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The Erdős Distance Problem
 
Alex Iosevich University of Rochester, Rochester, NY
Steven Senger University of Missouri-Columbia, Columbia, MO, Columbia, MO
Front Cover for The Erdos Distance Problem
Available Formats:
Softcover ISBN: 978-0-8218-5281-1
Product Code: STML/56
150 pp 
List Price: $33.00
Individual Price: $26.40
Electronic ISBN: 978-1-4704-1639-3
Product Code: STML/56.E
150 pp 
List Price: $31.00
Individual Price: $24.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $49.50
Front Cover for The Erdos Distance Problem
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  • Front Cover for The Erdos Distance Problem
  • Back Cover for The Erdos Distance Problem
The Erdős Distance Problem
Alex Iosevich University of Rochester, Rochester, NY
Steven Senger University of Missouri-Columbia, Columbia, MO, Columbia, MO
Available Formats:
Softcover ISBN:  978-0-8218-5281-1
Product Code:  STML/56
150 pp 
List Price: $33.00
Individual Price: $26.40
Electronic ISBN:  978-1-4704-1639-3
Product Code:  STML/56.E
150 pp 
List Price: $31.00
Individual Price: $24.80
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $49.50
  • Book Details
     
     
    Student Mathematical Library
    Volume: 562011
    MSC: Primary 05; 11; 42; 51;

    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.

    Readership

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

  • Table of Contents
     
     
    • Chapters
    • Introduction
    • Chapter 1. The $\sqrt {n}$ theory
    • Chapter 2. The $n^{2/3}$ theory
    • Chapter 3. The Cauchy-Schwarz inequality
    • Chapter 4. Graph theory and incidences
    • Chapter 5. The $n^{4/5}$ theory
    • Chapter 6. The $n^{6/7}$ theory
    • Chapter 7. Beyond $n^{6/7}$
    • Chapter 8. Information theory
    • Chapter 9. Dot products
    • Chapter 10. Vector spaces over finite fields
    • Chapter 11. Distances in vector spaces over finite fields
    • Chapter 12. Applications of the Erdős distance problem
    • Appendix A. Hyperbolas in the plane
    • Appendix B. Basic probability theory
    • Appendix C. Jensen’s inequality
  • 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
  • Request Review Copy
  • Get Permissions
Volume: 562011
MSC: Primary 05; 11; 42; 51;

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.

Readership

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

  • Chapters
  • Introduction
  • Chapter 1. The $\sqrt {n}$ theory
  • Chapter 2. The $n^{2/3}$ theory
  • Chapter 3. The Cauchy-Schwarz inequality
  • Chapter 4. Graph theory and incidences
  • Chapter 5. The $n^{4/5}$ theory
  • Chapter 6. The $n^{6/7}$ theory
  • Chapter 7. Beyond $n^{6/7}$
  • Chapter 8. Information theory
  • Chapter 9. Dot products
  • Chapter 10. Vector spaces over finite fields
  • Chapter 11. Distances in vector spaces over finite fields
  • Chapter 12. Applications of the Erdős distance problem
  • Appendix A. Hyperbolas in the plane
  • Appendix B. Basic probability theory
  • Appendix C. Jensen’s inequality
  • 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
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