Synopses & Reviews
What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? Such questions are treated in geometric discrepancy theory. The book is an accessible and lively introduction to this area, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research. Including a wide variety of mathematical techniques (from harmonic analysis, combinatorics, algebra etc.) in action on non-trivial examples, the book is suitable for a "special topic" course for early graduates in mathematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collection of objects should be "uniformly" represented by a smaller sample (such as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.). From the reviews: "...The numerous illustrations are well placed and instructive. The clear and elegant exposition conveys a wealth of intuitive insights into the techniques utilized. Each section usually consists of text, historical remarks and references for the specialist, and exercises. Hints are provided for the more difficult exercises, with the exercise-hint format permitting inclusion of more results than otherwise would be possible in a book of this size..." Allen D. Rogers, Mathematical Reviews Clippings (2001)
Review
From the reviews: "The book gives a very useful introduction to geometric discrepancy theory. The style is quite informal and lively which makes the book easily readable." (Robert F. Tichy, Zentralblatt MATH, Vol. 1197, 2010)
Synopsis
Discrepancy theory is also called the theory of irregularities of distribution. Here are some typical questions: What is the "most uniform" way of dis tributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? For a precise formulation of these questions, we must quantify the irregularity of a given distribution, and discrepancy is a numerical parameter of a point set serving this purpose. Such questions were first tackled in the thirties, with a motivation com ing from number theory. A more or less satisfactory solution of the basic discrepancy problem in the plane was completed in the late sixties, and the analogous higher-dimensional problem is far from solved even today. In the meantime, discrepancy theory blossomed into a field of remarkable breadth and diversity. There are subfields closely connected to the original number theoretic roots of discrepancy theory, areas related to Ramsey theory and to hypergraphs, and also results supporting eminently practical methods and algorithms for numerical integration and similar tasks. The applications in clude financial calculations, computer graphics, and computational physics, just to name a few. This book is an introductory textbook on discrepancy theory. It should be accessible to early graduate students of mathematics or theoretical computer science. At the same time, about half of the book consists of material that up until now was only available in original research papers or in various surveys."
Synopsis
The book is a monograph in an important area of discrete mathematics and geometry. It is bound to become a standard reference.
Synopsis
What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.
Description
Includes bibliographical references (p. [245]-263) and index.
Table of Contents
1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2. Low-Discrepancy Sets for Axis-Parallel Boxes 2.1 Sets with Good Worst-Case Discrepancy 2.2 Sets with Good Average Discrepancy 2.3 More Constructions: b-ary Nets 2.4 Scrambled Nets and Their Average Discrepancy 2.5 More Constructions: Lattice Sets 3. Upper Bounds in the Lebesgue-Measure Setting 3.1 Circular Discs: a Probabilistic Construction 3.2 A Surprise for the L 1-Discrepancy for Halfplanes 4. Combinatorial Discrepancy 4.1 Basic Upper Bounds for General Set Systems 4.2 Matrices, Lower Bounds, and Eigenvalues 4.3 Linear Discrepancy and More Lower Bounds 4.4 On Set Systems with Very Small Discrepancy 4.5 The Partial Coloring Method 4.6 The Entropy Method 5. VC-Dimension and Discrepancy 5.1 Discrepancy and Shatter Functions 5.2 Set Systems of Bounded VC-Dimension 5.3 Packing Lemma 5.4 Matchings with Low Crossing Number 5.5 Primal Shatter Function and Partial Colorings 6. Lower Bounds 6.1 Axis-Parallel Rectangles: L 2-Discrepancy 6.2 Axis-Parallel Rectangles: the Tight Bound 6.3 A Reduction: Squares from Rectangles 6.4 Halfplanes: the Combinatorial Discrepancy 6.5 Combinatorial Discrepancy for Halfplanes Revisited 6.6 Halfplanes: the Lebesgue-Measure Discrepancy 6.7 A Glimpse of Positive Definite Functions 7. More Lower Bounds and the Fourier Transform 7.1 Arbitrarily Rotated Squares 7.2 Axis-Parallel Cubes 7.3 An Excursion to Euclidean Ramsey Theory A. Tables of Selected Discrepancy Bounds Bibliography Index Hints