The Elements of Cantor Sets: With Applications

232 Pages·2013·17.518 MB·English

by  Robert W. Vallin

THE ELEMENTS OF
CANTOR SETS—
WITH APPLICATIONS
THE ELEMENTS OF
CANTOR SETS—
WITH APPLICATIONS
ROBERT W. VALLIN
Slippery Rock University
W I L EY
Copyright © 2013 by John Wiley & Sons, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Vallin, Robert W.
The elements of Cantor sets : with applications / Robert W. Vallin, Department of Mathematics,
Slippery Rock University, Slippery Rock, PA. — First edition,
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-40571-0 (hardback)
1. Cantor sets. 2. Measure theory. 3. Mathematical analysis. 4. Cantor, George, 1941- I. Title.
QA612.V35 2013
515'.8—dc23 2013009452
Printed in the United States of America.
10 9 8 7 6 5 4 3 21
To Micah, Jessica,
Rachel, Sophie, and Sean.
And to Jackie, who
always believes in me.
CONTENTS IN BRIEF
1 A Quick Biography of Cantor 1
2 Basics 5
3 Introducing the Cantor Set 17
4 Cantor Sets and Continued Fractions 51
5 p-adic Numbers and Valuations 67
6 Self-Similar Objects 91
7 Various Notions of Dimension 119
8 Porosity and Thickness-Looking at the Gaps 143
9 Creating Pathological Functions via C 157
10 Generalizations and Applications 181
11 Epilogue 217
VII
CONTENTS
Foreword xiii
Preface xv
Acknowledgments xvii
Introduction xix
1 A Quick Biography of Cantor 1
2 Basics 5
2.1 Review 5
Exercises 14
3 Introducing the Cantor Set 17
3.1 Some Definitions and Basics 17
3.2 Size of a Cantor Set 21
3.2.1 Cardinality 22
3.2.2 Category 29
3.2.3 Measure 36
ix
X CONTENTS
3.3 Large and Small 47
Exercises 48
4 Cantor Sets and Continued Fractions 51
4.1 Introducing Continued Fractions 52
4.2 Constructing a Cantor Set 59
4.3 Diophantine Equations 60
4.4 Miscellaneous 63
Exercises 65
5 p-adic Numbers and Valuations 67
5.1 Some Abstract Algebra 67
5.2 p-adic Numbers 72
5.2.1 An Analysis Point of View 72
5.2.2 An Algebra Point of View 75
5.3 p-adic Integers and Cantor Sets 80
5.4 p-adic Rational Numbers 82
Exercises 88
6 Self-Similar Objects 91
6.1 The Meaning of Self-Similar 91
6.2 Metric Spaces 92
6.3 Sequences in (S, d) 97
6.4 Affine Transformations 106
6.5 An Application for an IFS 114
Exercises 116
7 Various Notions of Dimension 119
7.1 Limit Supremum and Limit Infimum 119
7.2 Topological Dimension 123
7.3 Similarity Dimension 127
7.4 Box-Counting Dimension 128
7.5 Hausdorff Measure and Dimension 131
7.6 Miscellaneous Notions of Dimension 136
Exercises 140
8 Porosity and Thickness-Looking at the Gaps 143
8.1 The Porosity of a Set 143
CONTENTS XI
8.2 Symmetric Sets and Symmetric Porosity 146
8.3 A New and Different Definition of Cantor Set 149
8.4 Thickness of a Cantor Set 150
8.5 Applying Thickness 151
8.6 A Bit More on Thickness 153
8.7 Porosity in a Metric Space 154
Exercises 156
9 Creating Pathological Functions via C 157
9.1 Sequences of Functions 157
9.2 The Cantor Function 161
9.3 Space-Filling Curves 168
9.4 Baire Class One Functions 171
9.5 Darboux Functions 173
9.6 Linearly Continuous Functions 177
Exercises 180
10 Generalizations and Applications 181
10.1 Generalizing Cantor Sets 181
10.2 Fat Cantor Sets 185
10.3 Sums of Cantor Sets 186
10.4 Differences of Cantor Sets 193
10.5 Products of Cantor Sets 195
10.6 Cantor Target 197
10.7 Ana Sets 198
10.8 Average Distance 201
10.9 Non-Averaging Sets 203
10.10 Cantor Series and Cantor Sets 205
10.11 Liouville Numbers and Irrationality Exponents 207
10.12 Sets of Sums of Convergent Alternating Series 209
10.13 The Monty Hall Problem 211
11 Epilogue 217
References 219
Index
FOREWORD
Among the delights of mathematics are those moments when it surprises, when
something unexpected emerges from a seemingly mundane construction that is pushed
beyond the bounds of the finite: staircases of finite length for which one can never
climb from one step to the next, continuous and bounded functions that can be in­
tegrated but whose integral does not have a derivative, space-filling curves and frac­
tional dimensions, and the Banach-Tarski paradox: the ability - in theory - to cut
a pea into finitely many pieces and reassemble those using only rigid motions (ro­
tations and translations) into a solid object the size of the sun. Such mathematical
surprises are the stuff of which great mathematical advances are made. They chal­
lenge us to broaden our understanding of the world in which we live.
The Cantor set provides one of these surprising moments. If we take an interval
of the real number line, say from 0 to 1, and remove finitely many little intervals
from it, we are left with finitely many little intervals. No surprise here. But what
if we remove infinitely many little intervals? Are we left with infinitely many little
intervals? No less a mathematician than Axel Harnack (1851-88) implicitly assumed
this obvious conclusion. He was very wrong.
To see how wrong he was, imagine that from our interval [0,1] we remove ev­
ery real number with 7 in its decimal expansion. This takes out everything from
0.7 to just below 0.8, one interval. It also takes out the nine intervals [0.07, 0.08),
[0.17,0.18), ..., [0.97,0.98). We are not done. We also have to take out the all of
XIII
XIV FOREWORD
the intervals of length 0.001 that start at a number with two digits that are not 7,
followed by a 7, such as 0.237. That adds 92 intervals of length 1/103. This goes on
forever. The total amount that we remove is
1 9 92 93 _ 1_ ( _9_ 92 \ _ 1 / 1 \
Io + io2+To3+To4+'''_ To V +io + To2+'"y™io l^i-9/ioJ ~
It appears that we have removed the entire interval. But, in fact, it is quite easy
to come up with numbers that are left behind. We still have 1/2 = 0.5. No 7s in that
decimal expansion. We still have 1/3 = 0.333 In some sense, we still have as many
real numbers as we started with. Who needs the symbol 7? If we represent our real
numbers in base 9, the remaining nine symbols are sufficient to express every real
number in [0,1].
Lots of numbers are left, but no intervals. You cannot get from 1/2 to 1/3 without
crossing a number with a 7 in its decimal expansion. More than this, between any
two distinct real numbers there will always be a real number with a 7 in its decimal
expansion. What is left is a very unintuitive set that is so full of holes that nothing is
left, and yet it still describes everything. It is what we call a Cantor Set.
The name Cantor Set is a bit of a misattribution. Henry J.S. Smith (1826-83) used
these sets in a paper published in 1875. Unfortunately, Smith was an Englishman
at a time when few expected any significant mathematics to come out of England.
Any really important mathematics would have been published in German. The next
paper to use these sets was by Vito Volterra in 1881 (1860-1940). Volterra would
come to be known as one of greatest of all mathematicians, but in 1881 he was a
mere student publishing his results in an obscure Italian journal. No one noticed.
When Georg Cantor rediscovered and described these sets in 1883, it was one piece
of a major and highly visible assault on our understanding of the structure of the real
number line. Now mathematicians paid attention.
In this book, Robert Vallin leads us through the many surprises of Cantor Sets.
This is only the beginning. The structure of these sets is echoed in much of 20th
century mathematics from p-adic numbers to fractals. This tour will explore the
connections to continued fractions, to self-similarity and non-integer dimensions, to
derivatives that are not continuous and space-filling curves that are, and to one of
my favorite mathematical curiosities, the Cantor function, more aptly known as The
Devils Staircase.
David M. Bressoud