
we put
\tau(C) = 
where the supremum is over all derivations $\myD$ of $C$ and the infimum
is over all bridges $A$ of $\myD$.  It is not difficult to show that the 
supremum is always attained (see, for example, \cite{astelst}, Lemma 

In 1979 Sheldon Newhouse \cite{newhouse} proved the following
Let $C_1$ and $C_2$ be Cantor sets derived from $I_1$ and $I_2$ 
with $\tau(C_1)\tau(C_2)>1$.  Then either $I_1\cap I_2=\emptyset$, 
$C_1$ is
contained in a gap of $C_2$, $C_2$ is contained in a gap of $C_1$ or
$C_1\cap C_2\neq\emptyset$.

In fact, if Newhouse's proof is slightly altered, then we may replace
the condition ``$\tau(C_1)\tau(C_2)>1$'' in Theorem \ref{th:newhouse} 
the weaker condition ``$\tau(C_1)\tau(C_2)\ge 1$''.
This strengthened version of Theorem \ref{th:newhouse} has the following
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|\le|I_2| \quad\text{and}\quad |O_2|\le|I_1|.
If $\tau(C_1)\tau(C_2)\ge1$, then $C_1 + C_2 = I_1 + I_2$.

If $\tau(C_1)\tau(C_2)<1$, then the work of Newhouse does not yield 
any non-trivial results.  This case was the main focus of the author
in \cite{astelst}, 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
{\em normalized thickness} of $C$, $\gamma(C)$, to be
\gamma(C) = \frac{\tau(C)}{\tau(C)+1}.
Let $k$ be a positive integer and for $j=1,2,\dots, k$ 
let $C_j$ be a Cantor set derived from $I_j$, with $O_j$ 
a gap of maximal size in $C_j$.
Let $S_\gamma=\gamma(C_1)+\dots+\gamma(C_k)$.
\item If $S_\gamma\ge 1$, then $C_1+\dots+C_k$ contains an interval.  
Otherwise $C_1+\dots+C_k$ contains a Cantor set of thickness at least
|I_{r+1}| & \ge|O_j| & \qquad & \text{for $r=1,\dots,k-1$ and 
|I_1|+\dots+|I_r| & \ge|O_{r+1}| & \qquad & \text{for $r=1,\dots, 
and $S_\gamma\ge 1$, then
If \eqref{eq:gs1} and \eqref{eq:gs2} hold and $S_\gamma < 1$, then

See \cite{astelst}, Theorem 2.4.

Note that in the case $k=2$ the condition ``$\gamma(C_1)+\gamma(C_2)\ge 
is equivalent to the condition ``$\tau(C_1)\tau(C_2)\ge 1$'', hence 
Theorem \ref{th:genarbsum} implies Theorem \ref{th:gen2sum}.

Let $\hdim(E)$ denote the Hausdorff dimension of the set $E$.  There is 
connection between thickness and Hausdorff dimension, as illustrated by 
next theorem.

If $C$ is a Cantor set, then
\hdim(C)\ge\frac{\log 2}{\log\left(2+\frac 1{\tau(C)}\right)}.
See \cite{palis}, p. 77.

Using Theorem \ref{th:genarbsum} and Theorem \ref{th:palhaus} we can 
the following lower bound for the Hausdorff dimension of a sum of 
Cantor sets.

For $k\in\Z^+$ let $C_1,\dots,C_k$ be Cantor sets.  Then
\hdim(C_1+\dots+C_k)\ge \frac{\log 2}{\log\left(1+
\frac1{\min\{\gamma(C_1)+\dots+\gamma(C_k), 1\}}\right)}.

It may be the case that $C_1+\dots+C_k$ contains an interval yet 
In this case better results may be gained by employing a concept known 
as {\em maximal

\section{Maximal thickness}

We define the {\em maximal thickness} and {\em normalized maximal 
of a Cantor set $C$ to be
\tau_M(C)=\sup_{C'\subseteq C}\tau(C')\quad\text{and}\quad 
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 
Using Theorems \ref{th:genarbsum} and \ref{th:cantdim} we may establish 
the following result.
Let $C_1,\dots,C_k$ be Cantor sets.  If
then $C_1+\dots+C_k$ contains an interval.  Otherwise
\gamma_M(C_1+\dots+C_k)\ge \gamma_M(C_1)+\dots+\gamma_M(C_k)
\hdim(C_1+\dots+C_k)\ge\frac{\log 2}
{\log\left(1+\frac 1 {\gamma_M(C_1)+\dots+\gamma_M(C_k)}\right)}.


S. Astels.
\textit{Cantor sets and numbers with restricted partial quotients},
Trans. Amer. Math. Soc. (to appear).

Thomas W. Cusick and Mary E. Flahive.
\textit{The Markoff and Lagrange spectra},
Amer. Math. Soc., Providence, RI, 1989.
Sheldon E. Newhouse.
\textit{The abundance of wild hyperbolic sets and non-smooth stable 
sets for diffeomorphisms},
Inst. Hautes \'Etudes Sci. Publ. Math. 50
(1979), 101--151.
J. Palis and F. Takens.
\textit{Hyperbolicity and sensitive chaotic dynamics at
homoclinic bifurcations}, 
Cambridge University Press, Cambridge, 1993.