Preface
1 Minkowski and Hausdorff dimensions ........................... 1
1.1 Minkowski dimension ..................................... 1
1.2 Hausdorff dimension and the Mass Distribution
Principle ............................................... 4
1.3 Sets defined by digit restrictions ...................... 9
1.4 Billingsley's Lemma and the dimension of measures ...... 17
1.5 Sets defined by digit frequency ........................ 21
1.6 Slices ................................................. 26
1.7 Intersecting translates of Cantor sets ................. 29
1.8 Notes .................................................. 34
1.9 Exercises .............................................. 36
2 Self-similarity and packing dimension ....................... 45
2.1 Self-similar sets ...................................... 45
2.2 The open set condition is sufficient ................... 51
2.3 Homogeneous sets ....................................... 54
2.4 Microsets .............................................. 57
2.5 Poincare sets .......................................... 61
2.6 Alternative definitions of Minkowski dimension ......... 67
2.7 Packing measures and dimension ......................... 71
2.8 When do packing and Minkowski dimension agree? ......... 74
2.9 Notes .................................................. 76
2.10 Exercises .............................................. 78
3 Frostman's theory and capacity .............................. 83
3.1 Frostman's Lemma ....................................... 83
3.2 The dimension of product sets .......................... 88
3.3 Generalized Marstrand Slicing Theorem .................. 90
3.4 Capacity and dimension ................................. 93
3.5 Marstrand's Projection Theorem ......................... 95
3.6 Mapping a tree to Euclidean space preserves capacity .. 100
3.7 Dimension of random Cantor sets ....................... 104
3.8 Notes ................................................. 112
3.9 Exercises ............................................. 115
4 Self-affine sets ........................................... 119
4.1 Construction and Minkowski dimension .................. 119
4.2 The Hausdorff dimension of self-affine sets ........... 121
4.3 A dichotomy for Hausdorff measure ..................... 125
4.4 The Hausdorff measure is infinite ..................... 127
4.5 Notes ................................................. 131
4.6 Exercises ............................................. 133
5 Graphs of continuous functions ............................. 136
5.1 Holder continuous functions ........................... 136
5.2 The Weierstrass function is nowhere differentiable .... 140
5.3 Lower Holder estimates ................................ 145
5.4 Notes ................................................. 149
5.5 Exercises ............................................. 151
6 Brownian motion, Part I .................................... 160
6.1 Gaussian random variables ............................. 160
6.2 Levy's construction of Brownian motion ................ 163
6.3 Basic properties of Brownian motion ................... 167
6.4 Hausdorff dimension of the Brownian path and graph .... 172
6.5 Nowhere differentiability is prevalent ................ 176
6.6 Strong Markov property and the reflection principle ... 178
6.7 Local extrema of Brownian motion ...................... 180
6.8 Area of planar Brownian motion ........................ 181
6.9 General Markov processes .............................. 183
6.10 Zeros of Brownian motion .............................. 185
6.11 Harris'inequality and its consequences ................ 189
6.12 Points of increase .................................... 192
6.13 Notes ................................................. 196
6.14 Exercises ............................................. 199
7 Brownian motion, Part II ................................... 201
7.1 Dimension doubling .................................... 201
7.2 The Law of the Iterated Logarithm ..................... 206
7.3 Skorokhod's Representation ............................ 209
7.4 Donsker's Invariance Principle ........................ 216
7.5 Harmonic functions and Brownian motion in  4 .......... 221
7.6 The maximum principle for harmonic functions .......... 226
7.7 The Dirichlet problem ................................. 227
7.8 Polar points and recurrence ........................... 228
7.9 Conformal invariance .................................. 230
7.10 Capacity and harmonic functions ....................... 235
7.11 Notes ................................................. 239
7.12 Exercises ............................................. 241
8 Random walks, Markov chains and capacity ................... 244
8.1 Frostman's theory for discrete sets ................... 244
8.2 Markov chains and capacity ............................ 250
8.3 Intersection equivalence and return times ............. 254
8.4 Lyons'Theorem on percolation on trees ................. 258
8.5 Dimension of random Cantor sets (again) ............... 260
8.6 Brownian motion and Martin capacity ................... 264
8.7 Notes ................................................. 266
8.8 Exercises ............................................. 266
9 Besicovitcli-Kakeya sets ................................... 270
9.1 Existence and dimension ............................... 270
9.2 Splitting triangles ................................... 276
9.3 Fefferman's Disk Multiplier Theorem ................... 278
9.4 Random Besicovitch sets ............................... 286
9.5 Projections of self-similar Cantor sets ............... 290
9.6 The open set condition is necessary ................... 297
9.7 Notes ................................................. 302
9.8 Exercises ............................................. 305
10 The Traveling Salesman Theorem ............................. 313
10.1 Lines and length ...................................... 313
10.2 The β-numbers ......................................... 318
10.3 Counting with dyadic squares .......................... 322
10.4 β and μ are equivalent ................................ 325
10.5 β-sums estimate minimal paths ......................... 329
10.6 Notes ................................................. 334
10.7 Exercises ............................................. 337
Appendix A Banach's Fixed-Point Theorem ...................... 343
Appendix В Frostman's Lemma for analytic sets ................ 353
Appendix С Hints and solutions to selected exercises ......... 360
References .................................................... 379
Index ......................................................... 396
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