The Elements of Cantor Sets

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

3.3 Large and Small 46

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

5.3 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 112

Exercises 115

7 Various Notions of Dimension 117

7.1 Limit Supremum and Limit Infimum 117

7.2 Topological Dimension 121

7.3 Similarity Dimension 125

7.4 Box-Counting Dimension 126

7.5 Hausdorff Measure and Dimension 129

7.6 Miscellaneous Notions of Dimension 134

Exercises 138

8 Porosity and Thickness Looking

at the Gaps 141

8.1 The Porosity of a Set 141

8.2 Symmetric Sets and Symmetric Porosity 144

8.3 A New and Different Definition of Cantor Set 147

8.4 Thickness of a Cantor Set 148

8.5 Applying Thickness 149

8.6 A Bit More on Thickness 151

8.7 Porosity in a Metric Space 152

Exercises 154

9 Creating Pathological Functions via C 155

9.1 Sequences of Functions 155

9.2 The Cantor Function 159

9.3 Space-Filling Curves 165

9.4 Baire Class One Functions 169

9.5 Darboux Functions 171

9.6 Linearly Continuous Functions 175

Exercises 178

10 Generalizations and Applications 179

10.1 Generalizing Cantor Sets 179

10.2 Fat Cantor Sets 183

10.3 Sums of Cantor Sets 184

10.4 Differences of Cantor Sets 191

10.5 Products of Cantor Sets 193

10.6 Cantor Target 195

10.7 Ana Sets 196

10.8 Average Distance 199

10.9 Non-Averaging Sets 201

10.10 Cantor Series and Cantor Sets 203

10.11 Liouville Numbers and Irrationality Exponents 205

10.12 Sets of Sums of Convergent Alternating Series 207

10.13 The Monty Hall Problem 209

11 Epilogue 215

References 217