**Student Mathematical Library**

Volume: 56;
2011;
150 pp;
Softcover

MSC: Primary 05; 11; 42; 51;

Print ISBN: 978-0-8218-5281-1

Product Code: STML/56

List Price: $31.00

Individual Price: $24.80

**Electronic ISBN: 978-1-4704-1639-3
Product Code: STML/56.E**

List Price: $31.00

Individual Price: $24.80

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#### Supplemental Materials

# The Erdős Distance Problem

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*Julia Garibaldi; Alex Iosevich; Steven Senger*

The Erdős problem asks, What is the smallest possible number of distinct distances between points of a large finite subset of the Euclidean space in dimensions two and higher? The main goal of this book is to introduce the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience—from motivated high school students interested in mathematics to graduate students specializing in combinatorics and geometry—to absorb the content and appreciate its far-reaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.

The book is heavily problem oriented, following the authors' firm belief that most of the learning in mathematics is done by working through the exercises. Many of these problems are recently published results by mathematicians working in the area. The order of the exercises is designed both to reinforce the material presented in the text and, equally importantly, to entice the reader to leave all worldly concerns behind and launch head first into the multifaceted and rewarding world of Erdős combinatorics.

#### Table of Contents

# Table of Contents

## The Erdos Distance Problem

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Foreword ix10 free
- Acknowledgments xi12 free
- Introduction 114 free
- The √𝑛 theory 720 free
- The 𝑛^{2/3} theory 1528
- The Cauchy-Schwarz inequality 2336
- Graph theory and incidences 2942
- The 𝑛^{4/5} theory 4558
- The 𝑛^{6/7} theory 6578
- Beyond 𝑛^{6/7} 7184
- Information theory 8194
- Dot products 91104
- Vector spaces over finite fields 101114
- Distances in vector spaces over finite fields 119132
- Applications of the Erdős distance problem 127140
- Hyperbolas in the plane 131144
- Basic probability theory 135148
- Jensen’s inequality 139152
- Bibliography 143156
- Biographical information 147160
- Index of terminology 149162 free
- Back Cover Back Cover1166

#### Readership

Undergraduates, graduate students, and research mathematicians interested in geometric combinatorics and various topics in general combinatorics.

#### Reviews

The authors do an excellent job in bringing together the main techniques and results connected to the Erdős distance problem ... this is a useful book for the reader with sufficient mathematical experience who wishes to learn the principal techniques and results in the Erdős distance problem and related areas.

-- Mathematical Reviews

This book...achieves the remarkable feat of providing an extremely accessible treatment of a classic family of research problems. ...The book can be used for a reading course taken by an undergraduate student (parts of the book are accessible for talented high school students as well), or as introductory material for a graduate student who plans to investigate this area further...Highly recommended.

-- M. Bona, Choice