ISBN: 3-540-65528-X
TITLE: Geometric Discrepancy
AUTHOR: Matousek, Jiri
TOC:

Preface v
Notation xi
1. Introduction 1
1.1 Discrepancy for Rectangles and Uniform Distribution 1
1.2 Geometric Discrepancy in a More General Setting 9
1.3 Combinatorial Discrepancy 16
1.4 On Applications and Connections 22
2. Low-Discrepancy Sets for Axis-Parallel Boxes. 37
2.1 Sets with Good Worst-Case Discrepancy 38
2.2 Sets with Good Average Discrepancy 44
2.3 More Constructions: b-ary Nets 51
2.4 Scrambled Nets and Their Average Discrepancy 61
2.5 More Constructions: Lattice Sets 72
3. Upper Bounds in the Lebesgue-Measure Setting 83
3.1 Circular Discs: a Probabilistic Construction 84
3.2 A Surprise for the L1-Discrepancy for Halfplanes 93
4. Combinatorial Discrepancy 101
4.1 Basic Upper Bounds for General Set Systems 101
4.2 Matrices, Lower Bounds, and Eigenvalues105
4.3 Linear Discrepancy and More Lower Bounds 109
4.4 On Set Systems with Very Small Discrepancy 117
4.5 The Partial Coloring Method 120
4.6 The Entropy Method 128
5. VC-Dimension and Discrepancy 137
5.1 Discrepancy and Shatter Functions 137
5.2 Set Systems of Bounded VC-Dimension 145
5.3 Packing Lemma 155
5.4 Matchings with Low Crossing Number 159
5.5 Primal Shatter Function and Partial Colorings 164
6. Lower Bounds 171
6.1 Axis-Parallel Rectangles: L_2-Discrepancy 172
6.2 Axis-Parallel Rectangles: the Tight Bound 176
6.3 A Reduction: Squares from Rectangles 180
6.4 Halfplanes: Combinatorial Discrepancy 182
6.5 Combinatorial Discrepancy for Halfplanes Revisited 193
6.6 Halfplanes: the Lebesgue-Measure Discrepancy 197
6.7 A Glimpse of Positive Definite Functions 203
7. More Lower Bounds and the Fourier Transform 213
7.1 Arbitrarily Rotated Squares 213
7.2 Axis-Parallel Cubes 230
7.3 An Excursion to Euclidean Ramsey Theory 234
A. Tables of Selected Discrepancy Bounds 241
Bibliography 245
Index 265
Hints 275
END
