The Erdos Distance Problem: (Student Mathematical Library)

The Erd s 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 Erd s 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 Erd s combinatorics.

Table of Contents:
Foreword Acknowledgments Introduction The √𝑛 theory The 𝑛^{2/3} theory The Cauchy-Schwarz inequality Graph theory and incidences The 𝑛^{4/5} theory The 𝑛^{6/7} theory Beyond 𝑛^{6/7} Information theory Dot products Vector spaces over finite fields Distances in vector spaces over finite fields Applications of the Erdős distance problem Hyperbolas in the plane Basic probability theory Jensen’s inequality Bibliography Biographical information Index of terminology

About the Author :
Julia Garibaldi, Alex Iosevich University of Rochester, Rochester, NY, USA Steven Senger, University of Missouri-Columbia, Columbia, MO, Columbia, MO, USA

Review :
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